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- Committer: Bazaar Package Importer
- Author(s): Camm Maguire
- Date: 2010-04-30 13:30:33 UTC
- mto: This revision was merged to the branch mainline in revision 12.
- Revision ID: james.westby@ubuntu.com-20100430133033-wtewap0zdnmsix1y
Tags: upstream-5.21.1
Import upstream version 5.21.1
- files added:
- doc/info/figures/contour1.gif
- files removed:
- ChangeLog-5.20
- files modified:
- INSTALL.win32
added
removed
tests/rtest_integrate.mac
7
7
*/
8
8
9
9
(kill(all),
10
trigsign:false, /* Do not simplify erfc(-x) = 2 - erfc(x) !! */
10
/*trigsign:false,*/ /* Do not simplify erfc(-x) = 2 - erfc(x) !! */
11
11
declare(z,complex,n,integer),
12
12
assume(n>0),
13
13
done);
14
14
done;
15
15
16
assume(notequal(v,0),a > 0,b > 0,notequal(1+p,0),4*a*c-b^2 > 0);
17
[notequal(v,0),a > 0,b > 0,notequal(p,-1),4*a*c > b^2];
16
assume(notequal(v,0),a > 1,b > 0, d>1, notequal(1+p,0),4*a*c-b^2 > 0);
17
[notequal(v,0),a > 1,b > 0,d>1,notequal(p,-1),4*a*c > b^2];
18
18
19
assume(notequal((a-1)*(log(a)*d+c*log(h)),0));
20
[notequal((a-1)*(c*log(h)+log(a)*d),0)];
19
assume(notequal((a-1)*(log(a)*d+c*log(h)),0), h>0);
20
[notequal((a-1)*(c*log(h)+log(a)*d),0), h>0];
21
21
22
22
assume(notequal((a-1)*(log(a)*d+log(b)*e+f*log(c)),0));
23
23
[notequal((a-1)*(log(c)*f+log(b)*e+log(a)*d),0)];
47
47
-gamma_incomplete(1/r,-log(a)*b*z^r)*z/(b^(1/r)*r*(-log(a)*z^r)^(1/r));
48
48
49
49
/* Problem 11: OK */
50
integrate(a^(b*z^2),z);
51
sqrt(%pi)*erf(sqrt(-log(a))*sqrt(b)*z)/(2*sqrt(-log(a))*sqrt(b));
50
integrate(a^(b*z^2),z),erf_representation:erfi,trigsign:true;
51
sqrt(%pi)*erfi(sqrt(log(a))*sqrt(b)*z)/(2*sqrt(log(a))*sqrt(b));
52
52
53
53
/* Problem 12: OK */
54
54
integrate(a^(b*sqrt(z)),z);
59
59
-gamma_incomplete(1/(p*r),-log(a)*b*(z^r)^p)*z
60
60
/(b^(1/(p*r))*p*r*(-log(a)*(z^r)^p)^(1/(p*r)));
61
61
62
/* Problem 14: OK */
62
/* Problem 14: (OK) */
63
63
integrate(a^(b*(z^r)^(1/r)),z);
64
64
%e^(log(a)*b*z)/(log(a)*b);
65
65
66
/* Problem 15: OK */
67
integrate(a^(b*sqrt(z)),z);
68
2*(log(a)*b*sqrt(z)-1)*%e^(log(a)*b*sqrt(z))/(log(a)^2*b^2);
69
70
/* Problem 16: OK */
66
/* Problem 15: (OK) */
71
67
integrate(a^(b*sqrt(z^2)),z);
72
68
%e^(log(a)*b*z)/(log(a)*b);
73
69
74
/* Problem 17: NEW Type 1 */
70
/* Problem 16: NEW Type 1 */
75
71
integrate(%e^(a*z^r),z);
76
72
-gamma_incomplete(1/r,-a*z^r)*z/(a^(1/r)*r*(-z^r)^(1/r));
77
73
74
/* Problem 17: OK */
75
integrate(%e^(a*z^2),z),erf_representation:erfi;
76
sqrt(%pi)*erfi(sqrt(a)*z)/(2*sqrt(a));
77
78
78
/* Problem 18: OK */
79
integrate(%e^(a*z^2),z);
80
-sqrt(%pi)*%i*erf(%i*sqrt(a)*z)/(2*sqrt(a));
81
82
/* Problem 19: OK */
83
79
integrate(%e^(a*sqrt(z)),z);
84
80
2*(a*sqrt(z)-1)*%e^(a*sqrt(z))/a^2;
85
81
86
/* Problem 20: NEW Type 1 */
82
/* Problem 19: NEW Type 1 */
87
83
integrate(%e^(a*(z^r)^p),z);
88
84
-gamma_incomplete(1/(p*r),-a*(z^r)^p)*z/(a^(1/(p*r))*p*r*(-(z^r)^p)^(1/(p*r)));
89
85
90
/* Problem 21: OK */
86
/* Problem 20: (OK) */
91
87
integrate(%e^(a*(z^r)^(1/r)),z);
92
88
%e^(a*z)/a;
93
89
94
/* Problem 22: OK */
90
/* Problem 21: (OK) */
95
91
integrate(%e^(a*sqrt(z^2)),z);
96
92
%e^(a*z)/a;
97
93
98
/* Problem 23: NEW Type 2 */
94
/* Problem 22: NEW Type 2 */
99
95
integrate(z^(v-1)*a^(b*z),z);
100
96
-gamma_incomplete(v,-log(a)*b*z)*z^v/(b^v*(-log(a)*z)^v);
101
97
98
/* Problem 23: NEW Type 2 */
99
integrate(a^(b*z)/z,z),gamma_expand:true,logcontract;
100
expintegral_ei(b*z*log(a));
101
102
102
/* Problem 24: NEW Type 2 */
103
integrate(a^(b*z)/z,z);
104
-gamma_incomplete(0,-log(a)*b*z);
105
106
/* Problem 25: NEW Type 2 */
107
103
integrate(z^(v-1)*a^z,z);
108
104
-gamma_incomplete(v,-log(a)*z)*z^v/(-log(a)*z)^v;
109
105
106
/* Problem 25: NEW Type 2 */
107
integrate(a^z/z,z),gamma_expand:true,logcontract;
108
expintegral_ei(z*log(a));
109
110
110
/* Problem 26: NEW Type 2 */
111
integrate(a^z/z,z);
112
-gamma_incomplete(0,-log(a)*z);
113
114
/* Problem 27: NEW Type 2 */
115
111
integrate(z^(v-1)*%e^(a*z),z);
116
112
-gamma_incomplete(v,-a*z)*z^v/(a^v*(-z)^v);
117
113
118
/* Problem 28: NEW Type 2 */
114
/* Problem 27: NEW Type 2 */
119
115
integrate(z^(v-1)*%e^z,z);
120
116
-gamma_incomplete(v,-z)*z^v/(-z)^v;
121
117
122
/* Problem 29: NEW Type 2 */
118
/* Problem 28: NEW Type 2 */
123
119
integrate(z^n*%e^(a*z),z);
124
120
-a^(-n-1)*(-1)^(n+1)*gamma_incomplete(n+1,-a*z);
125
121
122
/* Problem 29: New Type 2 */
123
integrate(%e^(a*z)/z^n,z);
124
-a^(n-1)*gamma_incomplete(1-n,-a*z)*(-1)^(n-1);
125
126
126
/* Problem 30: OK */
127
127
integrate(z*%e^(a*z),z);
128
128
(a*z-1)*%e^(a*z)/a^2;
129
129
130
/* Problem 31: OK */
130
/* Problem 31: (OK) */
131
131
integrate(sqrt(z^2)*%e^(a*z),z);
132
132
(a*z-1)*%e^(a*z)/a^2;
133
133
168
168
/a^9;
169
169
170
170
/* Problem 39: NEW Type 2 */
171
integrate(%e^(a*z)/z,z);
172
-gamma_incomplete(0,-a*z);
171
integrate(%e^(a*z)/z,z),gamma_expand:true,logcontract;
172
expintegral_ei(a*z);
173
173
174
174
/* Problem 40: NEW Type 2 */
175
integrate(%e^(a*z)/z^2,z);
176
gamma_incomplete(-1,-a*z)*a;
175
integrate(%e^(a*z)/z^2,z),gamma_expand:true,logcontract,ratsimp;
176
(a*z*expintegral_ei(a*z)-%e^(a*z))/z;
177
177
178
178
/* Problem 41: NEW Type 2 */
179
integrate(%e^(a*z)/z^3,z);
180
-gamma_incomplete(-2,-a*z)*a^2;
179
integrate(%e^(a*z)/z^3,z),gamma_expand:true,logcontract,ratsimp;
180
(a^2*z^2*expintegral_ei(a*z)+(-a*z-1)*%e^(a*z))/(2*z^2);
181
181
182
182
/* Problem 42: NEW Type 2 */
183
integrate(%e^(a*z)/z^4,z);
184
gamma_incomplete(-3,-a*z)*a^3;
183
integrate(%e^(a*z)/z^4,z),gamma_expand:true,logcontract,ratsimp;
184
(a^3*z^3*expintegral_ei(a*z)+(-a^2*z^2-a*z-2)*%e^(a*z))/(6*z^3);
185
185
186
186
/* Problem 43: NEW Type 2 */
187
integrate(%e^(a*z)/z^5,z);
188
-gamma_incomplete(-4,-a*z)*a^4;
187
integrate(%e^(a*z)/z^5,z),gamma_expand:true,logcontract,ratsimp;
188
(a^4*z^4*expintegral_ei(a*z)+(-a^3*z^3-a^2*z^2-2*a*z-6)*%e^(a*z))/(24*z^4);
189
189
190
/* Problem 44: NEW Type 2 */
191
expand(integrate(z^(1/2+n)*%e^(a*z),z)-
192
-a^(-(2*n+1)/2-1)*gamma_incomplete((2*n+1)/2+1,-a*z)*(-z)^(-(2*n+1)/2-1)
193
*z^((2*n+1)/2+1));
194
0;
190
/* -----------------------------------------------------------------------------
191
* Problem 44: NEW Type 2 (OK)
192
* The following gives sqrt(z)/sqrt(-z) and not sqrt(-z)/sqrt(z),
193
* but the phase is correct.
194
*/
195
integrate(z^(1/2+n)*%e^(a*z),z),ratsimp;
196
a^(-n-3/2)*(-1)^n*gamma_incomplete((2*n+3)/2,-a*z)*sqrt(z)/sqrt(-z);
195
197
196
198
/* Problem 45: OK */
197
integrate(sqrt(z)*%e^(a*z),z);
198
2*(sqrt(z)*%e^(a*z)/(2*a)+sqrt(%pi)*%i*erf(%i*sqrt(a)*sqrt(z))
199
/(4*a^(3/2)));
199
integrate(sqrt(z)*%e^(a*z),z),erf_representation:erfi;
200
2*(sqrt(z)*%e^(a*z)/(2*a)-sqrt(%pi)*erfi(sqrt(a)*sqrt(z))/(4*a^(3/2)));
200
201
201
202
/* Problem 46: OK */
202
integrate(z^(3/2)*%e^(a*z),z);
203
integrate(z^(3/2)*%e^(a*z),z),erf_representation:erfi;
203
204
2*((2*a*z^(3/2)-3*sqrt(z))*%e^(a*z)/(4*a^2)
204
-3*sqrt(%pi)*%i*erf(%i*sqrt(a)*sqrt(z))/(8*a^(5/2)));
205
+3*sqrt(%pi)*erfi(sqrt(a)*sqrt(z))/(8*a^(5/2)));
205
206
206
207
/* Problem 47: OK */
207
integrate(z^(5/2)*%e^(a*z),z);
208
integrate(z^(5/2)*%e^(a*z),z),erf_representation:erfi;
208
209
2*((4*a^2*z^(5/2)-10*a*z^(3/2)+15*sqrt(z))*%e^(a*z)/(8*a^3)
209
+15*sqrt(%pi)*%i*erf(%i*sqrt(a)*sqrt(z))/(16*a^(7/2)));
210
-15*sqrt(%pi)*erfi(sqrt(a)*sqrt(z))/(16*a^(7/2)));
210
211
211
212
/* Problem 48: OK */
212
integrate(z^(7/2)*%e^(a*z),z);
213
2*((8*a^3*z^(7/2)-28*a^2*z^(5/2)+70*a*z^(3/2)-105*sqrt(z))*%e^(a*z)
214
/(16*a^4)
215
-105*sqrt(%pi)*%i*erf(%i*sqrt(a)*sqrt(z))/(32*a^(9/2)));
213
integrate(z^(7/2)*%e^(a*z),z),erf_representation:erfi;
214
2*((8*a^3*z^(7/2)-28*a^2*z^(5/2)+70*a*z^(3/2)-105*sqrt(z))*%e^(a*z)/(16*a^4)
215
+105*sqrt(%pi)*erfi(sqrt(a)*sqrt(z))/(32*a^(9/2)));
216
216
217
217
/* Problem 49: OK */
218
integrate(z^(9/2)*%e^(a*z),z);
219
2*((16*a^4*z^(9/2)-72*a^3*z^(7/2)+252*a^2*z^(5/2)-630*a*z^(3/2)
220
+945*sqrt(z))
221
*%e^(a*z)
222
/(32*a^5)
223
+945*sqrt(%pi)*%i*erf(%i*sqrt(a)*sqrt(z))/(64*a^(11/2)));
218
integrate(z^(9/2)*%e^(a*z),z),erf_representation:erfi;
219
2*((16*a^4*z^(9/2)-72*a^3*z^(7/2)+252*a^2*z^(5/2)-630*a*z^(3/2)+945*sqrt(z))
220
*%e^(a*z)
221
/(32*a^5)
222
-945*sqrt(%pi)*erfi(sqrt(a)*sqrt(z))/(64*a^(11/2)));
224
223
225
224
/* Problem 50: OK */
226
integrate(%e^(a*z)/sqrt(z),z);
227
-sqrt(%pi)*%i*erf(%i*sqrt(a)*sqrt(z))/sqrt(a);
228
229
/* Problem 51: NEW Type 2 */
230
integrate(%e^(a*z)/z^(3/2),z);
231
-%i*gamma_incomplete(-1/2,-a*z)*sqrt(a);
232
233
/* Problem 52: NEW Type 2 */
234
integrate(%e^(a*z)/sqrt(z^3),z);
235
-%i*gamma_incomplete(-1/2,-a*z)*sqrt(a);
236
237
/* Problem 53: NEW Type 2 */
238
integrate(%e^(a*z)/z^(5/2),z);
239
%i*gamma_incomplete(-3/2,-a*z)*a^(3/2);
240
241
/* Problem 54: NEW Type 2 */
242
integrate(%e^(a*z)/z^(7/2),z);
243
-%i*gamma_incomplete(-5/2,-a*z)*a^(5/2);
244
245
/* Problem 55: NEW Type 2 */
246
integrate(%e^(a*z)/z^(9/2),z);
247
%i*gamma_incomplete(-7/2,-a*z)*a^(7/2);
225
integrate(%e^(a*z)/sqrt(z),z),erf_representation:erfi;
226
sqrt(%pi)*erfi(sqrt(a)*sqrt(z))/sqrt(a);
227
228
/* -----------------------------------------------------------------------------
229
* Problem 51: NEW Type 2 (OK)
230
* We have a problem to simplify this integral to an erfi representation,
231
* because of expression like sqrt(z)/sqrt(-z).
232
* The same for No. 48, 49, 50, 51
233
*/
234
integrate(%e^(a*x)/x^(3/2),x);
235
-%i*gamma_incomplete(-1/2,-a*x)*sqrt(a);
236
237
/* Problem 52: NEW Type 2 (OK) */
238
integrate(%e^(a*x)/sqrt(x^3),x);
239
-%i*gamma_incomplete(-1/2,-a*x)*sqrt(a);
240
241
/* Problem 53: NEW Type 2 (OK) */
242
integrate(%e^(a*x)/x^(5/2),x);
243
%i*gamma_incomplete(-3/2,-a*x)*a^(3/2);
244
245
/* Problem 54: NEW Type 2 (OK) */
246
integrate(%e^(a*x)/x^(7/2),x);
247
-%i*gamma_incomplete(-5/2,-a*x)*a^(5/2);
248
249
/* Problem 55: NEW Type 2 (OK) */
250
integrate(%e^(a*x)/x^(9/2),x);
251
%i*gamma_incomplete(-7/2,-a*x)*a^(7/2);
248
252
249
253
/* Problem 56: NEW Type 2 */
250
254
integrate(z^(v-1)*a^(b*z+d),z);
251
255
-a^d*gamma_incomplete(v,-log(a)*b*z)*z^v/(b^v*(-log(a)*z)^v);
252
256
253
257
/* Problem 57: NEW 2 */
254
integrate(a^(b*z+d)/z,z);
255
-gamma_incomplete(0,-log(a)*b*z)*a^d;
258
integrate(a^(b*z+d)/z,z),gamma_expand:true,logcontract;
259
a^d*expintegral_ei(b*z*log(a));
256
260
257
261
/* Problem 58: NEW 2 */
258
262
integrate(z^(v-1)*%e^(a*z+b),z);
263
267
-a^(-n-1)*%e^b*(-1)^(n+1)*gamma_incomplete(n+1,-a*z);
264
268
265
269
/* Problem 60: OK */
266
integrate(z*%e^(a*z+b),z);
267
(a*%e^b*z-%e^b)*%e^(a*z)/a^2;
270
integrate(z*%e^(a*z+b),z),factor;
271
(a*z-1)*%e^(a*z+b)/a^2;
268
272
269
273
/* Problem 61: NEW 2 */
270
integrate(%e^(a*z+b)/z,z);
271
-gamma_incomplete(0,-a*z)*%e^b;
274
integrate(%e^(a*z+b)/z,z),gamma_expand:true,logcontract;
275
%e^b*expintegral_ei(a*z);
272
276
273
277
/* Problem 62: OK */
274
integrate(%e^(a*z+b)/sqrt(z),z);
275
-sqrt(%pi)*%i*%e^b*erf(%i*sqrt(a)*sqrt(z))/sqrt(a);
278
integrate(%e^(a*z+b)/sqrt(z),z),erf_representation:erfi;
279
sqrt(%pi)*%e^b*erfi(sqrt(a)*sqrt(z))/sqrt(a);
276
280
277
281
/* Problem 63: NEW 2 */
278
282
integrate(z^(v-1)*a^(b*z^r),z);
279
283
-gamma_incomplete(v/r,-log(a)*b*z^r)*z^v/(b^(v/r)*r*(-log(a)*z^r)^(v/r));
280
284
281
285
/* Problem 64: NEW 2 */
282
integrate(a^(b*z^r)/z,z);
283
-gamma_incomplete(0,-log(a)*b*z^r)/r;
286
integrate(a^(b*z^r)/z,z),gamma_expand:true,logcontract;
287
expintegral_ei(log(a)*b*z^r)/r;
284
288
285
289
/* Problem 65: NEW 2 */
286
290
integrate(z^(v-1)*%e^(a*z^r),z);
287
291
-gamma_incomplete(v/r,-a*z^r)*z^v/(a^(v/r)*r*(-z^r)^(v/r));
288
292
289
293
/* Problem 66: NEW 2 */
290
integrate(%e^(a*z^r)/z,z);
291
-gamma_incomplete(0,-a*z^r)/r;
294
integrate(%e^(a*z^r)/z,z),gamma_expand:true,logcontract;
295
expintegral_ei(a*z^r)/r;
292
296
293
297
/* Problem 67: NEW 2 */
294
integrate(z^(2*n)*%e^(a*z^2),z);
295
-gamma_incomplete((2*n+1)/2,-a*z^2)*z^(2*n+1)
296
/(2*a^((2*n+1)/2)*(-1)^((2*n+1)/2)*(z^2)^((2*n+1)/2));
298
integrate(z^(2*n)*%e^(a*z^2),z),ratsimp;
299
-a^(-n-1/2)*(-1)^n*gamma_incomplete((2*n+1)/2,-a*z^2)*z/(2*sqrt(-z^2));
297
300
298
301
/* Problem 68: NEW 2 */
299
302
integrate(z^(2*n-1)*%e^(a*z^2),z);
304
307
%e^(a*z^2)/(2*a);
305
308
306
309
/* Problem 70: OK */
307
integrate(z^2*%e^(a*z^2),z);
308
sqrt(%pi)*%i*erf(%i*sqrt(a)*z)/(4*a^(3/2))+z*%e^(a*z^2)/(2*a);
310
integrate(z^2*%e^(a*z^2),z),erf_representation:erfi;
311
z*%e^(a*z^2)/(2*a)-sqrt(%pi)*erfi(sqrt(a)*z)/(4*a^(3/2));
309
312
310
313
/* Problem 71: OK */
311
314
integrate(z^3*%e^(a*z^2),z);
312
315
(a*z^2-1)*%e^(a*z^2)/(2*a^2);
313
316
314
317
/* Problem 72: OK */
315
integrate(z^4*%e^(a*z^2),z);
316
(2*a*z^3-3*z)*%e^(a*z^2)/(4*a^2)-3*sqrt(%pi)*%i*erf(%i*sqrt(a)*z)
317
/(8*a^(5/2));
318
integrate(z^4*%e^(a*z^2),z),erf_representation:erfi;
319
3*sqrt(%pi)*erfi(sqrt(a)*z)/(8*a^(5/2))+(2*a*z^3-3*z)*%e^(a*z^2)/(4*a^2);
318
320
319
321
/* Problem 73: OK */
320
322
integrate(z^5*%e^(a*z^2),z);
321
323
(a^2*z^4-2*a*z^2+2)*%e^(a*z^2)/(2*a^3);
322
324
323
325
/* Problem 74: NEW 2 */
324
integrate(%e^(a*z^2)/z,z);
325
-gamma_incomplete(0,-a*z^2)/2;
326
integrate(%e^(a*z^2)/z,z),gamma_expand:true,logcontract;
327
expintegral_ei(a*z^2)/2;
326
328
327
329
/* Problem 75: NEW 2 */
328
integrate(%e^(a*z^2)/z^2,z);
329
-%i*gamma_incomplete(-1/2,-a*z^2)*sqrt(a)*sqrt(z^2)/(2*z);
330
integrate(%e^(a*z^2)/z^2,z), gamma_expand:true, erf_representation:erfi, radexpand:all, expand;
331
sqrt(%pi)*sqrt(a)*erfi(sqrt(a)*z)-%e^(a*z^2)/z+sqrt(%pi)*%i*sqrt(a);
330
332
331
333
/* Problem 76: NEW 2 */
332
integrate(%e^(a*z^2)/z^3,z);
333
gamma_incomplete(-1,-a*z^2)*a/2;
334
integrate(%e^(a*z^2)/z^3,z),gamma_expand:true,logcontract;
335
(2*a*z^2*expintegral_ei(a*z^2)-2*%e^(a*z^2))/(4*z^2);
334
336
335
337
/* Problem 77: NEW 2 */
336
integrate(%e^(a*z^2)/z^4,z);
337
%i*gamma_incomplete(-3/2,-a*z^2)*a^(3/2)*(z^2)^(3/2)/(2*z^3);
338
integrate(%e^(a*z^2)/z^4,z), gamma_expand:true, erf_representation:erfi, radexpand:all, expand;
339
2*sqrt(%pi)*a^(3/2)*erfi(sqrt(a)*z)/3-2*a*%e^(a*z^2)/(3*z)-%e^(a*z^2)/(3*z^3)
340
+2*sqrt(%pi)*%i*a^(3/2)/3;
338
341
339
342
/* Problem 78: NEW 2 */
340
integrate(%e^(a*z^2)/z^5,z);
341
-gamma_incomplete(-2,-a*z^2)*a^2/2;
343
integrate(%e^(a*z^2)/z^5,z),gamma_expand:true,logcontract;
344
(2*a^2*z^4*expintegral_ei(a*z^2)+(-2*a*z^2-2)*%e^(a*z^2))/(8*z^4);
342
345
343
346
/* Problem 79: NEW 2 */
344
347
integrate(z^n*%e^(a*sqrt(z)),z);
376
379
/a^12;
377
380
378
381
/* Problem 85: NEW 2 */
379
integrate(%e^(a*sqrt(z))/z,z);
380
-2*gamma_incomplete(0,-a*sqrt(z));
382
integrate(%e^(a*sqrt(z))/z,z),gamma_expand:true,logcontract;
383
2*expintegral_ei(a*sqrt(z));
381
384
382
385
/* Problem 86: NEW 2 */
383
integrate(%e^(a*sqrt(z))/z^2,z);
384
-2*gamma_incomplete(-2,-a*sqrt(z))*a^2;
386
integrate(%e^(a*sqrt(z))/z^2,z),gamma_expand:true,logcontract;
387
((2*a*z+2*sqrt(z))*%e^(a*sqrt(z))-2*a^2*expintegral_ei(a*sqrt(z))*z^(3/2))
388
/-(2*z^(3/2));
385
389
386
390
/* Problem 87: NEW 2 */
387
integrate(%e^(a*sqrt(z))/z^3,z);
388
-2*gamma_incomplete(-4,-a*sqrt(z))*a^4;
391
integrate(%e^(a*sqrt(z))/z^3,z),gamma_expand:true,logcontract;
392
((2*a^3*z^3+2*a^2*z^(5/2)+4*a*z^2+12*z^(3/2))*%e^(a*sqrt(z))
393
-2*a^4*expintegral_ei(a*sqrt(z))*z^(7/2))
394
/-(24*z^(7/2));
389
395
390
396
/* Problem 88: NEW 2 */
391
integrate(%e^(a*sqrt(z))/z^4,z);
392
-2*gamma_incomplete(-6,-a*sqrt(z))*a^6;
397
integrate(%e^(a*sqrt(z))/z^4,z),gamma_expand:true,logcontract;
398
((2*a^5*z^5+2*a^4*z^(9/2)+4*a^3*z^4+12*a^2*z^(7/2)+48*a*z^3+240*z^(5/2))
399
*%e^(a*sqrt(z))
400
-2*a^6*expintegral_ei(a*sqrt(z))*z^(11/2))
401
/-(720*z^(11/2));
393
402
394
403
/* Problem 89: NEW 2 */
395
integrate(%e^(a*sqrt(z))/z^5,z);
396
-2*gamma_incomplete(-8,-a*sqrt(z))*a^8;
404
integrate(%e^(a*sqrt(z))/z^5,z),gamma_expand:true,logcontract;
405
((2*a^7*z^7+2*a^6*z^(13/2)+4*a^5*z^6+12*a^4*z^(11/2)+48*a^3*z^5
406
+240*a^2*z^(9/2)+1440*a*z^4+10080*z^(7/2))
407
*%e^(a*sqrt(z))
408
-2*a^8*expintegral_ei(a*sqrt(z))*z^(15/2))
409
/-(40320*z^(15/2));
397
410
398
411
/* Problem 90: NEW 2 */
399
412
integrate(z^(v-1)*a^(d+b*z^r),z);
400
413
-a^d*gamma_incomplete(v/r,-log(a)*b*z^r)*z^v/(b^(v/r)*r*(-log(a)*z^r)^(v/r));
401
414
402
415
/* Problem 91: NEW 2 */
403
integrate(a^(d+b*z^r)/z,z);
404
-gamma_incomplete(0,-log(a)*b*z^r)*a^d/r;
416
integrate(a^(d+b*z^r)/z,z),gamma_expand:true,logcontract;
417
a^d*expintegral_ei(log(a)*b*z^r)/r;
405
418
406
419
/* Problem 92: NEW 2 */
407
420
integrate(z^(v-1)*%e^(b+a*z^r),z);
408
421
-%e^b*gamma_incomplete(v/r,-a*z^r)*z^v/(a^(v/r)*r*(-z^r)^(v/r));
409
422
410
423
/* Problem 93: NEW 2 */
411
integrate(%e^(b+a*z^r)/z,z);
412
-gamma_incomplete(0,-a*z^r)*%e^b/r;
424
integrate(%e^(b+a*z^r)/z,z),gamma_expand:true,logcontract;
425
%e^b*expintegral_ei(a*z^r)/r;
413
426
414
427
/* Problem 94: NEW 3 */
415
428
integrate(%e^(c*z)/(b+a*z),z);
416
429
-expintegral_e(1,-c*(a*z+b)/a)*%e^-(b*c/a)/a;
417
430
418
/* Problem 95: NEW 3
419
Problem the integrator expands the expression. We no longer match.
420
We factor again the expression at the beginning of the routine.
421
No it works. We have no other problems */
431
/* Problem 95: NEW 3 */
422
432
integrate(%e^(c*z)/(b+a*z)^2,z);
423
433
-expintegral_e(2,-c*(a*z+b)/a)*%e^-(b*c/a)/(a*(a*z+b));
424
434
545
555
-expintegral_e(3/2,-c*(a*z+b)/a)*%e^-(b*c/a)/(a*sqrt(a*z+b));
546
556
547
557
/* Problem 120: OK */
548
integrate(d^(c+b*z+a*z^2),z);
549
sqrt(%pi)*d^c*%e^-(b^2*log(d)/(4*a))
550
*erf(sqrt(a)*sqrt(-log(d))*z
551
-b*log(d)/(2*sqrt(a)*sqrt(-log(d))))
552
/(2*sqrt(a)*sqrt(-log(d)));
558
integrate(d^(c+b*z+a*z^2),z),erf_representation:erfi,ratsimp;
559
sqrt(%pi)*d^c*%e^-(b^2*log(d)/(4*a))*erfi(sqrt(log(d))*(2*a*z+b)/(2*sqrt(a)))
560
/(2*sqrt(a)*sqrt(log(d)));
553
561
554
562
/* Problem 121: OK */
555
integrate(%e^(c+b*z+a*z^2),z);
556
-sqrt(%pi)*%i*%e^(c-b^2/(4*a))*erf(%i*sqrt(a)*z+%i*b/(2*sqrt(a)))
557
/(2*sqrt(a));
563
integrate(%e^(c+b*z+a*z^2),z),erf_representation:erfi,ratsimp;
564
sqrt(%pi)*%e^(c-b^2/(4*a))*erfi((2*a*z+b)/(2*sqrt(a)))/(2*sqrt(a));
558
565
559
566
/* Problem 122: OK */
560
integrate(d^(b*z+a*z^2),z);
561
sqrt(%pi)*%e^-(b^2*log(d)/(4*a))
562
*erf(sqrt(a)*sqrt(-log(d))*z
563
-b*log(d)/(2*sqrt(a)*sqrt(-log(d))))
564
/(2*sqrt(a)*sqrt(-log(d)));
567
integrate(d^(b*z+a*z^2),z),erf_representation:erfi,ratsimp;
568
sqrt(%pi)*%e^-(b^2*log(d)/(4*a))*erfi(sqrt(log(d))*(2*a*z+b)/(2*sqrt(a)))
569
/(2*sqrt(a)*sqrt(log(d)));
565
570
566
571
/* Problem 123: OK */
567
integrate(%e^(b*z+a*z^2),z);
568
-sqrt(%pi)*%i*%e^-(b^2/(4*a))*erf(%i*sqrt(a)*z+%i*b/(2*sqrt(a)))
569
/(2*sqrt(a));
572
integrate(%e^(b*z+a*z^2),z),erf_representation:erfi,ratsimp;
573
sqrt(%pi)*%e^-(b^2/(4*a))*erfi((2*a*z+b)/(2*sqrt(a)))/(2*sqrt(a));
570
574
571
575
/* Problem 124: OK */
572
integrate(d^(c+a*z^2),z);
573
sqrt(%pi)*d^c*erf(sqrt(a)*sqrt(-log(d))*z)/(2*sqrt(a)*sqrt(-log(d)));
576
integrate(d^(c+a*z^2),z),erf_representation:erfi,ratsimp;
577
sqrt(%pi)*d^c*erfi(sqrt(a)*sqrt(log(d))*z)/(2*sqrt(a)*sqrt(log(d)));
574
578
575
579
/* Problem 125: OK */
576
integrate(%e^(c+a*z^2),z);
577
-sqrt(%pi)*%i*%e^c*erf(%i*sqrt(a)*z)/(2*sqrt(a));
580
integrate(%e^(c+a*z^2),z),erf_representation:erfi,ratsimp;
581
sqrt(%pi)*%e^c*erfi(sqrt(a)*z)/(2*sqrt(a));
578
582
579
583
/* Problem 126: NEW 4 */
580
expand(integrate(d^(b/z^2+a*z^2),z)-
581
sqrt(%pi)*%e^(2*sqrt(a)*sqrt(b)*log(d))
582
*(erfc(sqrt(b)*sqrt(-log(d))/z-sqrt(a)*sqrt(-log(d))*z)
583
-%e^-(4*sqrt(a)*sqrt(b)*log(d))
584
*erfc(sqrt(a)*sqrt(-log(d))*z+sqrt(b)*sqrt(-log(d))/z))
585
/(4*sqrt(a)*sqrt(-log(d))));
586
0;
584
integrate(d^(b/z^2+a*z^2),z),ratsimp;
585
sqrt(%pi)*%e^-(2*sqrt(a)*sqrt(b)*log(d))
586
*(%i*erfc(sqrt(log(d))*(%i*sqrt(a)*z^2+%i*sqrt(b))/z)
587
-%i*%e^(4*sqrt(a)*sqrt(b)*log(d))
588
*erfc(sqrt(log(d))*(%i*sqrt(b)-%i*sqrt(a)*z^2)/z))
589
/(4*sqrt(a)*sqrt(log(d)));
587
590
588
591
/* Problem 127: NEW 4 */
589
integrate(%e^(b/z^2+a*z^2),z);
590
-sqrt(%pi)*%i*%e^(2*sqrt(a)*sqrt(b))
591
*('erfc(%i*sqrt(b)/z-%i*sqrt(a)*z)
592
-%e^-(4*sqrt(a)*sqrt(b))*'erfc(%i*sqrt(a)*z+%i*sqrt(b)/z))
592
integrate(%e^(b/z^2+a*z^2),z),ratsimp;
593
sqrt(%pi)*%e^-(2*sqrt(a)*sqrt(b))
594
*(%i*erfc((%i*sqrt(a)*z^2+%i*sqrt(b))/z)
595
-%i*%e^(4*sqrt(a)*sqrt(b))*erfc((%i*sqrt(b)-%i*sqrt(a)*z^2)/z))
593
596
/(4*sqrt(a));
594
597
595
598
/* Problem 128: NEW 4 */
596
expand(integrate(d^(c+b/z^2+a*z^2),z) -
597
sqrt(%pi)*d^c*%e^(2*sqrt(a)*sqrt(b)*log(d))
598
*(erfc(sqrt(b)*sqrt(-log(d))/z-sqrt(a)*sqrt(-log(d))*z)
599
-%e^-(4*sqrt(a)*sqrt(b)*log(d))
600
*erfc(sqrt(a)*sqrt(-log(d))*z+sqrt(b)*sqrt(-log(d))/z))
601
/(4*sqrt(a)*sqrt(-log(d))));
602
0;
599
integrate(d^(c+b/z^2+a*z^2),z),ratsimp;
600
sqrt(%pi)*%e^-(2*sqrt(a)*sqrt(b)*log(d))
601
*(%i*d^c*erfc(sqrt(log(d))*(%i*sqrt(a)*z^2+%i*sqrt(b))/z)
602
-%i*d^c*%e^(4*sqrt(a)*sqrt(b)*log(d))
603
*erfc(sqrt(log(d))*(%i*sqrt(b)-%i*sqrt(a)*z^2)/z))
604
/(4*sqrt(a)*sqrt(log(d)));
603
605
604
606
/* Problem 129: NEW 4 */
605
integrate(%e^(c+b/z^2+a*z^2),z);
606
-sqrt(%pi)*%i*%e^(c+2*sqrt(a)*sqrt(b))
607
*('erfc(%i*sqrt(b)/z-%i*sqrt(a)*z)
608
-%e^-(4*sqrt(a)*sqrt(b))*'erfc(%i*sqrt(a)*z+%i*sqrt(b)/z))
607
integrate(%e^(c+b/z^2+a*z^2),z),ratsimp;
608
sqrt(%pi)*%e^-(2*sqrt(a)*sqrt(b))
609
*(%i*%e^c*erfc((%i*sqrt(a)*z^2+%i*sqrt(b))/z)
610
-%i*%e^(c+4*sqrt(a)*sqrt(b))*erfc((%i*sqrt(b)-%i*sqrt(a)*z^2)/z))
609
611
/(4*sqrt(a));
610
612
611
613
/* Problem 130: NEW 5 */
618
620
/(sqrt(a)*sqrt(log(d)));
619
621
620
622
/* Problem 131: NEW 5 */
621
expand(integrate(%e^(c+a*x+sqrt(x)*b),x) -
622
-(%i*gamma_incomplete(1/2,-(2*a*sqrt(x)+b)^2/(4*a))*b/(2*a)
623
-gamma_incomplete(1,-(2*a*sqrt(x)+b)^2/(4*a))/sqrt(a))
624
*%e^(c-b^2/(4*a))
625
/sqrt(a));
626
0;
623
integrate(%e^(c+a*x+sqrt(x)*b),x), gamma_expand:true, erf_representation:erfi, ratsimp;
624
%e^-(b^2/(4*a))*(sqrt(%pi)*sqrt(a)
625
*(b*%e^c*erfi((2*a*sqrt(x)+b)/(2*sqrt(a)))
626
+%i*b*%e^c)
627
-2*a*%e^(a*x+b*sqrt(x)+c+b^2/(4*a)))
628
/-(2*a^2);
627
629
628
630
/* Problem 132: NEW 5 */
629
expand(integrate(d^(a*x+sqrt(x)*b),x) -
630
-(gamma_incomplete(1/2,-log(d)*(2*a*sqrt(x)+b)^2/(4*a))*b*sqrt(-log(d))
631
/(2*a*sqrt(log(d)))
632
-gamma_incomplete(1,-log(d)*(2*a*sqrt(x)+b)^2/(4*a))/(sqrt(a)*sqrt(log(d))))
633
/(sqrt(a)*d^(b^2/(4*a))*sqrt(log(d))));
634
0;
631
integrate(d^(a*x+sqrt(x)*b),x),ratsimp;
632
(%i*gamma_incomplete(1/2,
633
(4*a^2*log(d)*x+4*a*b*log(d)*sqrt(x)+b^2*log(d))/-(4*a))
634
*sqrt(a)*b*sqrt(log(d))
635
-2*gamma_incomplete(1,
636
(4*a^2*log(d)*x+4*a*b*log(d)*sqrt(x)+b^2*log(d))/-(4*a))
637
*a)
638
/-(2*a^2*d^(b^2/(4*a))*log(d));
635
639
636
640
/* Problem 133: NEW 5 */
637
expand(integrate(%e^(a*x+sqrt(x)*b),x) -
638
-(%i*gamma_incomplete(1/2,-(2*a*sqrt(x)+b)^2/(4*a))*b/(2*a)
639
-gamma_incomplete(1,-(2*a*sqrt(x)+b)^2/(4*a))/sqrt(a))
640
*%e^-(b^2/(4*a))
641
/sqrt(a));
642
0;
641
integrate(%e^(a*x+sqrt(x)*b),x), gamma_expand:true, erf_representation:erfi, ratsimp;
642
-%e^-(b^2/(4*a))*(sqrt(%pi)*sqrt(a)*(b*erfi((2*a*sqrt(x)+b)/(2*sqrt(a)))+%i*b)
643
-2*a*%e^(a*x+b*sqrt(x)+b^2/(4*a)))
644
/(2*a^2);
643
645
644
646
/* Problem 134: NEW 1 */
645
647
integrate(d^(c+a*z^r),z);
650
652
-%e^c*gamma_incomplete(1/r,-a*z^r)*z/(a^(1/r)*r*(-z^r)^(1/r));
651
653
652
654
/* Problem 136: OK */
653
integrate(%e^(c+a*z^2),z);
654
-sqrt(%pi)*%i*%e^c*erf(%i*sqrt(a)*z)/(2*sqrt(a));
655
integrate(%e^(c+a*z^2),z),erf_representation:erfi;
656
sqrt(%pi)*%e^c*erfi(sqrt(a)*z)/(2*sqrt(a));
655
657
656
658
/* Problem 137: OK */
657
659
integrate(%e^(c+a*sqrt(z)),z);
658
660
2*(a*%e^c*sqrt(z)-%e^c)*%e^(a*sqrt(z))/a^2;
659
661
660
662
/* Problem 138: NEW 1 */
661
integrate(a^b^z,z);
662
-gamma_incomplete(0,-log(a)*b^z)/log(b);
663
664
/* Problem 139: NEW 1 */
665
integrate(%e^a^z,z);
666
-gamma_incomplete(0,-a^z)/log(a);
667
668
/* Problem 140: NEW 1 */
669
integrate(%e^%e^z,z);
670
-gamma_incomplete(0,-%e^z);
663
integrate(a^b^z,z),gamma_expand:true,logcontract;
664
expintegral_ei(log(a)*b^z)/log(b);
665
666
/* Problem 139: NEW 1
667
* We get a constant term log(-1), which Maxima does not simplify
668
*/
669
integrate(%e^a^z,z),gamma_expand:true,logexpand:all,logcontract;
670
(expintegral_ei(a^z)+log(-1))/log(a);
671
672
/* Problem 140: NEW 1
673
* We get a constant term log(-1), which Maxima does not simplify
674
*/
675
integrate(%e^%e^z,z),gamma_expand:true,logexpand:all,logcontract;
676
expintegral_ei(%e^z)+log(-1);
671
677
672
678
/* Problem 141: NOUN */
673
679
integrate(%e^tan(z),z);
749
755
integrate(%e^(a*asec(x)),x);
750
756
'integrate(%e^(a*asec(x)),x);
751
757
752
/* Problem 161: NOUN */
758
/* Problem 161: NEW 12/2009 */
753
759
integrate(%e^asinh(x),x);
754
'integrate(sqrt(x^2+1),x)+x^2/2;
760
log(sqrt(x^2+1)+x)/2+x*sqrt(x^2+1)/2+x^2/2;
755
761
756
762
/* Problem 162: NOUN */
757
763
integrate(%e^(a*asinh(x)),x);
758
764
'integrate(%e^(a*asinh(x)),x);
759
765
760
/* Problem 163: NOUN */
766
/* Problem 163: NEW 12/2009 */
761
767
integrate(%e^acosh(x),x);
762
'integrate(%e^acosh(x),x);
768
-log(2*sqrt(x^2-1)+2*x)/2+x*sqrt(x^2-1)/2+(x^2/2+x)/2+(x^2/2-x)/2;
763
769
764
770
/* Problem 164: NOUN */
765
771
integrate(%e^(a*acosh(x)),x);
789
795
integrate(%e^(a*acsch(x)),x);
790
796
'integrate(%e^(a*acsch(x)),x);
791
797
792
/* Problem 171: NOUN */
798
/* Problem 171: NEW 12/2009 */
793
799
integrate(%e^asech(x),x);
794
'integrate(%e^asech(x),x);
795
800
-log(2*sqrt(1-x^2)/abs(x)+2/abs(x))+(log(x)+x)/2+(log(x)-x)/2
801
+sqrt(1-x^2);
796
802
/* Problem 172: NOUN */
797
803
integrate(%e^(a*asech(x)),x);
798
804
'integrate(%e^(a*asech(x)),x);
799
805
800
806
/* Problem 173: NEW 5 */
801
expand(integrate(z^n*d^(b*z+a*z^2),z)-
802
-('sum(a^(-1/2+(-n)+1/2*(1+i5))*b^(n+(-1)*i5)
803
*(-log(d))^(n+(-1)*i5+(-1/2)*(1+i5))
804
*log(d)^(1/2+(-n)+i5)*binomial(n,i5)
805
*(2*a*z+b)^(1+i5)*abs(2*a*z+b)^(-1+(-1)*i5)
806
*2^(-n+i5)
807
*gamma_incomplete(1/2*(1+i5),
808
-log(d)*(2*a*z+b)^2/(4*a)),
809
i5,0,n))
810
/(2*sqrt(a)*d^(b^2/(4*a))*sqrt(log(d))));
807
expand(niceindices(integrate(x^n*d^(b*x+a*x^2),x)) -
808
(-('sum(a^(-n+(i+1)/2-1/2)*b^(n-i)*log(d)^(1/2-(i+1)/2)
809
*gamma_incomplete((i+1)/2,-log(d)*(2*a*x+b)^2/(4*a))
810
*2^(i-n)*(-1)^(n-(i+1)/2-i)*binomial(n,i)
811
*(2*a*x+b)^(i+1)*abs(2*a*x+b)^(-i-1),i,0,n))
812
/(2*sqrt(a)*d^(b^2/(4*a))*sqrt(log(d)))));
811
813
0;
812
814
813
815
/* Problem 174: NEW 5 */
814
expand(integrate(z^n*%e^(b*z+a*z^2),z)-
815
-%e^-(b^2/(4*a))*('sum(a^(-1/2+(-n)+1/2*(1+i6))*b^(n+(-1)*i6)*binomial(n,i6)
816
*(2*a*z+b)^(1+i6)
817
*abs(2*a*z+b)^(-1+(-1)*i6)
818
*2^(-n+i6)
819
*(-1)^(n+(-1)*i6+(-1/2)*(1+i6))
820
*gamma_incomplete(
821
1/2*(1+i6),
822
-(2*a*z+b)^2/(4*a)),i6,0,n))
823
/(2*sqrt(a)));
816
expand(niceindices(integrate(x^n*%e^(b*x+a*x^2),x)) -
817
(-%e^-(b^2/(4*a))*('sum(a^(-n+(i+1)/2-1/2)*b^(n-i)
818
*gamma_incomplete((i+1)/2,
819
-(2*a*x+b)^2/(4*a))
820
*2^(i-n)*(-1)^(n-(i+1)/2-i)
821
*binomial(n,i)*(2*a*x+b)^(i+1)
822
*abs(2*a*x+b)^(-i-1),i,0,n))
823
/(2*sqrt(a))));
824
824
0;
825
825
826
826
/* Problem 175: NEW 6 */
827
expand(integrate(x^n*d^(b*x+a*sqrt(x)),x) -
828
('sum(log(d)^(2*n+1)*binomial(n,i7)*(1/(2*b*sqrt(x)+a))*4^i7
829
*'sum(a^(-i8+2*n+(-1)*i7)*b^(1/2*(1+i8+i7))
830
*(-log(d))^((-1/2)*(1+i8+i7))
831
*(-1)^(i8+(-1)*i7)
832
*binomial(i7,i8)
833
*(a*log(d)*(2*b*sqrt(x)+a)
834
*gamma_incomplete(
835
1/2*(1+i8+i7),
836
-log(d)*(2*b*sqrt(x)+a)^2
837
/(4*b))
838
+2*sqrt(b)*sqrt(-log(d))
839
*(2*b*sqrt(x)+a)
840
*gamma_incomplete(
841
1/2*(2+i8+i7),
842
-log(d)*(2*b*sqrt(x)+a)^2
843
/(4*b))),i8,0,i7),i7,0,n))
844
*2^(-n-1)
845
/(b^(2*(n+1))*d^(a^2/(4*b))*log(d)^(2*(n+1))));
846
0;
827
expand(niceindices(niceindices(integrate(x^n*d^(b*x+a*sqrt(x)),x))) -
828
(2^(-n-1)*('sum(4^j*binomial(n,j)
829
*'sum(a^(2*n-j-i)*b^((j+i+1)/2)*log(d)^(2*n-(j+i+1)/2+1)
830
*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
831
*(2*%i*sqrt(b)*sqrt(log(d))
832
*gamma_incomplete((j+i+2)/2,
833
-log(d)
834
*(2*b*sqrt(x)+a)^2
835
/(4*b))
836
*(2*b*sqrt(x)+a)
837
+a*log(d)
838
*gamma_incomplete((j+i+1)/2,
839
-log(d)
840
*(2*b*sqrt(x)+a)^2
841
/(4*b))
842
*(2*b*sqrt(x)+a)),i,0,j),j,0,n))
843
/(b^(2*(n+1))*d^(a^2/(4*b))*log(d)^(2*(n+1))*(2*b*sqrt(x)+a))));
844
0;
847
845
848
846
/* Problem 176: NEW 6 */
849
expand(integrate(x^n*%e^(b*x+a*sqrt(x)),x) -
850
%e^-(a^2/(4*b))*('sum(binomial(n,i9)*(1/(2*b*sqrt(x)+a))*4^i9
851
*'sum(
852
a^(-i10+2*n+(-1)*i9)*b^(1/2*(1+i10+i9))
853
*(-1)
854
^(-i10+i9
855
+(-1/2)
856
*(1+i10+i9))
857
*binomial(i9,i10)
858
*(a*(2*b*sqrt(x)+a)
859
*gamma_incomplete(
860
1/2*(1+i10+i9),
861
-(2*b*sqrt(x)+a)
862
^2
863
/(4*b))
864
+2*%i*sqrt(b)
865
*(2*b*sqrt(x)+a)
866
*gamma_incomplete(
867
1/2*(2+i10+i9),
868
-(2*b*sqrt(x)+a)
869
^2
870
/(4*b))),i10,0,
871
i9),i9,0,n))*2^(-n-1)
872
/b^(2*(n+1)));
847
expand(niceindices(niceindices(integrate(x^n*%e^(b*x+a*sqrt(x)),x))) -
848
(%e^-(a^2/(4*b))*2^(-n-1)
849
*('sum(4^j*binomial(n,j)
850
*'sum(a^(2*n-j-i)*b^((j+i+1)/2)*(-1)^(-(j+i+1)/2+j-i)
851
*binomial(j,i)
852
*(2*%i*sqrt(b)
853
*gamma_incomplete(
854
(j+i+2)/2,
855
-(2*b*sqrt(x)+a)^2/(4*b))
856
*(2*b*sqrt(x)+a)
857
+a
858
*gamma_incomplete(
859
(j+i+1)/2,
860
-(2*b*sqrt(x)+a)^2/(4*b))
861
*(2*b*sqrt(x)+a)),i,0,j),j,0,n))
862
/(b^(2*(n+1))*(2*b*sqrt(x)+a))));
873
863
0;
874
864
875
865
/* Problem 177: NEW 5 */
876
expand(integrate(z^n*d^(c+b*z+a*z^2),z) -
877
-d^(c-b^2/(4*a))*('sum(a^(-1/2+(-n)+1/2*(1+i11))*b^(n+(-1)*i11)
878
*(-log(d))
879
^(n+(-1)*i11+(-1/2)*(1+i11))
880
*log(d)^(1/2+(-n)+i11)
881
*binomial(n,i11)
882
*(2*a*z+b)^(1+i11)
883
*abs(2*a*z+b)^(-1+(-1)*i11)
884
*2^(-n+i11)
885
*gamma_incomplete(
886
1/2*(1+i11),
887
-log(d)*(2*a*z+b)^2/(4*a)),i11,
888
0,n))
889
/(2*sqrt(a)*sqrt(log(d))));
866
expand(niceindices(integrate(x^n*d^(c+b*x+a*x^2),x)) -
867
(-d^(c-b^2/(4*a))*('sum(a^(-n+(i+1)/2-1/2)*b^(n-i)*log(d)^(1/2-(i+1)/2)
868
*gamma_incomplete((i+1)/2,
869
-log(d)*(2*a*x+b)^2
870
/(4*a))*2^(i-n)
871
*(-1)^(n-(i+1)/2-i)*binomial(n,i)
872
*(2*a*x+b)^(i+1)*abs(2*a*x+b)^(-i-1),
873
i,0,n))
874
/(2*sqrt(a)*sqrt(log(d)))));
890
875
0;
891
876
892
877
/* Problem 178: NEW 5 */
893
expand(integrate(z^n*%e^(c+b*z+a*z^2),z) -
894
-%e^(c-b^2/(4*a))*('sum(a^(-1/2+(-n)+1/2*(1+i12))*b^(n+(-1)*i12)
895
*binomial(n,i12)
896
*(2*a*z+b)^(1+i12)
897
*abs(2*a*z+b)^(-1+(-1)*i12)
898
*2^(-n+i12)
899
*(-1)
900
^(n+(-1)*i12+(-1/2)*(1+i12))
901
*gamma_incomplete(
902
1/2*(1+i12),
903
-(2*a*z+b)^2/(4*a)),i12,0,n))
904
/(2*sqrt(a)));
878
expand(niceindices(integrate(x^n*%e^(c+b*x+a*x^2),x)) -
879
(-%e^(c-b^2/(4*a))*('sum(a^(-n+(i+1)/2-1/2)*b^(n-i)
880
*gamma_incomplete((i+1)/2,
881
-(2*a*x+b)^2
882
/(4*a))*2^(i-n)
883
*(-1)^(n-(i+1)/2-i)*binomial(n,i)
884
*(2*a*x+b)^(i+1)
885
*abs(2*a*x+b)^(-i-1),i,0,n))
886
/(2*sqrt(a))));
905
887
0;
906
888
907
889
/* Problem 179: NEW 6 */
908
expand(integrate(x^n*d^(c+b*x+a*sqrt(x)),x) -
909
d^(c-a^2/(4*b))*('sum(log(d)^(2*n+1)*binomial(n,i13)*(1/(2*b*sqrt(x)+a))*4^i13
910
*'sum(
911
a^(-i14+2*n+(-1)*i13)
912
*b^(1/2*(1+i14+i13))
913
*(-log(d))^((-1/2)*(1+i14+i13))
914
*(-1)^(i14+(-1)*i13)*binomial(i13,i14)
915
*(a*log(d)*(2*b*sqrt(x)+a)
916
*gamma_incomplete(1/2*(1+i14+i13),
917
-log(d)
918
*(2*b*sqrt(x)+a)^2
919
/(4*b))
920
+2*sqrt(b)*sqrt(-log(d))
921
*(2*b*sqrt(x)+a)
922
*gamma_incomplete(1/2*(2+i14+i13),
923
-log(d)
924
*(2*b*sqrt(x)+a)^2
925
/(4*b))),i14,0,
926
i13),i13,0,n))*2^(-n-1)
927
/(b^(2*(n+1))*log(d)^(2*(n+1))));
890
expand(niceindices(niceindices(integrate(x^n*d^(c+b*x+a*sqrt(x)),x))) -
891
(d^(c-a^2/(4*b))*2^(-n-1)
892
*('sum(4^j*binomial(n,j)
893
*'sum(a^(2*n-j-i)*b^((j+i+1)/2)
894
*log(d)^(2*n-(j+i+1)/2+1)
895
*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
896
*(2*%i*sqrt(b)*sqrt(log(d))
897
*gamma_incomplete(
898
(j+i+2)/2,
899
-log(d)*(2*b*sqrt(x)+a)^2/(4*b))
900
*(2*b*sqrt(x)+a)
901
+a*log(d)
902
*gamma_incomplete(
903
(j+i+1)/2,
904
-log(d)*(2*b*sqrt(x)+a)^2/(4*b))
905
*(2*b*sqrt(x)+a)),i,0,j),j,0,n))
906
/(b^(2*(n+1))*log(d)^(2*(n+1))*(2*b*sqrt(x)+a))));
928
907
0;
929
908
930
909
/* Problem 180: NEW 6 */
931
expand(integrate(x^n*%e^(c+b*x+a*sqrt(x)),x) -
932
%e^(c-a^2/(4*b))*('sum(binomial(n,i15)*(1/(2*b*sqrt(x)+a))*4^i15
933
*'sum(
934
a^(-i16+2*n+(-1)*i15)
935
*b^(1/2*(1+i16+i15))
936
*(-1)^(-i16+i15+(-1/2)*(1+i16+i15))
937
*binomial(i15,i16)
938
*(a*(2*b*sqrt(x)+a)
939
*gamma_incomplete(
940
1/2*(1+i16+i15),
941
-(2*b*sqrt(x)+a)^2/(4*b))
942
+2*%i*sqrt(b)*(2*b*sqrt(x)+a)
943
*gamma_incomplete(
944
1/2*(2+i16+i15),
945
-(2*b*sqrt(x)+a)^2/(4*b))),i16,0,
946
i15),i15,0,n))*2^(-n-1)
947
/b^(2*(n+1)));
910
expand(niceindices(niceindices(integrate(x^n*%e^(c+b*x+a*sqrt(x)),x))) -
911
(%e^(c-a^2/(4*b))*2^(-n-1)
912
*('sum(4^j*binomial(n,j)
913
*'sum(a^(2*n-j-i)*b^((j+i+1)/2)
914
*(-1)^(-(j+i+1)/2+j-i)
915
*binomial(j,i)
916
*(2*%i*sqrt(b)
917
*gamma_incomplete(
918
(j+i+2)/2,
919
-(2*b*sqrt(x)+a)^2/(4*b))
920
*(2*b*sqrt(x)+a)
921
+a
922
*gamma_incomplete(
923
(j+i+1)/2,
924
-(2*b*sqrt(x)+a)^2/(4*b))
925
*(2*b*sqrt(x)+a)),i,0,j),j,0,n))
926
/(b^(2*(n+1))*(2*b*sqrt(x)+a))));
948
927
0;
949
928
950
929
/* Problem 181: NEW 2 */
957
936
958
937
/* Problem 183: NEW 2 */
959
938
integrate(z^(v-1)*d^(c+a*z^2),z);
960
-d^c*gamma_incomplete(v/2,-a*log(d)*z^2)*z^v
961
/(2*a^(v/2)*(-log(d))^(v/2)*(z^2)^(v/2));
939
d^c*gamma_incomplete(v/2,-a*log(d)*z^2)*z^v
940
/-(2*a^(v/2)*log(d)^(v/2)*(-z^2)^(v/2));
962
941
963
942
/* Problem 184: NEW 2 */
964
943
integrate(z^(v-1)*%e^(c+a*z^2),z);
965
-%e^c*gamma_incomplete(v/2,-a*z^2)*z^v/(2*a^(v/2)*(-1)^(v/2)*(z^2)^(v/2));
944
%e^c*gamma_incomplete(v/2,-a*z^2)*z^v/-(2*a^(v/2)*(-z^2)^(v/2));
966
945
967
946
/* Problem 185: NEW 2 */
968
947
integrate(z^(v-1)*d^(c+a*sqrt(z)),z);
969
-2*d^c*gamma_incomplete(2*v,-a*log(d)*sqrt(z))/(a^(2*v)*(-log(d))^(2*v));
948
2*d^c*gamma_incomplete(2*v,-a*log(d)*sqrt(z))*z^v
949
/-(a^(2*v)*log(d)^(2*v)*(-sqrt(z))^(2*v));
970
950
971
951
/* Problem 186: NEW 2 */
972
952
integrate(z^(v-1)*%e^(c+a*sqrt(z)),z);
973
-2*%e^c*gamma_incomplete(2*v,-a*sqrt(z))/(a^(2*v)*(-1)^(2*v));
953
2*%e^c*gamma_incomplete(2*v,-a*sqrt(z))*z^v/-(a^(2*v)*(-sqrt(z))^(2*v));
974
954
975
955
/* Problem 187: NOUN */
976
956
integrate(z^(2*n)*d^(b/z^2+a*z^2),z);
1139
1119
/* Problem 227: NEW 4 */
1140
1120
expand(integrate((%e^(b/z^2+a*z^2))^v,z) -
1141
1121
sqrt(%pi)*%e^(2*sqrt(a)*sqrt(b)*v)
1142
*('erfc(sqrt(b)*sqrt(-v)/z-sqrt(a)*sqrt(-v)*z)
1143
-%e^-(4*sqrt(a)*sqrt(b)*v)*'erfc(
1122
*(erfc(sqrt(b)*sqrt(-v)/z-sqrt(a)*sqrt(-v)*z)
1123
-%e^-(4*sqrt(a)*sqrt(b)*v)*erfc(
1144
1124
sqrt(a)*sqrt(-v)*z+sqrt(b)*sqrt(-v)/z))
1145
1125
/(4*sqrt(a)*sqrt(-v)));
1146
1126
0;
1148
1128
/* Problem 228: NEW 4 */
1149
1129
expand(integrate((%e^(c+b/z^2+a*z^2))^v,z) -
1150
1130
sqrt(%pi)*%e^(c*v+2*sqrt(a)*sqrt(b)*v)
1151
*('erfc(sqrt(b)*sqrt(-v)/z-sqrt(a)*sqrt(-v)*z)
1152
-%e^-(4*sqrt(a)*sqrt(b)*v)*'erfc(
1131
*(erfc(sqrt(b)*sqrt(-v)/z-sqrt(a)*sqrt(-v)*z)
1132
-%e^-(4*sqrt(a)*sqrt(b)*v)*erfc(
1153
1133
sqrt(a)*sqrt(-v)*z+sqrt(b)*sqrt(-v)/z))
1154
1134
/(4*sqrt(a)*sqrt(-v)));
1155
1135
0;
1203
1183
/2);
1204
1184
0;
1205
1185
1186
/* ---------- a^(b*z^r)*h^(c*z^r) ------------------------------------------- */
1187
1206
1188
/* Problem 236: NEW 7 */
1207
1189
expand(integrate(a^(b*z^r)*h^(c*z^r),z) -
1208
1190
-gamma_incomplete(1/r,-(c*log(h)+log(a)*b)*z^r)*z
1219
1201
*%e^(c*log(h)*sqrt(z)+log(a)*b*sqrt(z))
1220
1202
/(c^2*log(h)^2+2*log(a)*b*c*log(h)+log(a)^2*b^2);
1221
1203
1204
/* ---------- a^(d*z)*h^(c*z^r+g) ------------------------------------------- */
1205
1222
1206
/* Problem 239: OK */
1223
1207
integrate(a^(d*z)*h^(g+c*z^2),z);
1224
1208
sqrt(%pi)*h^g*%e^-(log(a)^2*d^2/(4*c*log(h)))
1239
1223
/2);
1240
1224
0;
1241
1225
1226
/* ---------- a^(d*z+e)*h^(c*z^r+g) ----------------------------------------- */
1227
1242
1228
/* Problem 241: OK */
1243
1229
integrate(a^(d*z+e)*h^(g+c*z^2),z);
1244
1230
sqrt(%pi)*a^e*h^g*%e^-(log(a)^2*d^2/(4*c*log(h)))
1328
1314
/2);
1329
1315
0;
1330
1316
1317
/* ---------- a^(b*z^r)*h^(c*z^r+f*z) --------------------------------------- */
1318
1331
1319
/* Problem 253: OK */
1332
1320
integrate(a^(b*z^2)*h^(f*z+c*z^2),z);
1333
1321
sqrt(%pi)*%e^(f^2*log(h)^2/(4*(-c*log(h)-log(a)*b)))
1335
1323
-f*log(h)/(2*sqrt(-c*log(h)-log(a)*b)))
1336
1324
/(2*sqrt(-c*log(h)-log(a)*b));
1337
1325
1338
/* Problem 254: NEW 10 (8) */
1326
/* Problem 254: NEW 9 */
1339
1327
expand(integrate(a^(b*sqrt(x))*h^(f*x+c*sqrt(x)),x)-
1340
-%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))
1341
*(gamma_incomplete(1/2,-(log(h)*(2*f*sqrt(x)+c)+log(a)*b)^2/(4*f*log(h)))
1342
*sqrt(-f*log(h))*(-c*log(h)-log(a)*b)*(log(h)*(2*f*sqrt(x)+c)+log(a)*b)
1328
(-%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))
1329
*(-%i*gamma_incomplete(1/2,(log(h)*(2*f*sqrt(x)+c)+log(a)*b)^2/-(4*f*log(h)))
1330
*sqrt(f*log(h))*(-c*log(h)-log(a)*b)*(log(h)*(2*f*sqrt(x)+c)+log(a)*b)
1343
1331
/(2*abs(log(h)*(2*f*sqrt(x)+c)+log(a)*b))
1344
-gamma_incomplete(1,-(log(h)*(2*f*sqrt(x)+c)+log(a)*b)^2/(4*f*log(h)))
1332
-gamma_incomplete(1,(log(h)*(2*f*sqrt(x)+c)+log(a)*b)^2/-(4*f*log(h)))
1345
1333
*f*log(h))
1346
/(f^2*log(h)^2));
1334
/(f^2*log(h)^2)));
1347
1335
0;
1348
1336
1337
1349
1338
/* Problem 255: OK */
1350
1339
integrate(a^(e+b*z^2)*h^(f*z+c*z^2),z);
1351
1340
sqrt(%pi)*a^e*%e^(f^2*log(h)^2/(4*(-c*log(h)-log(a)*b)))
1353
1342
-f*log(h)/(2*sqrt(-c*log(h)-log(a)*b)))
1354
1343
/(2*sqrt(-c*log(h)-log(a)*b));
1355
1344
1356
/* Problem 256: NEW 10 (8) */
1357
expand(integrate(a^(e+b*sqrt(x))*h^(f*x+c*sqrt(x)),x)-
1358
-a^e*%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))
1359
*(gamma_incomplete(1/2,-(log(h)*(2*f*sqrt(x)+c)+log(a)*b)^2/(4*f*log(h)))
1360
*sqrt(-f*log(h))*(-c*log(h)-log(a)*b)*(log(h)*(2*f*sqrt(x)+c)+log(a)*b)
1345
/* Problem 256: NEW 9 */
1346
expand(integrate(a^(e+b*sqrt(x))*h^(f*x+c*sqrt(x)),x) -
1347
(-a^e*%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))
1348
*(-%i*gamma_incomplete(1/2,
1349
(log(h)*(2*f*sqrt(x)+c)+log(a)*b)^2/-(4*f*log(h)))
1350
*sqrt(f*log(h))*(-c*log(h)-log(a)*b)
1351
*(log(h)*(2*f*sqrt(x)+c)+log(a)*b)
1361
1352
/(2*abs(log(h)*(2*f*sqrt(x)+c)+log(a)*b))
1362
-gamma_incomplete(1,-(log(h)*(2*f*sqrt(x)+c)+log(a)*b)^2/(4*f*log(h)))
1353
-gamma_incomplete(1,(log(h)*(2*f*sqrt(x)+c)+log(a)*b)^2/-(4*f*log(h)))
1363
1354
*f*log(h))
1364
/(f^2*log(h)^2));
1355
/(f^2*log(h)^2)));
1365
1356
0;
1366
1357
1367
1358
/* Problem 257: OK */
1372
1363
/(2*sqrt(-c*log(h)-log(a)*b));
1373
1364
1374
1365
/* Problem 258: NEW 10 */
1375
expand(integrate(a^(d*x+b*sqrt(x))*h^(f*x+c*sqrt(x)),x)-
1376
-%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d)))
1377
*(gamma_incomplete(1/2,
1378
-(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b))^2
1379
/(4*(f*log(h)+log(a)*d)))
1380
*(-c*log(h)-log(a)*b)*sqrt(-f*log(h)-log(a)*d)
1381
*(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b))
1366
expand(integrate(a^(d*x+b*sqrt(x))*h^(f*x+c*sqrt(x)),x) -
1367
(-%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d)))
1368
*(-%i*gamma_incomplete(1/2,
1369
(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b))^2
1370
/-(4*(f*log(h)+log(a)*d)))*(-c*log(h)-log(a)*b)
1371
*sqrt(f*log(h)+log(a)*d)*(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b))
1382
1372
/(2*abs(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b)))
1383
+gamma_incomplete(1,
1384
-(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b))^2
1385
/(4*(f*log(h)+log(a)*d)))
1386
*(-f*log(h)-log(a)*d))
1387
/(f*log(h)+log(a)*d)^2);
1373
-gamma_incomplete(1,
1374
(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b))^2
1375
/-(4*(f*log(h)+log(a)*d)))
1376
*(f*log(h)+log(a)*d))
1377
/(f*log(h)+log(a)*d)^2));
1388
1378
0;
1389
1379
1390
1380
/* Problem 259: OK */
1429
1419
-f*log(h)/(2*sqrt(-c*log(h)-log(a)*b)))
1430
1420
/(2*sqrt(-c*log(h)-log(a)*b));
1431
1421
1432
/* Problem 264: NEW 10 (8) */
1422
/* Problem 264: NEW 9 */
1433
1423
expand(integrate(a^(b*sqrt(x))*h^(g+f*x+c*sqrt(x)),x)-
1434
-h^g*%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))
1435
*(gamma_incomplete(1/2,-(log(h)*(2*f*sqrt(x)+c)+log(a)*b)^2/(4*f*log(h)))
1436
*sqrt(-f*log(h))*(-c*log(h)-log(a)*b)*(log(h)*(2*f*sqrt(x)+c)+log(a)*b)
1424
(-h^g*%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))
1425
*(-%i*gamma_incomplete(1/2,
1426
(log(h)*(2*f*sqrt(x)+c)+log(a)*b)^2/-(4*f*log(h)))
1427
*sqrt(f*log(h))*(-c*log(h)-log(a)*b)
1428
*(log(h)*(2*f*sqrt(x)+c)+log(a)*b)
1437
1429
/(2*abs(log(h)*(2*f*sqrt(x)+c)+log(a)*b))
1438
-gamma_incomplete(1,-(log(h)*(2*f*sqrt(x)+c)+log(a)*b)^2/(4*f*log(h)))
1430
-gamma_incomplete(1,(log(h)*(2*f*sqrt(x)+c)+log(a)*b)^2/-(4*f*log(h)))
1439
1431
*f*log(h))
1440
/(f^2*log(h)^2));
1432
/(f^2*log(h)^2)));
1441
1433
0;
1442
1434
1443
1435
/* Problem 265: OK */
1447
1439
-f*log(h)/(2*sqrt(-c*log(h)-log(a)*b)))
1448
1440
/(2*sqrt(-c*log(h)-log(a)*b));
1449
1441
1450
/* Problem 266: NEW 10 (8) */
1442
/* Problem 266: NEW 9 */
1451
1443
expand(integrate(a^(e+b*sqrt(x))*h^(g+f*x+c*sqrt(x)),x)-
1452
-a^e*h^g*%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))
1453
*(gamma_incomplete(1/2,-(log(h)*(2*f*sqrt(x)+c)+log(a)*b)^2/(4*f*log(h)))
1454
*sqrt(-f*log(h))*(-c*log(h)-log(a)*b)*(log(h)*(2*f*sqrt(x)+c)+log(a)*b)
1444
(-a^e*h^g*%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))
1445
*(-%i*gamma_incomplete(1/2,
1446
(log(h)*(2*f*sqrt(x)+c)+log(a)*b)^2/-(4*f*log(h)))
1447
*sqrt(f*log(h))*(-c*log(h)-log(a)*b)
1448
*(log(h)*(2*f*sqrt(x)+c)+log(a)*b)
1455
1449
/(2*abs(log(h)*(2*f*sqrt(x)+c)+log(a)*b))
1456
-gamma_incomplete(1,-(log(h)*(2*f*sqrt(x)+c)+log(a)*b)^2/(4*f*log(h)))
1450
-gamma_incomplete(1,(log(h)*(2*f*sqrt(x)+c)+log(a)*b)^2/-(4*f*log(h)))
1457
1451
*f*log(h))
1458
/(f^2*log(h)^2));
1452
/(f^2*log(h)^2)));
1459
1453
0;
1460
1454
1461
1455
/* Problem 267: OK */
1465
1459
+(-f*log(h)-log(a)*d)/(2*sqrt(-c*log(h)-log(a)*b)))
1466
1460
/(2*sqrt(-c*log(h)-log(a)*b));
1467
1461
1468
/* Problem 268: NEW 10 */
1462
/* Problem 268: NEW 9 */
1469
1463
expand(integrate(a^(d*x+b*sqrt(x))*h^(g+f*x+c*sqrt(x)),x)-
1470
-h^g*%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d)))
1471
*(gamma_incomplete(1/2,
1472
-(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b))^2
1473
/(4*(f*log(h)+log(a)*d)))
1474
*(-c*log(h)-log(a)*b)*sqrt(-f*log(h)-log(a)*d)
1475
*(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b))
1464
(-h^g*%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d)))
1465
*(-%i*gamma_incomplete(1/2,
1466
(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b))^2
1467
/-(4*(f*log(h)+log(a)*d)))*(-c*log(h)-log(a)*b)
1468
*sqrt(f*log(h)+log(a)*d)
1469
*(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b))
1476
1470
/(2*abs(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b)))
1477
+gamma_incomplete(1,
1478
-(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b))^2
1479
/(4*(f*log(h)+log(a)*d)))
1480
*(-f*log(h)-log(a)*d))
1481
/(f*log(h)+log(a)*d)^2);
1471
-gamma_incomplete(1,
1472
(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b))^2
1473
/-(4*(f*log(h)+log(a)*d)))
1474
*(f*log(h)+log(a)*d))
1475
/(f*log(h)+log(a)*d)^2));
1482
1476
0;
1483
1477
1484
1478
/* Problem 269: OK */
1488
1482
+(-f*log(h)-log(a)*d)/(2*sqrt(-c*log(h)-log(a)*b)))
1489
1483
/(2*sqrt(-c*log(h)-log(a)*b));
1490
1484
1491
/* Problem 270: NEW 10 */
1485
/* Problem 270: NEW 9 */
1492
1486
expand(integrate(a^(e+d*x+b*sqrt(x))*h^(g+f*x+c*sqrt(x)),x)-
1493
-a^e*h^g*%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d)))
1494
*(gamma_incomplete(1/2,
1495
-(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b))^2
1496
/(4*(f*log(h)+log(a)*d)))
1497
*(-c*log(h)-log(a)*b)*sqrt(-f*log(h)-log(a)*d)
1498
*(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b))
1487
(-a^e*h^g*%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d)))
1488
*(-%i*gamma_incomplete(1/2,
1489
(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b))^2
1490
/-(4*(f*log(h)+log(a)*d)))*(-c*log(h)-log(a)*b)
1491
*sqrt(f*log(h)+log(a)*d)
1492
*(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b))
1499
1493
/(2*abs(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b)))
1500
+gamma_incomplete(1,
1501
-(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b))^2
1502
/(4*(f*log(h)+log(a)*d)))
1503
*(-f*log(h)-log(a)*d))
1504
/(f*log(h)+log(a)*d)^2);
1494
-gamma_incomplete(1,
1495
(log(h)*(2*f*sqrt(x)+c)+log(a)*(2*d*sqrt(x)+b))^2
1496
/-(4*(f*log(h)+log(a)*d)))
1497
*(f*log(h)+log(a)*d))
1498
/(f*log(h)+log(a)*d)^2));
1505
1499
0;
1506
1500
1507
1501
/* Problem 271: OK */
1532
1526
1533
1527
/* Problem 277: NEW 5 */
1534
1528
expand(integrate(%e^(b*sqrt(x))*(%e^(c*x))^v,x) -
1535
-%e^-(b^2/(4*c*v))*(-gamma_incomplete(1/2,-(2*c*v*sqrt(x)+b)^2/(4*c*v))
1536
*b*sqrt(-c*v)*(2*c*v*sqrt(x)+b)
1537
/(2*(c*v)^(3/2)*abs(2*c*v*sqrt(x)+b))
1538
-gamma_incomplete(1,-(2*c*v*sqrt(x)+b)^2/(4*c*v))*c*v
1539
/(c*v)^(3/2))
1540
/sqrt(c*v));
1529
(-%e^-(b^2/(4*c*v))*(%i*gamma_incomplete(1/2,-(2*c*v*sqrt(x)+b)^2/(4*c*v))*b
1530
*(2*c*v*sqrt(x)+b)
1531
/(2*c*v*abs(2*c*v*sqrt(x)+b))
1532
-gamma_incomplete(1,-(2*c*v*sqrt(x)+b)^2/(4*c*v))
1533
/sqrt(c*v))
1534
/sqrt(c*v)));
1541
1535
0;
1542
1536
1543
1537
/* Problem 278: OK */
1547
1541
/(2*sqrt(b));
1548
1542
1549
1543
/* Problem 279: NEW 5 */
1550
expand(integrate(%e^(e+b*sqrt(x))*(%e^(c*x))^v,x)-
1551
-%e^(e-b^2/(4*c*v))*(-gamma_incomplete(1/2,-(2*c*v*sqrt(x)+b)^2/(4*c*v))
1552
*b*sqrt(-c*v)*(2*c*v*sqrt(x)+b)
1553
/(2*(c*v)^(3/2)*abs(2*c*v*sqrt(x)+b))
1554
-gamma_incomplete(1,-(2*c*v*sqrt(x)+b)^2/(4*c*v))*c*v
1555
/(c*v)^(3/2))
1556
/sqrt(c*v));
1544
expand(integrate(%e^(e+b*sqrt(x))*(%e^(c*x))^v,x) -
1545
(-%e^(e-b^2/(4*c*v))*(%i*gamma_incomplete(1/2,-(2*c*v*sqrt(x)+b)^2/(4*c*v))*b
1546
*(2*c*v*sqrt(x)+b)
1547
/(2*c*v*abs(2*c*v*sqrt(x)+b))
1548
-gamma_incomplete(1,-(2*c*v*sqrt(x)+b)^2/(4*c*v))
1549
/sqrt(c*v))
1550
/sqrt(c*v)));
1557
1551
0;
1558
1552
1559
1553
/* Problem 280: OK */
1562
1556
*erf(%i*sqrt(b)*z-%i*(-c*v-d)/(2*sqrt(b)))
1563
1557
/(2*sqrt(b));
1564
1558
1565
/* Problem 281: NEW 10 (8) */
1566
expand(integrate(%e^(d*x+b*sqrt(x))*(%e^(c*x))^v,x)-
1567
-%e^-(b^2/(4*(c*v+d)))*(gamma_incomplete(1,
1568
-(2*(c*v+d)*sqrt(x)+b)^2/(4*(c*v+d)))
1569
*(-c*v-d)
1570
-gamma_incomplete(1/2,
1571
-(2*(c*v+d)*sqrt(x)+b)^2/(4*(c*v+d)))
1572
*b*sqrt(-c*v-d)*(2*(c*v+d)*sqrt(x)+b)
1573
/(2*abs(2*(c*v+d)*sqrt(x)+b)))
1574
/(c*v+d)^2);
1559
/* Problem 281: NEW 5 */
1560
expand(integrate(%e^(d*x+b*sqrt(x))*(%e^(c*x))^v,x) -
1561
(-%e^-(b^2/(4*(c*v+d)))*(%i*gamma_incomplete(1/2,
1562
-(2*(c*v+d)*sqrt(x)+b)^2
1563
/(4*(c*v+d)))*b
1564
*(2*(c*v+d)*sqrt(x)+b)
1565
/(2*(c*v+d)*abs(2*(c*v+d)*sqrt(x)+b))
1566
-gamma_incomplete(1,
1567
-(2*(c*v+d)*sqrt(x)+b)^2/(4*(c*v+d)))
1568
/sqrt(c*v+d))
1569
/sqrt(c*v+d)));
1575
1570
0;
1576
1571
1577
1572
/* Problem 282: OK */
1580
1575
*erf(%i*sqrt(b)*z-%i*(-c*v-d)/(2*sqrt(b)))
1581
1576
/(2*sqrt(b));
1582
1577
1583
/* Problem 283: NEW 10 (8) */
1578
/* Problem 283: NEW 5 */
1584
1579
expand(integrate(%e^(e+d*x+b*sqrt(x))*(%e^(c*x))^v,x)-
1585
-%e^(e-b^2/(4*(c*v+d)))*(gamma_incomplete(1,
1586
-(2*(c*v+d)*sqrt(x)+b)^2
1587
/(4*(c*v+d)))
1588
*(-c*v-d)
1589
-gamma_incomplete(1/2,
1590
-(2*(c*v+d)*sqrt(x)+b)^2
1591
/(4*(c*v+d)))
1592
*b*sqrt(-c*v-d)*(2*(c*v+d)*sqrt(x)+b)
1593
/(2*abs(2*(c*v+d)*sqrt(x)+b)))
1594
/(c*v+d)^2);
1580
(-%e^(e-b^2/(4*(c*v+d)))*(%i*gamma_incomplete(1/2,
1581
(2*(c*v+d)*sqrt(x)+b)^2
1582
/-(4*(c*v+d)))*b
1583
*(2*(c*v+d)*sqrt(x)+b)
1584
/(2*(c*v+d)*abs(2*(c*v+d)*sqrt(x)+b))
1585
-gamma_incomplete(1,
1586
(2*(c*v+d)*sqrt(x)+b)^2
1587
/-(4*(c*v+d)))
1588
/sqrt(c*v+d))
1589
/sqrt(c*v+d)));
1595
1590
0;
1596
1591
1597
1592
/* Problem 284: OK */
1602
1597
1603
1598
/* Problem 285: NEW 5 */
1604
1599
expand(integrate(%e^(b*sqrt(x))*(%e^(g+f*x))^v,x) -
1605
-%e^(g*v-b^2/(4*f*v))*(-gamma_incomplete(1/2,-(2*f*v*sqrt(x)+b)^2/(4*f*v))
1606
*b*sqrt(-f*v)*(2*f*v*sqrt(x)+b)
1607
/(2*(f*v)^(3/2)*abs(2*f*v*sqrt(x)+b))
1608
-gamma_incomplete(1,-(2*f*v*sqrt(x)+b)^2/(4*f*v))*f*v
1609
/(f*v)^(3/2))
1610
/sqrt(f*v));
1600
(-%e^(g*v-b^2/(4*f*v))*(%i*gamma_incomplete(1/2,-(2*f*v*sqrt(x)+b)^2/(4*f*v))*b
1601
*(2*f*v*sqrt(x)+b)
1602
/(2*f*v*abs(2*f*v*sqrt(x)+b))
1603
-gamma_incomplete(1,-(2*f*v*sqrt(x)+b)^2/(4*f*v))
1604
/sqrt(f*v))
1605
/sqrt(f*v)));
1611
1606
0;
1612
1607
1613
1608
/* Problem 286: OK */
1618
1613
1619
1614
/* Problem 287: NEW 5 */
1620
1615
expand(integrate(%e^(e+b*sqrt(x))*(%e^(g+f*x))^v,x) -
1621
-%e^(g*v-b^2/(4*f*v)+e)*(-gamma_incomplete(1/2,-(2*f*v*sqrt(x)+b)^2/(4*f*v))
1622
*b*sqrt(-f*v)*(2*f*v*sqrt(x)+b)
1623
/(2*(f*v)^(3/2)*abs(2*f*v*sqrt(x)+b))
1624
-gamma_incomplete(1,-(2*f*v*sqrt(x)+b)^2/(4*f*v))*f*v
1625
/(f*v)^(3/2))
1626
/sqrt(f*v));
1616
(-%e^(g*v-b^2/(4*f*v)+e)*(%i*gamma_incomplete(1/2,-(2*f*v*sqrt(x)+b)^2/(4*f*v))
1617
*b*(2*f*v*sqrt(x)+b)
1618
/(2*f*v*abs(2*f*v*sqrt(x)+b))
1619
-gamma_incomplete(1,-(2*f*v*sqrt(x)+b)^2/(4*f*v))
1620
/sqrt(f*v))
1621
/sqrt(f*v)));
1627
1622
0;
1628
1623
1629
1624
/* Problem 288: OK */
1632
1627
*erf(%i*sqrt(b)*z-%i*(-f*v-d)/(2*sqrt(b)))
1633
1628
/(2*sqrt(b));
1634
1629
1635
/* Problem 289: NEW 10 (8) */
1630
/* Problem 289: NEW 5 */
1636
1631
expand(integrate(%e^(d*x+b*sqrt(x))*(%e^(g+f*x))^v,x)-
1637
-%e^(g*v-b^2/(4*(f*v+d)))*(gamma_incomplete(1,
1638
-(2*(f*v+d)*sqrt(x)+b)^2
1639
/(4*(f*v+d)))
1640
*(-f*v-d)
1641
-gamma_incomplete(1/2,
1642
-(2*(f*v+d)*sqrt(x)+b)^2
1643
/(4*(f*v+d)))
1644
*b*sqrt(-f*v-d)*(2*(f*v+d)*sqrt(x)+b)
1645
/(2*abs(2*(f*v+d)*sqrt(x)+b)))
1646
/(f*v+d)^2);
1632
(-%e^(g*v-b^2/(4*(f*v+d)))*(%i*gamma_incomplete(1/2,
1633
-(2*(f*v+d)*sqrt(x)+b)^2
1634
/(4*(f*v+d)))*b
1635
*(2*(f*v+d)*sqrt(x)+b)
1636
/(2*(f*v+d)*abs(2*(f*v+d)*sqrt(x)+b))
1637
-gamma_incomplete(1,
1638
-(2*(f*v+d)*sqrt(x)+b)^2
1639
/(4*(f*v+d)))
1640
/sqrt(f*v+d))
1641
/sqrt(f*v+d)));
1647
1642
0;
1648
1643
1649
1644
/* Problem 290: OK */
1652
1647
*erf(%i*sqrt(b)*z-%i*(-f*v-d)/(2*sqrt(b)))
1653
1648
/(2*sqrt(b));
1654
1649
1655
/* Problem 291: NEW 10 (8) */
1650
/* Problem 291: NEW 5 */
1656
1651
expand(integrate(%e^(e+d*x+b*sqrt(x))*(%e^(g+f*x))^v,x)-
1657
-%e^(-b^2/(4*(f*v+d))+g*v+e)*(gamma_incomplete(1,
1658
-(2*(f*v+d)*sqrt(x)+b)^2
1659
/(4*(f*v+d)))
1660
*(-f*v-d)
1661
-gamma_incomplete(1/2,
1662
-(2*(f*v+d)*sqrt(x)+b)^2
1663
/(4*(f*v+d)))
1664
*b*sqrt(-f*v-d)*(2*(f*v+d)*sqrt(x)+b)
1665
/(2*abs(2*(f*v+d)*sqrt(x)+b)))
1666
/(f*v+d)^2);
1652
(-%e^(-b^2/(4*(f*v+d))+g*v+e)*(%i*gamma_incomplete(1/2,
1653
-(2*(f*v+d)*sqrt(x)+b)^2
1654
/(4*(f*v+d)))*b
1655
*(2*(f*v+d)*sqrt(x)+b)
1656
/(2*(f*v+d)*abs(2*(f*v+d)*sqrt(x)+b))
1657
-gamma_incomplete(1,
1658
-(2*(f*v+d)*sqrt(x)+b)^2
1659
/(4*(f*v+d)))
1660
/sqrt(f*v+d))
1661
/sqrt(f*v+d)));
1667
1662
0;
1668
1663
1664
1669
1665
/* Problem 292: OK */
1670
1666
integrate(%e^(b*z)*(%e^(c*z^2))^v,z);
1671
1667
sqrt(%pi)*%e^-(b^2/(4*c*v))*erf(sqrt(-c*v)*z-b/(2*sqrt(-c*v)))
1687
1683
1688
1684
/* Problem 295: NEW 5 */
1689
1685
expand(integrate(%e^(d*x+e)*(%e^(c*sqrt(x)))^v,x) -
1690
-%e^(e-c^2*v^2/(4*d))*(-gamma_incomplete(1/2,-(2*d*sqrt(x)+c*v)^2/(4*d))
1691
*c*sqrt(-d)*v*(2*d*sqrt(x)+c*v)
1692
/(2*d^(3/2)*abs(2*d*sqrt(x)+c*v))
1686
(-%e^(e-c^2*v^2/(4*d))*(%i*gamma_incomplete(1/2,-(2*d*sqrt(x)+c*v)^2/(4*d))*c*v
1687
*(2*d*sqrt(x)+c*v)
1688
/(2*d*abs(2*d*sqrt(x)+c*v))
1693
1689
-gamma_incomplete(1,-(2*d*sqrt(x)+c*v)^2/(4*d))/sqrt(d))
1694
/sqrt(d));
1690
/sqrt(d)));
1695
1691
0;
1696
1692
1697
1693
/* Problem 296: NEW 9 */
1726
1722
sqrt(%pi)*%e^(d^2/(4*(-c*v-b)))*erf(sqrt(-c*v-b)*z-d/(2*sqrt(-c*v-b)))
1727
1723
/(2*sqrt(-c*v-b));
1728
1724
1729
/* Problem 303: NEW 10 (8) */
1725
/* Problem 303: NEW 5 */
1730
1726
expand(integrate(%e^(d*x+b*sqrt(x))*(%e^(c*sqrt(x)))^v,x)-
1731
-%e^-((c*v+b)^2/(4*d))*(gamma_incomplete(1/2,-(2*d*sqrt(x)+c*v+b)^2/(4*d))
1732
*sqrt(-d)*(-c*v-b)*(2*d*sqrt(x)+c*v+b)
1733
/(2*abs(2*d*sqrt(x)+c*v+b))
1734
-gamma_incomplete(1,-(2*d*sqrt(x)+c*v+b)^2/(4*d))*d)
1735
/d^2);
1727
(-%e^-((c*v+b)^2/(4*d))*(-%i*gamma_incomplete(1/2,-(2*d*sqrt(x)+c*v+b)^2/(4*d))
1728
*(-c*v-b)*(2*d*sqrt(x)+c*v+b)
1729
/(2*d*abs(2*d*sqrt(x)+c*v+b))
1730
-gamma_incomplete(1,-(2*d*sqrt(x)+c*v+b)^2/(4*d))
1731
/sqrt(d))
1732
/sqrt(d)));
1736
1733
0;
1737
1734
1738
1735
/* Problem 304: OK */
1741
1738
*erf(sqrt(-c*v-b)*z-d/(2*sqrt(-c*v-b)))
1742
1739
/(2*sqrt(-c*v-b));
1743
1740
1744
/* Problem 305: NEW 10 (8) */
1741
/* Problem 305: NEW 5 */
1745
1742
expand(integrate(%e^(e+d*x+b*sqrt(x))*(%e^(c*sqrt(x)))^v,x)-
1746
-%e^(e-(c*v+b)^2/(4*d))*(gamma_incomplete(1/2,-(2*d*sqrt(x)+c*v+b)^2/(4*d))
1747
*sqrt(-d)*(-c*v-b)*(2*d*sqrt(x)+c*v+b)
1748
/(2*abs(2*d*sqrt(x)+c*v+b))
1749
-gamma_incomplete(1,-(2*d*sqrt(x)+c*v+b)^2/(4*d))*d)
1750
/d^2);
1743
(-%e^(e-(c*v+b)^2/(4*d))*(-%i*gamma_incomplete(1/2,
1744
-(2*d*sqrt(x)+c*v+b)^2/(4*d))
1745
*(-c*v-b)*(2*d*sqrt(x)+c*v+b)
1746
/(2*d*abs(2*d*sqrt(x)+c*v+b))
1747
-gamma_incomplete(1,-(2*d*sqrt(x)+c*v+b)^2/(4*d))
1748
/sqrt(d))
1749
/sqrt(d)));
1751
1750
0;
1752
1751
1753
1752
/* Problem 306: OK */
1757
1756
1758
1757
/* Problem 307: NEW 5 */
1759
1758
expand(integrate(%e^(d*x)*(%e^(g+c*sqrt(x)))^v,x)-
1760
-%e^(g*v-c^2*v^2/(4*d))*(-gamma_incomplete(1/2,-(2*d*sqrt(x)+c*v)^2/(4*d))
1761
*c*sqrt(-d)*v*(2*d*sqrt(x)+c*v)
1762
/(2*d^(3/2)*abs(2*d*sqrt(x)+c*v))
1759
(-%e^(g*v-c^2*v^2/(4*d))*(%i*gamma_incomplete(1/2,-(2*d*sqrt(x)+c*v)^2/(4*d))*c
1760
*v*(2*d*sqrt(x)+c*v)
1761
/(2*d*abs(2*d*sqrt(x)+c*v))
1763
1762
-gamma_incomplete(1,-(2*d*sqrt(x)+c*v)^2/(4*d))
1764
1763
/sqrt(d))
1765
/sqrt(d));
1764
/sqrt(d)));
1766
1765
0;
1767
1766
1768
1767
/* Problem 308: OK */
1772
1771
1773
1772
/* Problem 309: NEW 5 */
1774
1773
expand(integrate(%e^(d*x+e)*(%e^(g+c*sqrt(x)))^v,x) -
1775
-%e^(-c^2*v^2/(4*d)+g*v+e)*(-gamma_incomplete(1/2,-(2*d*sqrt(x)+c*v)^2/(4*d))
1776
*c*sqrt(-d)*v*(2*d*sqrt(x)+c*v)
1777
/(2*d^(3/2)*abs(2*d*sqrt(x)+c*v))
1774
(-%e^(-c^2*v^2/(4*d)+g*v+e)*(%i*gamma_incomplete(1/2,
1775
-(2*d*sqrt(x)+c*v)^2/(4*d))*c
1776
*v*(2*d*sqrt(x)+c*v)
1777
/(2*d*abs(2*d*sqrt(x)+c*v))
1778
1778
-gamma_incomplete(1,-(2*d*sqrt(x)+c*v)^2/(4*d))
1779
1779
/sqrt(d))
1780
/sqrt(d));
1780
/sqrt(d)));
1781
1781
0;
1782
1782
1783
1783
/* Problem 310: NEW 9 */
1816
1816
*erf(sqrt(-c*v-b)*z-d/(2*sqrt(-c*v-b)))
1817
1817
/(2*sqrt(-c*v-b));
1818
1818
1819
/* Problem 317: NEW 10 (8) */
1819
/* Problem 317: NEW 5 */
1820
1820
expand(integrate(%e^(d*x+b*sqrt(x))*(%e^(g+c*sqrt(x)))^v,x)-
1821
-%e^(g*v-(c*v+b)^2/(4*d))*(gamma_incomplete(1/2,-(2*d*sqrt(x)+c*v+b)^2/(4*d))
1822
*sqrt(-d)*(-c*v-b)*(2*d*sqrt(x)+c*v+b)
1823
/(2*abs(2*d*sqrt(x)+c*v+b))
1824
-gamma_incomplete(1,-(2*d*sqrt(x)+c*v+b)^2/(4*d))*d)
1825
/d^2);
1821
(-%e^(g*v-(c*v+b)^2/(4*d))*(-%i*gamma_incomplete(1/2,
1822
-(2*d*sqrt(x)+c*v+b)^2/(4*d))
1823
*(-c*v-b)*(2*d*sqrt(x)+c*v+b)
1824
/(2*d*abs(2*d*sqrt(x)+c*v+b))
1825
-gamma_incomplete(1,-(2*d*sqrt(x)+c*v+b)^2/(4*d))
1826
/sqrt(d))
1827
/sqrt(d)));
1826
1828
0;
1827
1829
1828
1830
/* Problem 318: OK */
1831
1833
*erf(sqrt(-c*v-b)*z-d/(2*sqrt(-c*v-b)))
1832
1834
/(2*sqrt(-c*v-b));
1833
1835
1834
/* Problem 319: NEW 10 (8) */
1836
/* Problem 319: NEW 5 */
1835
1837
expand(integrate(%e^(e+d*x+b*sqrt(x))*(%e^(g+c*sqrt(x)))^v,x)-
1836
-%e^(-(c*v+b)^2/(4*d)+g*v+e)*(gamma_incomplete(1/2,
1837
-(2*d*sqrt(x)+c*v+b)^2/(4*d))
1838
*sqrt(-d)*(-c*v-b)*(2*d*sqrt(x)+c*v+b)
1839
/(2*abs(2*d*sqrt(x)+c*v+b))
1838
(-%e^(-(c*v+b)^2/(4*d)+g*v+e)*(-%i*gamma_incomplete(1/2,
1839
-(2*d*sqrt(x)+c*v+b)^2
1840
/(4*d))*(-c*v-b)
1841
*(2*d*sqrt(x)+c*v+b)
1842
/(2*d*abs(2*d*sqrt(x)+c*v+b))
1840
1843
-gamma_incomplete(1,-(2*d*sqrt(x)+c*v+b)^2/(4*d))
1841
*d)
1842
/d^2);
1844
/sqrt(d))
1845
/sqrt(d)));
1843
1846
0;
1844
1847
1845
1848
/* Problem 320: OK */
1848
1851
*erf(sqrt(-c*v)*z+(-f*v-d)/(2*sqrt(-c*v)))
1849
1852
/(2*sqrt(-c*v));
1850
1853
1851
/* Problem 321: NEW 10 (8) */
1854
/* Problem 321: NEW 5 */
1852
1855
expand(integrate(%e^(d*x)*(%e^(f*x+c*sqrt(x)))^v,x)-
1853
-%e^-(c^2*v^2/(4*(f*v+d)))*(gamma_incomplete(1,
1854
-(2*(f*v+d)*sqrt(x)+c*v)^2
1855
/(4*(f*v+d)))
1856
*(-f*v-d)
1857
-gamma_incomplete(1/2,
1858
-(2*(f*v+d)*sqrt(x)+c*v)^2
1859
/(4*(f*v+d)))
1860
*c*v*sqrt(-f*v-d)*(2*(f*v+d)*sqrt(x)+c*v)
1861
/(2*abs(2*(f*v+d)*sqrt(x)+c*v)))
1862
/(f*v+d)^2);
1856
(-%e^-(c^2*v^2/(4*(f*v+d)))*(%i*gamma_incomplete(1/2,
1857
-(2*(f*v+d)*sqrt(x)+c*v)^2
1858
/(4*(f*v+d)))*c*v
1859
*(2*(f*v+d)*sqrt(x)+c*v)
1860
/(2*(f*v+d)*abs(2*(f*v+d)*sqrt(x)+c*v))
1861
-gamma_incomplete(1,
1862
-(2*(f*v+d)*sqrt(x)+c*v)^2
1863
/(4*(f*v+d)))
1864
/sqrt(f*v+d))
1865
/sqrt(f*v+d)));
1863
1866
0;
1864
1867
1865
1868
/* Problem 322: OK */
1868
1871
*erf(sqrt(-c*v)*z+(-f*v-d)/(2*sqrt(-c*v)))
1869
1872
/(2*sqrt(-c*v));
1870
1873
1871
/* Problem 323: NEW 10 (8) */
1874
/* Problem 323: NEW 5 */
1872
1875
expand(integrate(%e^(d*x+e)*(%e^(f*x+c*sqrt(x)))^v,x)-
1873
-%e^(e-c^2*v^2/(4*(f*v+d)))*(gamma_incomplete(1,
1874
-(2*(f*v+d)*sqrt(x)+c*v)^2
1875
/(4*(f*v+d)))
1876
*(-f*v-d)
1877
-gamma_incomplete(1/2,
1878
-(2*(f*v+d)*sqrt(x)+c*v)^2
1879
/(4*(f*v+d)))
1880
*c*v*sqrt(-f*v-d)*(2*(f*v+d)*sqrt(x)+c*v)
1881
/(2*abs(2*(f*v+d)*sqrt(x)+c*v)))
1882
/(f*v+d)^2);
1876
(-%e^(e-c^2*v^2/(4*(f*v+d)))*(%i*gamma_incomplete(1/2,
1877
-(2*(f*v+d)*sqrt(x)+c*v)^2
1878
/(4*(f*v+d)))*c*v
1879
*(2*(f*v+d)*sqrt(x)+c*v)
1880
/(2*(f*v+d)*abs(2*(f*v+d)*sqrt(x)+c*v))
1881
-gamma_incomplete(1,
1882
-(2*(f*v+d)*sqrt(x)+c*v)^2
1883
/(4*(f*v+d)))
1884
/sqrt(f*v+d))
1885
/sqrt(f*v+d)));
1883
1886
0;
1884
1887
1885
1888
/* Problem 324: OK */
1888
1891
*erf(sqrt(-c*v-b)*z-f*v/(2*sqrt(-c*v-b)))
1889
1892
/(2*sqrt(-c*v-b));
1890
1893
1891
/* Problem 325: NEW 10 (8) */
1894
/* Problem 325: NEW 5 */
1892
1895
expand(integrate(%e^(b*sqrt(x))*(%e^(f*x+c*sqrt(x)))^v,x)-
1893
-%e^-((c*v+b)^2/(4*f*v))*(gamma_incomplete(1/2,
1894
-(2*f*v*sqrt(x)+c*v+b)^2/(4*f*v))
1895
*sqrt(-f*v)*(-c*v-b)*(2*f*v*sqrt(x)+c*v+b)
1896
/(2*abs(2*f*v*sqrt(x)+c*v+b))
1896
(-%e^-((c*v+b)^2/(4*f*v))*(-%i*gamma_incomplete(1/2,
1897
-(2*f*v*sqrt(x)+c*v+b)^2
1898
/(4*f*v))*(-c*v-b)
1899
*(2*f*v*sqrt(x)+c*v+b)
1900
/(2*f*v*abs(2*f*v*sqrt(x)+c*v+b))
1897
1901
-gamma_incomplete(1,-(2*f*v*sqrt(x)+c*v+b)^2/(4*f*v))
1898
*f*v)
1899
/(f^2*v^2));
1902
/sqrt(f*v))
1903
/sqrt(f*v)));
1900
1904
0;
1901
1905
1902
1906
/* Problem 326: OK */
1905
1909
*erf(sqrt(-c*v-b)*z-f*v/(2*sqrt(-c*v-b)))
1906
1910
/(2*sqrt(-c*v-b));
1907
1911
1908
/* Problem 327: NEW 10 (8) */
1912
/* Problem 327: NEW 5 */
1909
1913
expand(integrate(%e^(e+b*sqrt(x))*(%e^(f*x+c*sqrt(x)))^v,x)-
1910
-%e^(e-(c*v+b)^2/(4*f*v))*(gamma_incomplete(1/2,
1911
-(2*f*v*sqrt(x)+c*v+b)^2/(4*f*v))
1912
*sqrt(-f*v)*(-c*v-b)*(2*f*v*sqrt(x)+c*v+b)
1913
/(2*abs(2*f*v*sqrt(x)+c*v+b))
1914
(-%e^(e-(c*v+b)^2/(4*f*v))*(-%i*gamma_incomplete(1/2,
1915
-(2*f*v*sqrt(x)+c*v+b)^2
1916
/(4*f*v))*(-c*v-b)
1917
*(2*f*v*sqrt(x)+c*v+b)
1918
/(2*f*v*abs(2*f*v*sqrt(x)+c*v+b))
1914
1919
-gamma_incomplete(1,
1915
1920
-(2*f*v*sqrt(x)+c*v+b)^2/(4*f*v))
1916
*f*v)
1917
/(f^2*v^2));
1921
/sqrt(f*v))
1922
/sqrt(f*v)));
1918
1923
0;
1919
1924
1920
1925
/* Problem 328: OK */
1923
1928
*erf(sqrt(-c*v-b)*z+(-f*v-d)/(2*sqrt(-c*v-b)))
1924
1929
/(2*sqrt(-c*v-b));
1925
1930
1926
/* Problem 329: NEW 10 (8) */
1931
/* Problem 329: NEW 5 */
1927
1932
expand(integrate(%e^(d*x+b*sqrt(x))*(%e^(f*x+c*sqrt(x)))^v,x)-
1928
-%e^-((c*v+b)^2/(4*(f*v+d)))*(gamma_incomplete(1/2,
1929
-(2*(f*v+d)*sqrt(x)+c*v+b)^2
1930
/(4*(f*v+d)))
1931
*(-c*v-b)*sqrt(-f*v-d)*(2*(f*v+d)*sqrt(x)+c*v+b)
1932
/(2*abs(2*(f*v+d)*sqrt(x)+c*v+b))
1933
+gamma_incomplete(1,
1934
-(2*(f*v+d)*sqrt(x)+c*v+b)^2
1935
/(4*(f*v+d)))
1936
*(-f*v-d))
1937
/(f*v+d)^2);
1933
(-%e^-((c*v+b)^2/(4*(f*v+d)))*(-%i*gamma_incomplete(1/2,
1934
-(2*(f*v+d)*sqrt(x)+c*v+b)
1935
^2
1936
/(4*(f*v+d)))*(-c*v-b)
1937
*(2*(f*v+d)*sqrt(x)+c*v+b)
1938
/(2*(f*v+d)*abs(2*(f*v+d)*sqrt(x)+c*v+b))
1939
-gamma_incomplete(1,
1940
-(2*(f*v+d)*sqrt(x)+c*v+b)^2
1941
/(4*(f*v+d)))
1942
/sqrt(f*v+d))
1943
/sqrt(f*v+d)));
1938
1944
0;
1939
1945
1940
1946
/* Problem 330: OK */
1943
1949
*erf(sqrt(-c*v-b)*z+(-f*v-d)/(2*sqrt(-c*v-b)))
1944
1950
/(2*sqrt(-c*v-b));
1945
1951
1946
/* Problem 331: NEW 10 (8) */
1952
/* Problem 331: NEW 5 */
1947
1953
expand(integrate(%e^(e+d*x+b*sqrt(x))*(%e^(f*x+c*sqrt(x)))^v,x)-
1948
-%e^(e-(c*v+b)^2/(4*(f*v+d)))*(gamma_incomplete(1/2,
1949
-(2*(f*v+d)*sqrt(x)+c*v+b)^2
1950
/(4*(f*v+d)))
1951
*(-c*v-b)*sqrt(-f*v-d)*(2*(f*v+d)*sqrt(x)+c*v+b)
1952
/(2*abs(2*(f*v+d)*sqrt(x)+c*v+b))
1953
+gamma_incomplete(1,
1954
-(2*(f*v+d)*sqrt(x)+c*v+b)^2
1955
/(4*(f*v+d)))
1956
*(-f*v-d))
1957
/(f*v+d)^2);
1954
(-%e^(e-(c*v+b)^2/(4*(f*v+d)))*(-%i*gamma_incomplete(1/2,
1955
-(2*(f*v+d)*sqrt(x)+c*v+b)
1956
^2
1957
/(4*(f*v+d)))*(-c*v-b)
1958
*(2*(f*v+d)*sqrt(x)+c*v+b)
1959
/(2*(f*v+d)*abs(2*(f*v+d)*sqrt(x)+c*v+b))
1960
-gamma_incomplete(1,
1961
-(2*(f*v+d)*sqrt(x)+c*v+b)^2
1962
/(4*(f*v+d)))
1963
/sqrt(f*v+d))
1964
/sqrt(f*v+d)));
1958
1965
0;
1959
1966
1960
1967
/* Problem 332: OK */
1963
1970
*erf(sqrt(-c*v)*z+(-f*v-d)/(2*sqrt(-c*v)))
1964
1971
/(2*sqrt(-c*v));
1965
1972
1966
/* Problem 333: NEW 10 (8) */
1973
/* Problem 333: NEW 5 */
1967
1974
expand(integrate(%e^(d*x)*(%e^(g+f*x+c*sqrt(x)))^v,x)-
1968
-%e^(g*v-c^2*v^2/(4*(f*v+d)))*(gamma_incomplete(1,
1969
-(2*(f*v+d)*sqrt(x)+c*v)^2
1970
/(4*(f*v+d)))
1971
*(-f*v-d)
1972
-gamma_incomplete(1/2,
1973
-(2*(f*v+d)*sqrt(x)+c*v)^2
1974
/(4*(f*v+d)))
1975
*c*v*sqrt(-f*v-d)*(2*(f*v+d)*sqrt(x)+c*v)
1976
/(2*abs(2*(f*v+d)*sqrt(x)+c*v)))
1977
/(f*v+d)^2);
1975
(-%e^(g*v-c^2*v^2/(4*(f*v+d)))*(%i*gamma_incomplete(1/2,
1976
-(2*(f*v+d)*sqrt(x)+c*v)^2
1977
/(4*(f*v+d)))*c*v
1978
*(2*(f*v+d)*sqrt(x)+c*v)
1979
/(2*(f*v+d)*abs(2*(f*v+d)*sqrt(x)+c*v))
1980
-gamma_incomplete(1,
1981
-(2*(f*v+d)*sqrt(x)+c*v)^2
1982
/(4*(f*v+d)))
1983
/sqrt(f*v+d))
1984
/sqrt(f*v+d)));
1978
1985
0;
1979
1986
1980
1987
/* Problem 334: OK */
1983
1990
*erf(sqrt(-c*v)*z+(-f*v-d)/(2*sqrt(-c*v)))
1984
1991
/(2*sqrt(-c*v));
1985
1992
1986
/* Problem 335: NEW 10 (8) */
1993
/* Problem 335: NEW 5 */
1987
1994
expand(integrate(%e^(d*x+e)*(%e^(g+f*x+c*sqrt(x)))^v,x)-
1988
-%e^(-c^2*v^2/(4*(f*v+d))+g*v+e)*(gamma_incomplete(1,
1989
-(2*(f*v+d)*sqrt(x)+c*v)^2
1990
/(4*(f*v+d)))
1991
*(-f*v-d)
1992
-gamma_incomplete(1/2,
1993
-(2*(f*v+d)*sqrt(x)+c*v)^2
1994
/(4*(f*v+d)))
1995
*c*v*sqrt(-f*v-d)*(2*(f*v+d)*sqrt(x)+c*v)
1996
/(2*abs(2*(f*v+d)*sqrt(x)+c*v)))
1997
/(f*v+d)^2);
1995
(-%e^(-c^2*v^2/(4*(f*v+d))+g*v+e)*(%i*gamma_incomplete(1/2,
1996
-(2*(f*v+d)*sqrt(x)+c*v)
1997
^2
1998
/(4*(f*v+d)))*c*v
1999
*(2*(f*v+d)*sqrt(x)+c*v)
2000
/(2*(f*v+d)*abs(2*(f*v+d)*sqrt(x)+c*v))
2001
-gamma_incomplete(1,
2002
-(2*(f*v+d)*sqrt(x)+c*v)^2
2003
/(4*(f*v+d)))
2004
/sqrt(f*v+d))
2005
/sqrt(f*v+d)));
1998
2006
0;
1999
2007
2000
2008
/* Problem 336: OK */
2003
2011
*erf(sqrt(-c*v-b)*z-f*v/(2*sqrt(-c*v-b)))
2004
2012
/(2*sqrt(-c*v-b));
2005
2013
2006
/* Problem 337: NEW 10 (8) */
2014
/* Problem 337: NEW 5 */
2007
2015
expand(integrate(%e^(b*sqrt(x))*(%e^(g+f*x+c*sqrt(x)))^v,x)-
2008
-%e^(g*v-(c*v+b)^2/(4*f*v))*(gamma_incomplete(1/2,
2009
-(2*f*v*sqrt(x)+c*v+b)^2
2010
/(4*f*v))
2011
*sqrt(-f*v)*(-c*v-b)*(2*f*v*sqrt(x)+c*v+b)
2012
/(2*abs(2*f*v*sqrt(x)+c*v+b))
2016
(-%e^(g*v-(c*v+b)^2/(4*f*v))*(-%i*gamma_incomplete(1/2,
2017
-(2*f*v*sqrt(x)+c*v+b)^2
2018
/(4*f*v))*(-c*v-b)
2019
*(2*f*v*sqrt(x)+c*v+b)
2020
/(2*f*v*abs(2*f*v*sqrt(x)+c*v+b))
2013
2021
-gamma_incomplete(1,
2014
2022
-(2*f*v*sqrt(x)+c*v+b)^2
2015
2023
/(4*f*v))
2016
*f*v)
2017
/(f^2*v^2));
2024
/sqrt(f*v))
2025
/sqrt(f*v)));
2018
2026
0;
2019
2027
2020
2028
/* Problem 338: OK */
2023
2031
*erf(sqrt(-c*v-b)*z-f*v/(2*sqrt(-c*v-b)))
2024
2032
/(2*sqrt(-c*v-b));
2025
2033
2026
/* Problem 339: NEW 10 (8) */
2034
/* Problem 339: NEW 5 */
2027
2035
expand(integrate(%e^(e+b*sqrt(x))*(%e^(g+f*x+c*sqrt(x)))^v,x)-
2028
-%e^(-(c*v+b)^2/(4*f*v)+g*v+e)*(gamma_incomplete(1/2,
2029
-(2*f*v*sqrt(x)+c*v+b)^2
2030
/(4*f*v))
2031
*sqrt(-f*v)*(-c*v-b)*(2*f*v*sqrt(x)+c*v+b)
2032
/(2*abs(2*f*v*sqrt(x)+c*v+b))
2036
(-%e^(-(c*v+b)^2/(4*f*v)+g*v+e)*(-%i*gamma_incomplete(1/2,
2037
-(2*f*v*sqrt(x)+c*v+b)^2
2038
/(4*f*v))*(-c*v-b)
2039
*(2*f*v*sqrt(x)+c*v+b)
2040
/(2*f*v*abs(2*f*v*sqrt(x)+c*v+b))
2033
2041
-gamma_incomplete(1,
2034
2042
-(2*f*v*sqrt(x)+c*v+b)^2
2035
2043
/(4*f*v))
2036
*f*v)
2037
/(f^2*v^2));
2044
/sqrt(f*v))
2045
/sqrt(f*v)));
2038
2046
0;
2039
2047
2040
2048
/* Problem 340: OK */
2043
2051
*erf(sqrt(-c*v-b)*z+(-f*v-d)/(2*sqrt(-c*v-b)))
2044
2052
/(2*sqrt(-c*v-b));
2045
2053
2046
/* Problem 341: NEW 10 (8) */
2054
/* Problem 341: NEW 5 */
2047
2055
expand(integrate(%e^(d*x+b*sqrt(x))*(%e^(g+f*x+c*sqrt(x)))^v,x)-
2048
-%e^(g*v-(c*v+b)^2/(4*(f*v+d)))*(gamma_incomplete(1/2,
2049
-(2*(f*v+d)*sqrt(x)+c*v+b)^2
2050
/(4*(f*v+d)))
2051
*(-c*v-b)*sqrt(-f*v-d)
2052
*(2*(f*v+d)*sqrt(x)+c*v+b)
2053
/(2*abs(2*(f*v+d)*sqrt(x)+c*v+b))
2054
+gamma_incomplete(1,
2055
-(2*(f*v+d)*sqrt(x)+c*v+b)^2
2056
/(4*(f*v+d)))
2057
*(-f*v-d))
2058
/(f*v+d)^2);
2056
(-%e^(g*v-(c*v+b)^2/(4*(f*v+d)))*(-%i*gamma_incomplete(1/2,
2057
-(2*(f*v+d)*sqrt(x)
2058
+c*v+b)
2059
^2
2060
/(4*(f*v+d)))*(-c*v-b)
2061
*(2*(f*v+d)*sqrt(x)+c*v+b)
2062
/(2*(f*v+d)*abs(2*(f*v+d)*sqrt(x)+c*v+b))
2063
-gamma_incomplete(1,
2064
-(2*(f*v+d)*sqrt(x)+c*v+b)^2
2065
/(4*(f*v+d)))
2066
/sqrt(f*v+d))
2067
/sqrt(f*v+d)));
2059
2068
0;
2060
2069
2061
2070
/* Problem 342: OK */
2064
2073
*erf(sqrt(-c*v-b)*z+(-f*v-d)/(2*sqrt(-c*v-b)))
2065
2074
/(2*sqrt(-c*v-b));
2066
2075
2067
/* Problem 343: NEW 10 (8) */
2076
/* Problem 343: NEW 5 */
2068
2077
expand(integrate(%e^(e+d*x+b*sqrt(x))*(%e^(g+f*x+c*sqrt(x)))^v,x)-
2069
-%e^(-(c*v+b)^2/(4*(f*v+d))+g*v+e)*(gamma_incomplete(1/2,
2070
-(2*(f*v+d)*sqrt(x)
2071
+c*v+b)
2072
^2
2073
/(4*(f*v+d)))
2074
*(-c*v-b)*sqrt(-f*v-d)
2075
*(2*(f*v+d)*sqrt(x)+c*v+b)
2076
/(2*abs(2*(f*v+d)*sqrt(x)+c*v+b))
2077
+gamma_incomplete(1,
2078
-(2*(f*v+d)*sqrt(x)
2079
+c*v+b)
2080
^2
2081
/(4*(f*v+d)))
2082
*(-f*v-d))
2083
/(f*v+d)^2);
2078
(-%e^(-(c*v+b)^2/(4*(f*v+d))+g*v+e)*(-%i*gamma_incomplete(1/2,
2079
-(2*(f*v+d)*sqrt(x)
2080
+c*v+b)
2081
^2
2082
/(4*(f*v+d)))
2083
*(-c*v-b)*(2*(f*v+d)*sqrt(x)+c*v+b)
2084
/(2*(f*v+d)*abs(2*(f*v+d)*sqrt(x)+c*v+b))
2085
-gamma_incomplete(1,
2086
-(2*(f*v+d)*sqrt(x)
2087
+c*v+b)
2088
^2
2089
/(4*(f*v+d)))
2090
/sqrt(f*v+d))
2091
/sqrt(f*v+d)));
2084
2092
0;
2085
2093
2086
2094
/* Problem 344: OK */
2102
2110
2103
2111
/* Problem 348: NEW 5 */
2104
2112
expand(integrate((%e^(b*x))^u*(%e^(c*sqrt(x)))^v,x) -
2105
-%e^-(c^2*v^2/(4*b*u))*(-gamma_incomplete(1/2,-(2*b*u*sqrt(x)+c*v)^2/(4*b*u))
2106
*c*sqrt(-u)*v*(2*b*u*sqrt(x)+c*v)
2107
/(2*b*u^(3/2)*abs(2*b*u*sqrt(x)+c*v))
2113
(-%e^-(c^2*v^2/(4*b*u))*(%i*gamma_incomplete(1/2,
2114
-(2*b*u*sqrt(x)+c*v)^2/(4*b*u))*c
2115
*v*(2*b*u*sqrt(x)+c*v)
2116
/(2*b*u*abs(2*b*u*sqrt(x)+c*v))
2108
2117
-gamma_incomplete(1,-(2*b*u*sqrt(x)+c*v)^2/(4*b*u))
2109
2118
/(sqrt(b)*sqrt(u)))
2110
/(sqrt(b)*sqrt(u)));
2119
/(sqrt(b)*sqrt(u))));
2111
2120
0;
2112
2121
2113
2122
/* Problem 349: OK */
2118
2127
2119
2128
/* Problem 350: NEW 5 */
2120
2129
expand(integrate((%e^(d*x+e))^u*(%e^(c*sqrt(x)))^v,x)-
2121
-%e^(e*u-c^2*v^2/(4*d*u))*(-gamma_incomplete(1/2,
2122
-(2*d*u*sqrt(x)+c*v)^2/(4*d*u))
2123
*c*sqrt(-d*u)*v*(2*d*u*sqrt(x)+c*v)
2124
/(2*(d*u)^(3/2)*abs(2*d*u*sqrt(x)+c*v))
2130
(-%e^(e*u-c^2*v^2/(4*d*u))*(%i*gamma_incomplete(1/2,
2131
-(2*d*u*sqrt(x)+c*v)^2/(4*d*u))
2132
*c*v*(2*d*u*sqrt(x)+c*v)
2133
/(2*d*u*abs(2*d*u*sqrt(x)+c*v))
2125
2134
-gamma_incomplete(1,-(2*d*u*sqrt(x)+c*v)^2/(4*d*u))
2126
*d*u
2127
/(d*u)^(3/2))
2128
/sqrt(d*u));
2135
/(sqrt(d)*sqrt(u)))
2136
/(sqrt(d)*sqrt(u))));
2129
2137
0;
2130
2138
2131
2139
/* Problem 351: NEW 9 */
2150
2158
2151
2159
/* Problem 355: NEW 5 */
2152
2160
expand(integrate((%e^(d*x))^u*(%e^(g+c*sqrt(x)))^v,x) -
2153
-%e^(g*v-c^2*v^2/(4*d*u))*(-gamma_incomplete(1/2,
2154
-(2*d*u*sqrt(x)+c*v)^2/(4*d*u))
2155
*c*sqrt(-d*u)*v*(2*d*u*sqrt(x)+c*v)
2156
/(2*(d*u)^(3/2)*abs(2*d*u*sqrt(x)+c*v))
2161
(-%e^(g*v-c^2*v^2/(4*d*u))*(%i*gamma_incomplete(1/2,
2162
-(2*d*u*sqrt(x)+c*v)^2/(4*d*u))
2163
*c*v*(2*d*u*sqrt(x)+c*v)
2164
/(2*d*u*abs(2*d*u*sqrt(x)+c*v))
2157
2165
-gamma_incomplete(1,-(2*d*u*sqrt(x)+c*v)^2/(4*d*u))
2158
*d*u
2159
/(d*u)^(3/2))
2160
/sqrt(d*u));
2166
/(sqrt(d)*sqrt(u)))
2167
/(sqrt(d)*sqrt(u))));
2161
2168
0;
2162
2169
2163
2170
/* Problem 356: OK */
2168
2175
2169
2176
/* Problem 357: NEW 5 */
2170
2177
expand(integrate((%e^(d*x+e))^u*(%e^(g+c*sqrt(x)))^v,x)-
2171
-%e^(-c^2*v^2/(4*d*u)+g*v+e*u)*(-gamma_incomplete(1/2,
2172
-(2*d*u*sqrt(x)+c*v)^2
2173
/(4*d*u))
2174
*c*sqrt(-d*u)*v*(2*d*u*sqrt(x)+c*v)
2175
/(2*(d*u)^(3/2)*abs(2*d*u*sqrt(x)+c*v))
2178
(-%e^(-c^2*v^2/(4*d*u)+g*v+e*u)*(%i*gamma_incomplete(1/2,
2179
-(2*d*u*sqrt(x)+c*v)^2
2180
/(4*d*u))*c*v
2181
*(2*d*u*sqrt(x)+c*v)
2182
/(2*d*u*abs(2*d*u*sqrt(x)+c*v))
2176
2183
-gamma_incomplete(1,
2177
2184
-(2*d*u*sqrt(x)+c*v)^2
2178
2185
/(4*d*u))
2179
*d*u
2180
/(d*u)^(3/2))
2181
/sqrt(d*u));
2186
/(sqrt(d)*sqrt(u)))
2187
/(sqrt(d)*sqrt(u))));
2182
2188
0;
2183
2189
2184
2190
/* Problem 358: NEW 9 */
2218
2224
*erf(sqrt(-c*v)*z+(-f*v-d*u)/(2*sqrt(-c*v)))
2219
2225
/(2*sqrt(-c*v));
2220
2226
2221
/* Problem 365: NEW 10 (8) */
2227
/* Problem 365: NEW 5 */
2222
2228
expand(integrate((%e^(d*x))^u*(%e^(f*x+c*sqrt(x)))^v,x)-
2223
-%e^-(c^2*v^2/(4*(f*v+d*u)))*(gamma_incomplete(1,
2224
-(2*(f*v+d*u)*sqrt(x)+c*v)^2
2225
/(4*(f*v+d*u)))
2226
*(-f*v-d*u)
2227
-gamma_incomplete(1/2,
2228
-(2*(f*v+d*u)*sqrt(x)+c*v)^2
2229
/(4*(f*v+d*u)))
2230
*c*v*sqrt(-f*v-d*u)*(2*(f*v+d*u)*sqrt(x)+c*v)
2231
/(2*abs(2*(f*v+d*u)*sqrt(x)+c*v)))
2232
/(f*v+d*u)^2);
2229
(-%e^-(c^2*v^2/(4*(f*v+d*u)))*(%i*gamma_incomplete(1/2,
2230
-(2*(f*v+d*u)*sqrt(x)+c*v)^2
2231
/(4*(f*v+d*u)))*c*v
2232
*(2*(f*v+d*u)*sqrt(x)+c*v)
2233
/(2*(f*v+d*u)*abs(2*(f*v+d*u)*sqrt(x)+c*v))
2234
-gamma_incomplete(1,
2235
-(2*(f*v+d*u)*sqrt(x)+c*v)^2
2236
/(4*(f*v+d*u)))
2237
/sqrt(f*v+d*u))
2238
/sqrt(f*v+d*u)));
2233
2239
0;
2234
2240
2235
2241
/* Problem 366: OK */
2238
2244
*erf(sqrt(-c*v)*z+(-f*v-d*u)/(2*sqrt(-c*v)))
2239
2245
/(2*sqrt(-c*v));
2240
2246
2241
/* Problem 367: NEW 10 (8) */
2247
/* Problem 367: NEW 5 */
2242
2248
expand(integrate((%e^(e+d*x))^u*(%e^(f*x+c*sqrt(x)))^v,x)-
2243
-%e^(e*u-c^2*v^2/(4*(f*v+d*u)))*(gamma_incomplete(1,
2244
-(2*(f*v+d*u)*sqrt(x)+c*v)^2
2245
/(4*(f*v+d*u)))
2246
*(-f*v-d*u)
2247
-gamma_incomplete(1/2,
2248
-(2*(f*v+d*u)*sqrt(x)+c*v)^2
2249
/(4*(f*v+d*u)))
2250
*c*v*sqrt(-f*v-d*u)*(2*(f*v+d*u)*sqrt(x)+c*v)
2251
/(2*abs(2*(f*v+d*u)*sqrt(x)+c*v)))
2252
/(f*v+d*u)^2);
2249
(-%e^(e*u-c^2*v^2/(4*(f*v+d*u)))*(%i*gamma_incomplete(1/2,
2250
-(2*(f*v+d*u)*sqrt(x)
2251
+c*v)
2252
^2
2253
/(4*(f*v+d*u)))*c*v
2254
*(2*(f*v+d*u)*sqrt(x)+c*v)
2255
/(2*(f*v+d*u)*abs(2*(f*v+d*u)*sqrt(x)+c*v))
2256
-gamma_incomplete(1,
2257
-(2*(f*v+d*u)*sqrt(x)+c*v)^2
2258
/(4*(f*v+d*u)))
2259
/sqrt(f*v+d*u))
2260
/sqrt(f*v+d*u)));
2253
2261
0;
2254
2262
2255
2263
/* Problem 368: OK */
2258
2266
*erf(sqrt(-c*v-b*u)*z-f*v/(2*sqrt(-c*v-b*u)))
2259
2267
/(2*sqrt(-c*v-b*u));
2260
2268
2261
/* Problem 369: NEW 10 (8) */
2269
/* Problem 369: NEW 5 */
2262
2270
expand(integrate((%e^(b*sqrt(x)))^u*(%e^(f*x+c*sqrt(x)))^v,x)-
2263
-%e^-((c*v+b*u)^2/(4*f*v))*(gamma_incomplete(1/2,
2264
-(2*f*v*sqrt(x)+c*v+b*u)^2
2265
/(4*f*v))
2266
*sqrt(-f*v)*(-c*v-b*u)*(2*f*v*sqrt(x)+c*v+b*u)
2267
/(2*abs(2*f*v*sqrt(x)+c*v+b*u))
2271
(-%e^-((c*v+b*u)^2/(4*f*v))*(-%i*gamma_incomplete(1/2,
2272
-(2*f*v*sqrt(x)+c*v+b*u)^2
2273
/(4*f*v))*(-c*v-b*u)
2274
*(2*f*v*sqrt(x)+c*v+b*u)
2275
/(2*f*v*abs(2*f*v*sqrt(x)+c*v+b*u))
2268
2276
-gamma_incomplete(1,
2269
2277
-(2*f*v*sqrt(x)+c*v+b*u)^2
2270
2278
/(4*f*v))
2271
*f*v)
2272
/(f^2*v^2));
2279
/sqrt(f*v))
2280
/sqrt(f*v)));
2273
2281
0;
2274
2282
2275
2283
/* Problem 370: OK */
2278
2286
*erf(sqrt(-c*v-b*u)*z-f*v/(2*sqrt(-c*v-b*u)))
2279
2287
/(2*sqrt(-c*v-b*u));
2280
2288
2281
/* Problem 371: NEW 10 (8) */
2289
/* Problem 371: NEW 5 */
2282
2290
expand(integrate((%e^(e+b*sqrt(x)))^u*(%e^(f*x+c*sqrt(x)))^v,x)-
2283
-%e^(e*u-(c*v+b*u)^2/(4*f*v))*(gamma_incomplete(1/2,
2284
-(2*f*v*sqrt(x)+c*v+b*u)^2
2285
/(4*f*v))
2286
*sqrt(-f*v)*(-c*v-b*u)*(2*f*v*sqrt(x)+c*v+b*u)
2287
/(2*abs(2*f*v*sqrt(x)+c*v+b*u))
2291
(-%e^(e*u-(c*v+b*u)^2/(4*f*v))*(-%i*gamma_incomplete(1/2,
2292
-(2*f*v*sqrt(x)+c*v+b*u)^2
2293
/(4*f*v))*(-c*v-b*u)
2294
*(2*f*v*sqrt(x)+c*v+b*u)
2295
/(2*f*v*abs(2*f*v*sqrt(x)+c*v+b*u))
2288
2296
-gamma_incomplete(1,
2289
2297
-(2*f*v*sqrt(x)+c*v+b*u)^2
2290
2298
/(4*f*v))
2291
*f*v)
2292
/(f^2*v^2));
2299
/sqrt(f*v))
2300
/sqrt(f*v)));
2293
2301
0;
2294
2302
2295
2303
/* Problem 372: OK */
2298
2306
*erf(sqrt(-c*v-b*u)*z+(-f*v-d*u)/(2*sqrt(-c*v-b*u)))
2299
2307
/(2*sqrt(-c*v-b*u));
2300
2308
2301
/* Problem 373: NEW 10 (8) */
2309
/* Problem 373: NEW 5 */
2302
2310
expand(integrate((%e^(d*x+b*sqrt(x)))^u*(%e^(f*x+c*sqrt(x)))^v,x)-
2303
-%e^-((c*v+b*u)^2/(4*(f*v+d*u)))*(gamma_incomplete(1/2,
2304
-(2*(f*v+d*u)*sqrt(x)
2305
+c*v+b*u)
2306
^2
2307
/(4*(f*v+d*u)))
2308
*(-c*v-b*u)*sqrt(-f*v-d*u)
2309
*(2*(f*v+d*u)*sqrt(x)+c*v+b*u)
2310
/(2*abs(2*(f*v+d*u)*sqrt(x)+c*v+b*u))
2311
+gamma_incomplete(1,
2312
-(2*(f*v+d*u)*sqrt(x)
2313
+c*v+b*u)
2314
^2
2315
/(4*(f*v+d*u)))
2316
*(-f*v-d*u))
2317
/(f*v+d*u)^2);
2311
(-%e^-((c*v+b*u)^2/(4*(f*v+d*u)))*(-%i*gamma_incomplete(1/2,
2312
-(2*(f*v+d*u)*sqrt(x)
2313
+c*v+b*u)
2314
^2
2315
/(4*(f*v+d*u)))
2316
*(-c*v-b*u)*(2*(f*v+d*u)*sqrt(x)+c*v+b*u)
2317
/(2*(f*v+d*u)
2318
*abs(2*(f*v+d*u)*sqrt(x)+c*v+b*u))
2319
-gamma_incomplete(1,
2320
-(2*(f*v+d*u)*sqrt(x)
2321
+c*v+b*u)
2322
^2
2323
/(4*(f*v+d*u)))
2324
/sqrt(f*v+d*u))
2325
/sqrt(f*v+d*u)));
2318
2326
0;
2319
2327
2320
2328
/* Problem 374: OK */
2323
2331
*erf(sqrt(-c*v)*z+(-f*v-d*u)/(2*sqrt(-c*v)))
2324
2332
/(2*sqrt(-c*v));
2325
2333
2326
/* Problem 375: NEW 10 (8) */
2334
/* Problem 375: NEW 5 */
2327
2335
expand(integrate((%e^(d*x))^u*(%e^(g+f*x+c*sqrt(x)))^v,x)-
2328
-%e^(g*v-c^2*v^2/(4*(f*v+d*u)))*(gamma_incomplete(1,
2329
-(2*(f*v+d*u)*sqrt(x)+c*v)^2
2330
/(4*(f*v+d*u)))
2331
*(-f*v-d*u)
2332
-gamma_incomplete(1/2,
2333
-(2*(f*v+d*u)*sqrt(x)+c*v)^2
2334
/(4*(f*v+d*u)))
2335
*c*v*sqrt(-f*v-d*u)*(2*(f*v+d*u)*sqrt(x)+c*v)
2336
/(2*abs(2*(f*v+d*u)*sqrt(x)+c*v)))
2337
/(f*v+d*u)^2);
2336
(-%e^(g*v-c^2*v^2/(4*(f*v+d*u)))*(%i*gamma_incomplete(1/2,
2337
-(2*(f*v+d*u)*sqrt(x)
2338
+c*v)
2339
^2
2340
/(4*(f*v+d*u)))*c*v
2341
*(2*(f*v+d*u)*sqrt(x)+c*v)
2342
/(2*(f*v+d*u)*abs(2*(f*v+d*u)*sqrt(x)+c*v))
2343
-gamma_incomplete(1,
2344
-(2*(f*v+d*u)*sqrt(x)+c*v)^2
2345
/(4*(f*v+d*u)))
2346
/sqrt(f*v+d*u))
2347
/sqrt(f*v+d*u)));
2338
2348
0;
2339
2349
2340
2350
/* Problem 376: OK */
2343
2353
*erf(sqrt(-c*v)*z+(-f*v-d*u)/(2*sqrt(-c*v)))
2344
2354
/(2*sqrt(-c*v));
2345
2355
2346
/* Problem 377: NEW 10 (8) */
2356
/* Problem 377: NEW 5 */
2347
2357
expand(integrate((%e^(e+d*x))^u*(%e^(g+f*x+c*sqrt(x)))^v,x)-
2348
-%e^(-c^2*v^2/(4*(f*v+d*u))+g*v+e*u)*(gamma_incomplete(1,
2349
-(2*(f*v+d*u)*sqrt(x)
2350
+c*v)
2351
^2
2352
/(4*(f*v+d*u)))
2353
*(-f*v-d*u)
2354
-gamma_incomplete(1/2,
2355
-(2*(f*v+d*u)*sqrt(x)
2356
+c*v)
2357
^2
2358
/(4*(f*v+d*u)))
2359
*c*v*sqrt(-f*v-d*u)
2360
*(2*(f*v+d*u)*sqrt(x)+c*v)
2361
/(2*abs(2*(f*v+d*u)*sqrt(x)+c*v)))
2362
/(f*v+d*u)^2);
2358
(-%e^(-c^2*v^2/(4*(f*v+d*u))+g*v+e*u)*(%i
2359
*gamma_incomplete(1/2,
2360
-(2*(f*v+d*u)*sqrt(x)
2361
+c*v)
2362
^2
2363
/(4*(f*v+d*u)))*c*v
2364
*(2*(f*v+d*u)*sqrt(x)+c*v)
2365
/(2*(f*v+d*u)
2366
*abs(2*(f*v+d*u)*sqrt(x)+c*v))
2367
-gamma_incomplete(1,
2368
-(2*(f*v+d*u)*sqrt(x)
2369
+c*v)
2370
^2
2371
/(4*(f*v+d*u)))
2372
/sqrt(f*v+d*u))
2373
/sqrt(f*v+d*u)));
2363
2374
0;
2364
2375
2365
2376
/* Problem 378: OK */
2368
2379
*erf(sqrt(-c*v-b*u)*z-f*v/(2*sqrt(-c*v-b*u)))
2369
2380
/(2*sqrt(-c*v-b*u));
2370
2381
2371
/* Problem 379: NEW 10 (8) */
2382
/* Problem 379: NEW 5 */
2372
2383
expand(integrate((%e^(b*sqrt(x)))^u*(%e^(g+f*x+c*sqrt(x)))^v,x)-
2373
-%e^(g*v-(c*v+b*u)^2/(4*f*v))*(gamma_incomplete(1/2,
2374
-(2*f*v*sqrt(x)+c*v+b*u)^2
2375
/(4*f*v))
2376
*sqrt(-f*v)*(-c*v-b*u)*(2*f*v*sqrt(x)+c*v+b*u)
2377
/(2*abs(2*f*v*sqrt(x)+c*v+b*u))
2384
(-%e^(g*v-(c*v+b*u)^2/(4*f*v))*(-%i*gamma_incomplete(1/2,
2385
-(2*f*v*sqrt(x)+c*v+b*u)^2
2386
/(4*f*v))*(-c*v-b*u)
2387
*(2*f*v*sqrt(x)+c*v+b*u)
2388
/(2*f*v*abs(2*f*v*sqrt(x)+c*v+b*u))
2378
2389
-gamma_incomplete(1,
2379
2390
-(2*f*v*sqrt(x)+c*v+b*u)^2
2380
2391
/(4*f*v))
2381
*f*v)
2382
/(f^2*v^2));
2392
/sqrt(f*v))
2393
/sqrt(f*v)));
2383
2394
0;
2384
2395
2385
2396
/* Problem 380: OK */
2388
2399
*erf(sqrt(-c*v-b*u)*z-f*v/(2*sqrt(-c*v-b*u)))
2389
2400
/(2*sqrt(-c*v-b*u));
2390
2401
2391
/* Problem 381: NEW 10 (8) */
2402
/* Problem 381: NEW 5 */
2392
2403
expand(integrate((%e^(e+b*sqrt(x)))^u*(%e^(g+f*x+c*sqrt(x)))^v,x)-
2393
-%e^(-(c*v+b*u)^2/(4*f*v)+g*v+e*u)*(gamma_incomplete(1/2,
2394
-(2*f*v*sqrt(x)+c*v+b*u)
2395
^2
2396
/(4*f*v))
2397
*sqrt(-f*v)*(-c*v-b*u)
2398
*(2*f*v*sqrt(x)+c*v+b*u)
2399
/(2*abs(2*f*v*sqrt(x)+c*v+b*u))
2404
(-%e^(-(c*v+b*u)^2/(4*f*v)+g*v+e*u)*(-%i*gamma_incomplete(1/2,
2405
-(2*f*v*sqrt(x)
2406
+c*v+b*u)
2407
^2
2408
/(4*f*v))*(-c*v-b*u)
2409
*(2*f*v*sqrt(x)+c*v+b*u)
2410
/(2*f*v*abs(2*f*v*sqrt(x)+c*v+b*u))
2400
2411
-gamma_incomplete(1,
2401
2412
-(2*f*v*sqrt(x)+c*v+b*u)
2402
2413
^2
2403
2414
/(4*f*v))
2404
*f*v)
2405
/(f^2*v^2));
2415
/sqrt(f*v))
2416
/sqrt(f*v)));
2406
2417
0;
2407
2418
2408
2419
/* Problem 382: OK */
2411
2422
*erf(sqrt(-c*v-b*u)*z+(-f*v-d*u)/(2*sqrt(-c*v-b*u)))
2412
2423
/(2*sqrt(-c*v-b*u));
2413
2424
2414
/* Problem 383: NEW 10 (8) */
2425
/* Problem 383: NEW 5 */
2415
2426
expand(integrate((%e^(d*x+b*sqrt(x)))^u*(%e^(g+f*x+c*sqrt(x)))^v,x)-
2416
-%e^(g*v-(c*v+b*u)^2/(4*(f*v+d*u)))*(gamma_incomplete(1/2,
2417
-(2*(f*v+d*u)*sqrt(x)
2418
+c*v+b*u)
2419
^2
2420
/(4*(f*v+d*u)))
2421
*(-c*v-b*u)*sqrt(-f*v-d*u)
2422
*(2*(f*v+d*u)*sqrt(x)+c*v+b*u)
2423
/(2*abs(2*(f*v+d*u)*sqrt(x)+c*v+b*u))
2424
+gamma_incomplete(1,
2425
-(2*(f*v+d*u)*sqrt(x)
2426
+c*v+b*u)
2427
^2
2428
/(4*(f*v+d*u)))
2429
*(-f*v-d*u))
2430
/(f*v+d*u)^2);
2427
(-%e^(g*v-(c*v+b*u)^2/(4*(f*v+d*u)))*(-%i
2428
*gamma_incomplete(1/2,
2429
-(2*(f*v+d*u)*sqrt(x)
2430
+c*v+b*u)
2431
^2
2432
/(4*(f*v+d*u)))
2433
*(-c*v-b*u)*(2*(f*v+d*u)*sqrt(x)+c*v+b*u)
2434
/(2*(f*v+d*u)
2435
*abs(2*(f*v+d*u)*sqrt(x)+c*v+b*u))
2436
-gamma_incomplete(1,
2437
-(2*(f*v+d*u)*sqrt(x)
2438
+c*v+b*u)
2439
^2
2440
/(4*(f*v+d*u)))
2441
/sqrt(f*v+d*u))
2442
/sqrt(f*v+d*u)));
2431
2443
0;
2432
2444
2433
2445
/* Problem 384: OK */
2436
2448
*erf(sqrt(-c*v-b*u)*z+(-f*v-d*u)/(2*sqrt(-c*v-b*u)))
2437
2449
/(2*sqrt(-c*v-b*u));
2438
2450
2439
/* Problem 385: NEW 10 (8) */
2451
/* Problem 385: NEW 5 */
2440
2452
expand(integrate((%e^(e+d*x+b*sqrt(x)))^u*(%e^(g+f*x+c*sqrt(x)))^v,x)-
2441
-%e^(-(c*v+b*u)^2/(4*(f*v+d*u))+g*v+e*u)
2442
*(gamma_incomplete(1/2,-(2*(f*v+d*u)*sqrt(x)+c*v+b*u)^2/(4*(f*v+d*u)))
2443
*(-c*v-b*u)*sqrt(-f*v-d*u)*(2*(f*v+d*u)*sqrt(x)+c*v+b*u)
2444
/(2*abs(2*(f*v+d*u)*sqrt(x)+c*v+b*u))
2445
+gamma_incomplete(1,-(2*(f*v+d*u)*sqrt(x)+c*v+b*u)^2/(4*(f*v+d*u)))
2446
*(-f*v-d*u))
2447
/(f*v+d*u)^2);
2453
(-%e^(-(c*v+b*u)^2/(4*(f*v+d*u))+g*v+e*u)
2454
*(-%i*gamma_incomplete(1/2,-(2*(f*v+d*u)*sqrt(x)+c*v+b*u)^2/(4*(f*v+d*u)))
2455
*(-c*v-b*u)*(2*(f*v+d*u)*sqrt(x)+c*v+b*u)
2456
/(2*(f*v+d*u)*abs(2*(f*v+d*u)*sqrt(x)+c*v+b*u))
2457
-gamma_incomplete(1,-(2*(f*v+d*u)*sqrt(x)+c*v+b*u)^2/(4*(f*v+d*u)))
2458
/sqrt(f*v+d*u))
2459
/sqrt(f*v+d*u)));
2448
2460
0;
2449
2461
2450
2462
/* Problem 386: OK */
2692
2704
'integrate(((b*%e^(c*z)+a)^v)^p,z);
2693
2705
2694
2706
/* Problem 430: OK */
2695
integrate(((b*%e^(c*z)+a)^5)^(1/2),z);
2707
integrate(((b*%e^(c*z)+a)^5)^(1/2),z),radexpand:all;
2696
2708
a^(5/2)*log((2*sqrt(b*%e^(c*z)+a)-2*sqrt(a))
2697
2709
/(2*sqrt(b*%e^(c*z)+a)+2*sqrt(a)))
2698
2710
/c
2701
2713
/(15*c);
2702
2714
2703
2715
/* Problem 431: OK */
2704
integrate(((b*%e^(c*z)+a)^3)^(1/2),z);
2716
integrate(((b*%e^(c*z)+a)^3)^(1/2),z),radexpand:all;
2705
2717
a^(3/2)*log((2*sqrt(b*%e^(c*z)+a)-2*sqrt(a))
2706
2718
/(2*sqrt(b*%e^(c*z)+a)+2*sqrt(a)))
2707
2719
/c
2777
2789
2778
2790
/* Problem 448: OK */
2779
2791
integrate(((b*%e^(c*z)+a)^5)^(1/2)*%e^(c*z),z);
2780
2*(b*%e^(c*z)+a)^(7/2)/(7*b*c);
2792
(b*%e^(c*z)+a)^(5/2)*(2*b*%e^(c*z)+2*a)/(7*b*c);
2781
2793
2782
2794
/* Problem 449: OK */
2783
2795
integrate(((b*%e^(c*z)+a)^3)^(1/2)*%e^(c*z),z);
2784
2*(b*%e^(c*z)+a)^(5/2)/(5*b*c);
2796
(b*%e^(c*z)+a)^(3/2)*(2*b*%e^(c*z)+2*a)/(5*b*c);
2785
2797
2786
2798
/* Problem 450: OK */
2787
2799
integrate(((b*%e^(c*z)+a)^3)^((-1)/2)*%e^(c*z),z);
2865
2877
2866
2878
/* Problem 470: NOUN */
2867
2879
integrate(((b*%e^(2*c*z)+a)^5)^(1/2)*%e^(c*z),z);
2868
'integrate(%e^(c*z)*(b*%e^(2*c*z)+a)^(5/2),z);
2880
'integrate(%e^(c*z)*sqrt((b*%e^(2*c*z)+a)^5),z);
2869
2881
2870
2882
/* Problem 471: NOUN */
2871
2883
integrate(((b*%e^(2*c*z)+a)^3)^(1/2)*%e^(c*z),z);
2872
'integrate(%e^(c*z)*(b*%e^(2*c*z)+a)^(3/2),z);
2884
'integrate(%e^(c*z)*sqrt((b*%e^(2*c*z)+a)^3),z);
2873
2885
2874
2886
/* Problem 472: NOUN */
2875
2887
integrate(((b*%e^(2*c*z)+a)^3)^((-1)/2)*%e^(c*z),z);
2929
2941
2930
2942
/* Problem 486: NOUN */
2931
2943
integrate(sqrt((b*%e^(e*z)+a)/(d*%e^(e*z)+c)),z);
2932
'integrate(sqrt(b*%e^(e*z)+a)/sqrt(d*%e^(e*z)+c),z);
2944
'integrate(sqrt((b*%e^(e*z)+a)/(d*%e^(e*z)+c)),z);
2933
2945
2934
2946
/* Problem 487: NOUN */
2935
2947
integrate(sqrt(b*%e^(e*z)+a)/sqrt(d*%e^(e*z)+c),z);
2988
3000
-gamma_incomplete(0,-a*u*z^r)/r;
2989
3001
2990
3002
/* Problem 500: NEW */
2991
integrate(z^n*(%e^(a*z^2))^u,z);
2992
-gamma_incomplete((n+1)/2,-a*u*z^2)*z^(n+1)
2993
/(2*a^((n+1)/2)*(-u)^((n+1)/2)*(z^2)^((n+1)/2));
3003
integrate(z^n*(%e^(a*z^2))^u,z),ratsimp;
3004
-a^(-n/2-1/2)*gamma_incomplete((n+1)/2,-a*u*z^2)*z^(n+1)*(-u*z^2)^(-n/2-1/2)/2;
2994
3005
2995
3006
/* Problem 501: NEW */
2996
integrate(z^(2*n)*(%e^(a*z^2))^u,z);
2997
-gamma_incomplete((2*n+1)/2,-a*u*z^2)*z^(2*n+1)
2998
/(2*a^((2*n+1)/2)*(-u)^((2*n+1)/2)*(z^2)^((2*n+1)/2));
3007
integrate(z^(2*n)*(%e^(a*z^2))^u,z),ratsimp;
3008
-a^(-n-1/2)*(-1)^n*gamma_incomplete((2*n+1)/2,-a*u*z^2)*z/(2*u^n*sqrt(-u*z^2));
2999
3009
3000
3010
/* Problem 502: NEW */
3001
3011
integrate(z^(1+2*n)*(%e^(a*z^2))^u,z);
3015
3025
-gamma_incomplete(0,-a*u*z^r)*%e^(b*u)/r;
3016
3026
3017
3027
/* Problem 506: NEW */
3018
integrate(z^n*(%e^(b+a*z^2))^u,z);
3019
-gamma_incomplete((n+1)/2,-a*u*z^2)*%e^(b*u)*z^(n+1)
3020
/(2*a^((n+1)/2)*(-u)^((n+1)/2)*(z^2)^((n+1)/2));
3028
integrate(z^n*(%e^(b+a*z^2))^u,z),ratsimp;
3029
-a^(-n/2-1/2)*gamma_incomplete((n+1)/2,-a*u*z^2)*%e^(b*u)*z^(n+1)
3030
*(-u*z^2)^(-n/2-1/2)/2;
3021
3031
3022
3032
/* Problem 507: NEW */
3023
integrate(z^(2*n)*(%e^(b+a*z^2))^u,z);
3024
-gamma_incomplete((2*n+1)/2,-a*u*z^2)*%e^(b*u)*z^(2*n+1)
3025
/(2*a^((2*n+1)/2)*(-u)^((2*n+1)/2)*(z^2)^((2*n+1)/2));
3033
integrate(z^(2*n)*(%e^(b+a*z^2))^u,z),ratsimp;
3034
-a^(-n-1/2)*(-1)^n*gamma_incomplete((2*n+1)/2,-a*u*z^2)*%e^(b*u)*z
3035
/(2*u^n*sqrt(-u*z^2));
3026
3036
3027
3037
/* Problem 508: NEW */
3028
3038
integrate(z^(1+2*n)*(%e^(b+a*z^2))^u,z);
3035
3045
3036
3046
/* Problem 510: NEW 5 */
3037
3047
expand(niceindices(integrate(z^n*(%e^(b*z+a*z^2))^u,z))-
3038
-u^(-n-1)*%e^-(b^2*u/(4*a))
3039
*('sum(a^(-n+(i+1)/2-1/2)*b^(n-i)
3040
*gamma_incomplete((i+1)/2,
3041
-(2*a*u*z+b*u)^2/(4*a*u))
3042
*2^(i-n)*binomial(n,i)*(-u)^(n+(i+1)/2-i)
3043
*(2*a*u*z+b*u)^(i+1)
3044
*abs(2*a*u*z+b*u)^(-i-1),i,0,n))
3045
/(2*sqrt(a)));
3048
(-%e^-(b^2*u/(4*a))*('sum(a^(-n+(i+1)/2-1/2)*b^(n-i)
3049
*gamma_incomplete(
3050
(i+1)/2,-(2*a*u*z+b*u)^2/(4*a*u))
3051
*(-1)^(i-n)*2^(i-n)*binomial(n,i)
3052
*u^(-i-1/2)*(2*a*u*z+b*u)^(i+1)
3053
/(-(2*a*u*z+b*u)^2/u)^((i+1)/2),i,0,n))
3054
/(2*sqrt(a)*sqrt(u))));
3046
3055
0;
3047
3056
3048
3057
/* Problem 511: NEW 6 */
3049
3058
expand(niceindices(integrate(x^n*(%e^(b*x+a*sqrt(x)))^u,x))-
3050
2^(-n-1)*%e^-(a^2*u/(4*b))
3059
(2^(-n-1)*%e^-(a^2*u/(4*b))
3051
3060
*('sum(4^j*binomial(n,j)
3052
*'sum(a^(2*n-j-i)*b^((j+i+1)/2)*(-1)^(i-j)*binomial(j,i)
3053
*(-u)^((j+i+1)/2)*u^(2*n-j-i)
3061
*'sum(a^(2*n-j-i)*b^((j+i+1)/2)*(-1)^(-(j+i+1)/2+j-i)
3062
*binomial(j,i)*u^(2*n+(j+i+1)/2-j-i)
3054
3063
*abs(2*b*u*sqrt(x)+a*u)^(-j-i-1)
3055
3064
*(2*b*u*sqrt(x)+a*u)^(j+i)
3056
3065
*(a*gamma_incomplete((j+i+1)/2,
3057
3066
-(2*b*u*sqrt(x)+a*u)^2
3058
3067
/(4*b*u))*u
3059
3068
*(2*b*u*sqrt(x)+a*u)
3060
+2*sqrt(b)
3069
+2*%i*sqrt(b)
3061
3070
*gamma_incomplete((j+i+2)/2,
3062
3071
-(2*b*u*sqrt(x)+a*u)^2
3063
/(4*b*u))*u
3064
*abs(2*b*u*sqrt(x)+a*u)
3065
/sqrt(-u)),i,0,j),j,0,n))
3066
/(b^(2*(n+1))*u^(2*(n+1))));
3072
/(4*b*u))*sqrt(u)
3073
*abs(2*b*u*sqrt(x)+a*u)),i,0,j),j,0,n))
3074
/(b^(2*(n+1))*u^(2*(n+1)))));
3067
3075
0;
3068
3076
3069
3077
/* Problem 512: NEW 5 */
3070
3078
expand(niceindices(integrate(z^n*(%e^(c+b*z+a*z^2))^u,z))-
3071
-u^(-n-1)*%e^(c*u-b^2*u/(4*a))
3072
*('sum(a^(-n+(i+1)/2-1/2)*b^(n-i)
3073
*gamma_incomplete((i+1)/2,
3074
-(2*a*u*z+b*u)^2/(4*a*u))
3075
*2^(i-n)*binomial(n,i)*(-u)^(n+(i+1)/2-i)
3076
*(2*a*u*z+b*u)^(i+1)
3077
*abs(2*a*u*z+b*u)^(-i-1),i,0,n))
3078
/(2*sqrt(a)));
3079
(-%e^(c*u-b^2*u/(4*a))*('sum(a^(-n+(i+1)/2-1/2)*b^(n-i)
3080
*gamma_incomplete(
3081
(i+1)/2,
3082
-(2*a*u*z+b*u)^2/(4*a*u))
3083
*(-1)^(i-n)*2^(i-n)
3084
*binomial(n,i)*u^(-i-1/2)
3085
*(2*a*u*z+b*u)^(i+1)
3086
/(-(2*a*u*z+b*u)^2/u)^((i+1)/2),i,0,n))
3087
/(2*sqrt(a)*sqrt(u))));
3079
3088
0;
3080
3089
3081
3090
/* Problem 513: NEW 6 */
3082
3091
expand(niceindices(integrate(x^n*(%e^(c+b*x+a*sqrt(x)))^u,x))-
3083
2^(-n-1)*%e^(c*u-a^2*u/(4*b))
3092
(2^(-n-1)*%e^(c*u-a^2*u/(4*b))
3084
3093
*('sum(4^j*binomial(n,j)
3085
*'sum(a^(2*n-j-i)*b^((j+i+1)/2)*(-1)^(i-j)*binomial(j,i)
3086
*(-u)^((j+i+1)/2)*u^(2*n-j-i)
3094
*'sum(a^(2*n-j-i)*b^((j+i+1)/2)*(-1)^(-(j+i+1)/2+j-i)
3095
*binomial(j,i)*u^(2*n+(j+i+1)/2-j-i)
3087
3096
*abs(2*b*u*sqrt(x)+a*u)^(-j-i-1)
3088
3097
*(2*b*u*sqrt(x)+a*u)^(j+i)
3089
3098
*(a*gamma_incomplete((j+i+1)/2,
3090
3099
-(2*b*u*sqrt(x)+a*u)^2
3091
3100
/(4*b*u))*u
3092
3101
*(2*b*u*sqrt(x)+a*u)
3093
+2*sqrt(b)
3102
+2*%i*sqrt(b)
3094
3103
*gamma_incomplete((j+i+2)/2,
3095
3104
-(2*b*u*sqrt(x)+a*u)^2
3096
/(4*b*u))*u
3097
*abs(2*b*u*sqrt(x)+a*u)
3098
/sqrt(-u)),i,0,j),j,0,n))
3099
/(b^(2*(n+1))*u^(2*(n+1))));
3105
/(4*b*u))*sqrt(u)
3106
*abs(2*b*u*sqrt(x)+a*u)),i,0,j),j,0,n))
3107
/(b^(2*(n+1))*u^(2*(n+1)))));
3100
3108
0;
3101
3109
3102
3110
/* Problem 514: NEW 7 */
3151
3159
*(-(c*log(h)+log(a)*d)*z)^(-n-3/2));
3152
3160
0;
3153
3161
3154
/* Problem 523: NEW 10 */
3162
/* Problem 523: NEW 9 */
3155
3163
expand(niceindices(integrate(z^n*a^(b*z)*h^(c*z^2),z))-
3156
-c^(-n-1)*%e^-(log(a)^2*b^2/(4*c*log(h)))*log(h)^(-n-1)
3157
*('sum(log(a)^(n-i)*b^(n-i)*(-c*log(h))^((i+1)/2)
3164
(-c^(-n-1)*%e^-(log(a)^2*b^2/(4*c*log(h)))*log(h)^(-n-1)
3165
*('sum(log(a)^(n-i)*b^(n-i)
3158
3166
*gamma_incomplete((i+1)/2,
3159
3167
-(2*c*log(h)*z+log(a)*b)^2
3160
3168
/(4*c*log(h)))*(-1)^(i-n)
3161
3169
*2^(i-n)*binomial(n,i)
3162
3170
*(2*c*log(h)*z+log(a)*b)^(i+1)
3163
*abs(2*c*log(h)*z+log(a)*b)^(-i-1),i,0,n))
3164
/2);
3171
/(-(2*c*log(h)*z+log(a)*b)^2/(c*log(h)))^((i+1)/2),i,0,n))
3172
/2));
3165
3173
0;
3166
3174
3167
/* Problem 524: NEW 11 */
3175
/* Problem 524: NEW 10 */
3168
3176
expand(niceindices(integrate(x^n*a^(b*x)*h^(c*sqrt(x)),x))-
3169
%e^-(c^2*log(h)^2/(4*log(a)*b))*2^(-2*n-1)
3177
(%e^-(c^2*log(h)^2/(4*log(a)*b))*2^(-2*n-1)
3170
3178
*('sum(4^j*binomial(n,j)
3171
3179
*'sum(
3172
(-log(a))^((j+i+1)/2)
3173
*b^((j+i+1)/2)*c^(2*n-j-i)
3174
*log(h)^(2*n-j-i)*(-1)^(i-j)
3175
*binomial(j,i)
3176
*abs(2*log(a)*b*sqrt(x)+c*log(h))
3177
^(-j-i-1)
3178
*(2*log(a)*b*sqrt(x)+c*log(h))
3179
^(j+i)
3180
*(c*log(h)
3181
*gamma_incomplete(
3182
(j+i+1)/2,
3183
-(2*log(a)*b*sqrt(x)+c*log(h))
3184
^2
3185
/(4*log(a)*b))
3186
*(2*log(a)*b*sqrt(x)+c*log(h))
3187
+2*log(a)*sqrt(b)
3188
*gamma_incomplete(
3189
(j+i+2)/2,
3190
-(2*log(a)*b*sqrt(x)+c*log(h))
3191
^2
3192
/(4*log(a)*b))
3193
*abs(
3194
2*log(a)*b*sqrt(x)+c*log(h))
3195
/sqrt(-log(a))),i,0,j),j,0,n))
3196
/(log(a)^(2*(n+1))*b^(2*(n+1))));
3180
log(a)^((j+i+1)/2)*b^((j+i+1)/2)
3181
*c^(2*n-j-i)
3182
*log(h)^(2*n-j-i)
3183
*(-1)
3184
^(-(j+i+1)/2+j-i)
3185
*binomial(j,i)
3186
*abs(
3187
2*log(a)*b
3188
*sqrt(x)
3189
+c*log(h))
3190
^(-j-i-1)
3191
*(2*log(a)*b
3192
*sqrt(x)
3193
+c*log(h))
3194
^(j+i)
3195
*(c*log(h)
3196
*gamma_incomplete(
3197
(j+i+1)/2,
3198
-(2*log(a)*b
3199
*sqrt(x)
3200
+c*log(h))
3201
^2
3202
/(4*log(a)
3203
*b))
3204
*(2*log(a)*b
3205
*sqrt(x)
3206
+c*log(h))
3207
+2*%i
3208
*sqrt(log(a))
3209
*sqrt(b)
3210
*gamma_incomplete(
3211
(j+i+2)/2,
3212
-(2*log(a)*b
3213
*sqrt(x)
3214
+c*log(h))
3215
^2
3216
/(4*log(a)
3217
*b))
3218
*abs(
3219
2*log(a)*b
3220
*sqrt(x)
3221
+c*log(h))),
3222
i,0,j),j,0,n))
3223
/(log(a)^(2*(n+1))*b^(2*(n+1)))));
3197
3224
0;
3198
3225
3199
/* Problem 525: NEW 10 */
3226
/* Problem 525: NEW 9 */
3200
3227
expand(niceindices(integrate(z^n*a^(e+d*z)*h^(c*z^2),z))-
3201
-a^e*c^(-n-1)*%e^-(log(a)^2*d^2/(4*c*log(h)))*log(h)^(-n-1)
3202
*('sum(log(a)^(n-i)*d^(n-i)*(-c*log(h))^((i+1)/2)
3228
(-a^e*c^(-n-1)*%e^-(log(a)^2*d^2/(4*c*log(h)))*log(h)^(-n-1)
3229
*('sum(log(a)^(n-i)*d^(n-i)
3203
3230
*gamma_incomplete((i+1)/2,
3204
3231
-(2*c*log(h)*z+log(a)*d)^2
3205
3232
/(4*c*log(h)))*(-1)^(i-n)*2^(i-n)
3206
3233
*binomial(n,i)*(2*c*log(h)*z+log(a)*d)^(i+1)
3207
*abs(2*c*log(h)*z+log(a)*d)^(-i-1),i,0,n))
3208
/2);
3234
/(-(2*c*log(h)*z+log(a)*d)^2/(c*log(h)))^((i+1)/2),i,0,n))
3235
/2));
3209
3236
0;
3210
3237
3211
/* Problem 526: NEW 11 */
3238
/* Problem 526: NEW 10 */
3212
3239
expand(niceindices(integrate(x^n*a^(e+d*x)*h^(c*sqrt(x)),x))-
3213
a^e*%e^-(c^2*log(h)^2/(4*log(a)*d))*2^(-2*n-1)
3240
(a^e*%e^-(c^2*log(h)^2/(4*log(a)*d))*2^(-2*n-1)
3214
3241
*('sum(4^j*binomial(n,j)
3215
*'sum(c^(2*n-j-i)*(-log(a)*d)^((j+i+1)/2)*log(h)^(2*n-j-i)
3216
*(-1)^(i-j)*binomial(j,i)
3217
*abs(2*log(a)*d*sqrt(x)+c*log(h))^(-j-i-1)
3218
*(2*log(a)*d*sqrt(x)+c*log(h))^(j+i)
3219
*(c*log(h)
3220
*gamma_incomplete((j+i+1)/2,
3221
-(2*log(a)*d*sqrt(x)
3222
+c*log(h))
3223
^2
3224
/(4*log(a)*d))
3225
*(2*log(a)*d*sqrt(x)+c*log(h))
3226
+2*log(a)*d
3227
*gamma_incomplete((j+i+2)/2,
3228
-(2*log(a)*d*sqrt(x)
3229
+c*log(h))
3230
^2
3231
/(4*log(a)*d))
3232
*abs(2*log(a)*d*sqrt(x)+c*log(h))
3233
/sqrt(-log(a)*d)),i,0,j),j,0,n))
3234
/(log(a)^(2*(n+1))*d^(2*(n+1))));
3242
*'sum(log(a)^((j+i+1)/2)*c^(2*n-j-i)*d^((j+i+1)/2)
3243
*log(h)^(2*n-j-i)*(-1)^(-(j+i+1)/2+j-i)
3244
*binomial(j,i)
3245
*abs(2*log(a)*d*sqrt(x)+c*log(h))
3246
^(-j-i-1)
3247
*(2*log(a)*d*sqrt(x)+c*log(h))^(j+i)
3248
*(c*log(h)
3249
*gamma_incomplete((j+i+1)/2,
3250
-(2*log(a)*d*sqrt(x)
3251
+c*log(h))
3252
^2
3253
/(4*log(a)*d))
3254
*(2*log(a)*d*sqrt(x)+c*log(h))
3255
+2*%i*sqrt(log(a))*sqrt(d)
3256
*gamma_incomplete((j+i+2)/2,
3257
-(2*log(a)*d*sqrt(x)
3258
+c*log(h))
3259
^2
3260
/(4*log(a)*d))
3261
*abs(2*log(a)*d*sqrt(x)+c*log(h))),i,
3262
0,j),j,0,n))
3263
/(log(a)^(2*(n+1))*d^(2*(n+1)))));
3235
3264
0;
3236
3265
3237
3266
/* Problem 527: NEW 7 */
3242
3271
3243
3272
/* Problem 528: NEW 7 */
3244
3273
expand(integrate(z^n*a^(b*z^2)*h^(c*z^2),z)-
3245
-(-c*log(h)-log(a)*b)^((-n-1)/2)*gamma_incomplete((n+1)/2,
3246
-(c*log(h)+log(a)*b)*z^2)
3247
*z^(n+1)*(z^2)^((-n-1)/2)
3248
/2);
3274
(-gamma_incomplete((n+1)/2,-(c*log(h)+log(a)*b)*z^2)
3275
*z^(n+1)*(-(c*log(h)+log(a)*b)*z^2)^((-n-1)/2)
3276
/2));
3249
3277
0;
3250
3278
3251
3279
/* Problem 529: NEW 7 */
3252
3280
expand(integrate(z^(2*n)*a^(b*z^2)*h^(c*z^2),z)-
3253
-(-c*log(h)-log(a)*b)^((-2*n-1)/2)*gamma_incomplete((2*n+1)/2,
3254
-(c*log(h)+log(a)*b)*z^2)
3255
*z^(2*n+1)*(z^2)^((-2*n-1)/2)
3256
/2);
3281
(-gamma_incomplete((2*n+1)/2,-(c*log(h)+log(a)*b)*z^2)
3282
*z^(2*n+1)*(-(c*log(h)+log(a)*b)*z^2)^((-2*n-1)/2)
3283
/2));
3257
3284
0;
3258
3285
3259
3286
/* Problem 530: NEW 7 */
3270
3297
*gamma_incomplete(2*(n+1),-(c*log(h)+log(a)*b)*sqrt(x)));
3271
3298
0;
3272
3299
3273
/* Problem 532: NEW 10 */
3300
/* Problem 532: NEW 9 */
3274
3301
expand(niceindices(integrate(z^n*a^(d*z)*h^(g+c*z^2),z))-
3275
-c^(-n-1)*h^g*%e^-(log(a)^2*d^2/(4*c*log(h)))*log(h)^(-n-1)
3276
*('sum(log(a)^(n-i)*d^(n-i)*(-c*log(h))^((i+1)/2)
3302
(-c^(-n-1)*h^g*%e^-(log(a)^2*d^2/(4*c*log(h)))*log(h)^(-n-1)
3303
*('sum(log(a)^(n-i)*d^(n-i)
3277
3304
*gamma_incomplete((i+1)/2,
3278
3305
-(2*c*log(h)*z+log(a)*d)^2
3279
3306
/(4*c*log(h)))*(-1)^(i-n)
3280
3307
*2^(i-n)*binomial(n,i)
3281
3308
*(2*c*log(h)*z+log(a)*d)^(i+1)
3282
*abs(2*c*log(h)*z+log(a)*d)^(-i-1),i,0,n))
3283
/2);
3309
/(-(2*c*log(h)*z+log(a)*d)^2/(c*log(h)))^((i+1)/2),i,0,n))
3310
/2));
3284
3311
0;
3285
3312
3286
/* Problem 533: NEW 11 */
3313
/* Problem 533: NEW 10 */
3287
3314
expand(niceindices(integrate(x^n*a^(d*x)*h^(g+c*sqrt(x)),x))-
3288
h^g*%e^-(c^2*log(h)^2/(4*log(a)*d))*2^(-2*n-1)
3315
(h^g*%e^-(c^2*log(h)^2/(4*log(a)*d))*2^(-2*n-1)
3289
3316
*('sum(4^j*binomial(n,j)
3290
*'sum(c^(2*n-j-i)*(-log(a)*d)^((j+i+1)/2)*log(h)^(2*n-j-i)
3291
*(-1)^(i-j)*binomial(j,i)
3292
*abs(2*log(a)*d*sqrt(x)+c*log(h))^(-j-i-1)
3293
*(2*log(a)*d*sqrt(x)+c*log(h))^(j+i)
3294
*(c*log(h)
3295
*gamma_incomplete((j+i+1)/2,
3296
-(2*log(a)*d*sqrt(x)
3297
+c*log(h))
3298
^2
3299
/(4*log(a)*d))
3300
*(2*log(a)*d*sqrt(x)+c*log(h))
3301
+2*log(a)*d
3302
*gamma_incomplete((j+i+2)/2,
3303
-(2*log(a)*d*sqrt(x)
3304
+c*log(h))
3305
^2
3306
/(4*log(a)*d))
3307
*abs(2*log(a)*d*sqrt(x)+c*log(h))
3308
/sqrt(-log(a)*d)),i,0,j),j,0,n))
3309
/(log(a)^(2*(n+1))*d^(2*(n+1))));
3317
*'sum(log(a)^((j+i+1)/2)*c^(2*n-j-i)*d^((j+i+1)/2)
3318
*log(h)^(2*n-j-i)*(-1)^(-(j+i+1)/2+j-i)
3319
*binomial(j,i)
3320
*abs(2*log(a)*d*sqrt(x)+c*log(h))
3321
^(-j-i-1)
3322
*(2*log(a)*d*sqrt(x)+c*log(h))^(j+i)
3323
*(c*log(h)
3324
*gamma_incomplete((j+i+1)/2,
3325
-(2*log(a)*d*sqrt(x)
3326
+c*log(h))
3327
^2
3328
/(4*log(a)*d))
3329
*(2*log(a)*d*sqrt(x)+c*log(h))
3330
+2*%i*sqrt(log(a))*sqrt(d)
3331
*gamma_incomplete((j+i+2)/2,
3332
-(2*log(a)*d*sqrt(x)
3333
+c*log(h))
3334
^2
3335
/(4*log(a)*d))
3336
*abs(2*log(a)*d*sqrt(x)+c*log(h))),i,
3337
0,j),j,0,n))
3338
/(log(a)^(2*(n+1))*d^(2*(n+1)))));
3310
3339
0;
3311
3340
3312
/* Problem 534: NEW 10 */
3341
/* Problem 534: NEW 9 */
3313
3342
expand(niceindices(integrate(z^n*a^(e+d*z)*h^(g+c*z^2),z))-
3314
-a^e*c^(-n-1)*h^g*%e^-(log(a)^2*d^2/(4*c*log(h)))*log(h)^(-n-1)
3315
*('sum(log(a)^(n-i)*d^(n-i)*(-c*log(h))^((i+1)/2)
3343
(-a^e*c^(-n-1)*h^g*%e^-(log(a)^2*d^2/(4*c*log(h)))*log(h)^(-n-1)
3344
*('sum(log(a)^(n-i)*d^(n-i)
3316
3345
*gamma_incomplete((i+1)/2,
3317
3346
-(2*c*log(h)*z+log(a)*d)^2
3318
3347
/(4*c*log(h)))*(-1)^(i-n)*2^(i-n)
3319
3348
*binomial(n,i)*(2*c*log(h)*z+log(a)*d)^(i+1)
3320
*abs(2*c*log(h)*z+log(a)*d)^(-i-1),i,0,n))
3321
/2);
3349
/(-(2*c*log(h)*z+log(a)*d)^2/(c*log(h)))^((i+1)/2),i,0,n))
3350
/2));
3322
3351
0;
3323
3352
3324
/* Problem 535: NEW 11 */
3353
/* Problem 535: NEW 10 */
3325
3354
expand(niceindices(integrate(x^n*a^(e+d*x)*h^(g+c*sqrt(x)),x))-
3326
a^e*h^g*%e^-(c^2*log(h)^2/(4*log(a)*d))*2^(-2*n-1)
3355
(a^e*h^g*%e^-(c^2*log(h)^2/(4*log(a)*d))*2^(-2*n-1)
3327
3356
*('sum(4^j*binomial(n,j)
3328
*'sum(c^(2*n-j-i)*(-log(a)*d)^((j+i+1)/2)*log(h)^(2*n-j-i)
3329
*(-1)^(i-j)*binomial(j,i)
3330
*abs(2*log(a)*d*sqrt(x)+c*log(h))^(-j-i-1)
3331
*(2*log(a)*d*sqrt(x)+c*log(h))^(j+i)
3332
*(c*log(h)
3333
*gamma_incomplete((j+i+1)/2,
3334
-(2*log(a)*d*sqrt(x)
3335
+c*log(h))
3336
^2
3337
/(4*log(a)*d))
3338
*(2*log(a)*d*sqrt(x)+c*log(h))
3339
+2*log(a)*d
3340
*gamma_incomplete((j+i+2)/2,
3341
-(2*log(a)*d*sqrt(x)
3342
+c*log(h))
3343
^2
3344
/(4*log(a)*d))
3345
*abs(2*log(a)*d*sqrt(x)+c*log(h))
3346
/sqrt(-log(a)*d)),i,0,j),j,0,n))
3347
/(log(a)^(2*(n+1))*d^(2*(n+1))));
3357
*'sum(log(a)^((j+i+1)/2)*c^(2*n-j-i)*d^((j+i+1)/2)
3358
*log(h)^(2*n-j-i)*(-1)^(-(j+i+1)/2+j-i)
3359
*binomial(j,i)
3360
*abs(2*log(a)*d*sqrt(x)+c*log(h))
3361
^(-j-i-1)
3362
*(2*log(a)*d*sqrt(x)+c*log(h))^(j+i)
3363
*(c*log(h)
3364
*gamma_incomplete((j+i+1)/2,
3365
-(2*log(a)*d*sqrt(x)
3366
+c*log(h))
3367
^2
3368
/(4*log(a)*d))
3369
*(2*log(a)*d*sqrt(x)+c*log(h))
3370
+2*%i*sqrt(log(a))*sqrt(d)
3371
*gamma_incomplete((j+i+2)/2,
3372
-(2*log(a)*d*sqrt(x)
3373
+c*log(h))
3374
^2
3375
/(4*log(a)*d))
3376
*abs(2*log(a)*d*sqrt(x)+c*log(h))),i,
3377
0,j),j,0,n))
3378
/(log(a)^(2*(n+1))*d^(2*(n+1)))));
3348
3379
0;
3349
3380
3350
3381
/* Problem 536: NEW 7 */
3355
3386
3356
3387
/* Problem 537: NEW 7 */
3357
3388
expand(integrate(z^n*a^(b*z^2)*h^(g+c*z^2),z)-
3358
-h^g*(-c*log(h)-log(a)*b)^((-n-1)/2)
3359
*gamma_incomplete((n+1)/2,-(c*log(h)+log(a)*b)*z^2)*z^(n+1)
3360
*(z^2)^((-n-1)/2)
3361
/2);
3389
(-h^g*gamma_incomplete((n+1)/2,-(c*log(h)+log(a)*b)*z^2)*z^(n+1)
3390
*(-(c*log(h)+log(a)*b)*z^2)^((-n-1)/2)
3391
/2));
3362
3392
0;
3363
3393
3364
3394
/* Problem 538: NEW 7 */
3365
3395
expand(integrate(z^(2*n)*a^(b*z^2)*h^(g+c*z^2),z)-
3366
-h^g*(-c*log(h)-log(a)*b)^((-2*n-1)/2)
3367
*gamma_incomplete((2*n+1)/2,-(c*log(h)+log(a)*b)*z^2)*z^(2*n+1)
3368
*(z^2)^((-2*n-1)/2)
3369
/2);
3396
(-h^g*gamma_incomplete((2*n+1)/2,-(c*log(h)+log(a)*b)*z^2)*z^(2*n+1)
3397
*(-(c*log(h)+log(a)*b)*z^2)^((-2*n-1)/2)
3398
/2));
3370
3399
0;
3371
3400
3372
3401
/* Problem 539: NEW 7 */
3390
3419
3391
3420
/* Problem 542: NEW 7 */
3392
3421
expand(integrate(z^n*a^(e+b*z^2)*h^(g+c*z^2),z)-
3393
-a^e*h^g*(-c*log(h)-log(a)*b)^((-n-1)/2)
3394
*gamma_incomplete((n+1)/2,-(c*log(h)+log(a)*b)*z^2)*z^(n+1)
3395
*(z^2)^((-n-1)/2)
3396
/2);
3422
(-a^e*h^g*gamma_incomplete((n+1)/2,-(c*log(h)+log(a)*b)*z^2)*z^(n+1)
3423
*(-(c*log(h)+log(a)*b)*z^2)^((-n-1)/2)
3424
/2));
3397
3425
0;
3398
3426
3399
3427
/* Problem 543: NEW 7 */
3400
3428
expand(integrate(z^(2*n)*a^(e+b*z^2)*h^(g+c*z^2),z)-
3401
-a^e*h^g*(-c*log(h)-log(a)*b)^((-2*n-1)/2)
3402
*gamma_incomplete((2*n+1)/2,-(c*log(h)+log(a)*b)*z^2)*z^(2*n+1)
3403
*(z^2)^((-2*n-1)/2)
3404
/2);
3429
(-a^e*h^g*gamma_incomplete((2*n+1)/2,-(c*log(h)+log(a)*b)*z^2)*z^(2*n+1)
3430
*(-(c*log(h)+log(a)*b)*z^2)^((-2*n-1)/2)
3431
/2));
3405
3432
0;
3406
3433
3407
3434
/* Problem 544: NEW 7 */
3417
3444
*gamma_incomplete(2*(2*n+2),-(c*log(h)+log(a)*b)*sqrt(z)));
3418
3445
0;
3419
3446
3420
/* Problem 546: NEW 10 */
3447
/* Problem 546: NEW 9 */
3421
3448
expand(niceindices(integrate(z^n*a^(d*z)*h^(f*z+c*z^2),z))-
3422
-c^(-n-1)*log(h)^(-n-1)*%e^-((f*log(h)+log(a)*d)^2/(4*c*log(h)))
3423
*('sum((-c*log(h))^((i+1)/2)*(-f*log(h)-log(a)*d)^(n-i)
3424
*gamma_incomplete((i+1)/2,
3425
-(log(h)*(2*c*z+f)
3426
+log(a)*d)
3427
^2
3428
/(4*c*log(h)))*2^(i-n)
3429
*binomial(n,i)
3430
*(log(h)*(2*c*z+f)+log(a)*d)^(i+1)
3431
*abs(log(h)*(2*c*z+f)+log(a)*d)^(-i-1),i,
3432
0,n))
3433
/2);
3449
(-c^(-n-1)*%e^-((f*log(h)+log(a)*d)^2/(4*c*log(h)))*log(h)^(-n-1)
3450
*('sum((-f*log(h)-log(a)*d)^(n-i)
3451
*gamma_incomplete((i+1)/2,
3452
-(log(h)*(2*c*z+f)+log(a)*d)^2
3453
/(4*c*log(h)))*2^(i-n)*binomial(n,i)
3454
*(log(h)*(2*c*z+f)+log(a)*d)^(i+1)
3455
/(-(log(h)*(2*c*z+f)+log(a)*d)^2/(c*log(h)))^((i+1)/2),i,0,
3456
n))
3457
/2));
3434
3458
0;
3435
3459
3436
/* Problem 547: NEW 11 */
3437
expand(niceindices(integrate(x^n*a^(d*x)*h^(f*x+c*sqrt(x)),x))-
3438
%e^-(c^2*log(h)^2/(4*(f*log(h)+log(a)*d)))
3460
/* Problem 547: NEW 10 */
3461
expand(niceindices(integrate(x^n*a^(d*x)*h^(f*x+c*sqrt(x)),x)) -
3462
(%e^-(c^2*log(h)^2/(4*(f*log(h)+log(a)*d)))
3439
3463
*2^(-2*n-1)
3440
3464
*('sum(4^j*binomial(n,j)
3441
*'sum(c^(2*n-j-i)*log(h)^(2*n-j-i)*(-f*log(h)-log(a)*d)^((j+i+1)/2)
3442
*(-1)^(i-j)*binomial(j,i)
3465
*'sum(c^(2*n-j-i)*log(h)^(2*n-j-i)*(f*log(h)+log(a)*d)^((j+i+1)/2)
3466
*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
3443
3467
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h))
3444
3468
^(-j-i-1)
3445
3469
*(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h))^(j+i)
3451
3475
^2
3452
3476
/(4*(f*log(h)+log(a)*d)))
3453
3477
*(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h))
3454
+2*(f*log(h)+log(a)*d)
3478
+2*%i*sqrt(f*log(h)+log(a)*d)
3455
3479
*gamma_incomplete((j+i+2)/2,
3456
3480
-(2*(f*log(h)+log(a)*d)
3457
3481
*sqrt(x)
3458
3482
+c*log(h))
3459
3483
^2
3460
3484
/(4*(f*log(h)+log(a)*d)))
3461
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h))
3462
/sqrt(-f*log(h)-log(a)*d)),i,0,j),j,0,n))
3463
/(f*log(h)+log(a)*d)^(2*(n+1)));
3485
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h))),
3486
i,0,j),j,0,n))
3487
/(f*log(h)+log(a)*d)^(2*(n+1))));
3464
3488
0;
3465
3489
3466
/* Problem 548: NEW 10 */
3490
/* Problem 548: NEW 9 */
3467
3491
expand(niceindices(integrate(z^n*a^(e+d*z)*h^(f*z+c*z^2),z))-
3468
-a^e*c^(-n-1)*log(h)^(-n-1)*%e^-((f*log(h)+log(a)*d)^2/(4*c*log(h)))
3469
*('sum((-c*log(h))^((i+1)/2)*(-f*log(h)-log(a)*d)^(n-i)
3470
*gamma_incomplete((i+1)/2,
3471
-(log(h)*(2*c*z+f)+log(a)*d)
3472
^2
3473
/(4*c*log(h)))*2^(i-n)
3474
*binomial(n,i)
3475
*(log(h)*(2*c*z+f)+log(a)*d)^(i+1)
3476
*abs(log(h)*(2*c*z+f)+log(a)*d)^(-i-1),i,0,n))
3477
/2);
3492
(-a^e*c^(-n-1)*%e^-((f*log(h)+log(a)*d)^2/(4*c*log(h)))*log(h)^(-n-1)
3493
*('sum((-f*log(h)-log(a)*d)^(n-i)*gamma_incomplete((i+1)/2,
3494
-(log(h)*(2*c*z+f)
3495
+log(a)*d)
3496
^2
3497
/(4*c*log(h)))*2^(i-n)
3498
*binomial(n,i)
3499
*(log(h)*(2*c*z+f)+log(a)*d)^(i+1)
3500
/(-(log(h)*(2*c*z+f)+log(a)*d)^2/(c*log(h)))^((i+1)/2),i,0,n))
3501
/2));
3478
3502
0;
3479
3503
3480
/* Problem 549: NEW 11 */
3504
/* Problem 549: NEW 10 */
3481
3505
expand(niceindices(integrate(x^n*a^(e+d*x)*h^(f*x+c*sqrt(x)),x))-
3482
a^e*%e^-(c^2*log(h)^2/(4*(f*log(h)+log(a)*d)))*2^(-2*n-1)
3506
(a^e*%e^-(c^2*log(h)^2/(4*(f*log(h)+log(a)*d)))*2^(-2*n-1)
3483
3507
*('sum(4^j*binomial(n,j)
3484
3508
*'sum(c^(2*n-j-i)*log(h)^(2*n-j-i)
3485
*(-f*log(h)-log(a)*d)^((j+i+1)/2)*(-1)^(i-j)
3486
*binomial(j,i)
3509
*(f*log(h)+log(a)*d)^((j+i+1)/2)
3510
*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
3487
3511
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h))
3488
3512
^(-j-i-1)
3489
3513
*(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h))^(j+i)
3495
3519
^2
3496
3520
/(4*(f*log(h)+log(a)*d)))
3497
3521
*(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h))
3498
+2*(f*log(h)+log(a)*d)
3522
+2*%i*sqrt(f*log(h)+log(a)*d)
3499
3523
*gamma_incomplete((j+i+2)/2,
3500
3524
-(2*(f*log(h)+log(a)*d)
3501
3525
*sqrt(x)
3502
3526
+c*log(h))
3503
3527
^2
3504
3528
/(4*(f*log(h)+log(a)*d)))
3505
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h))
3506
/sqrt(-f*log(h)-log(a)*d)),i,0,j),j,0,n))
3507
/(f*log(h)+log(a)*d)^(2*(n+1)));
3529
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)
3530
+c*log(h))),i,0,j),j,0,n))
3531
/(f*log(h)+log(a)*d)^(2*(n+1))));
3508
3532
0;
3509
3533
3510
/* Problem 550: NEW 10 */
3534
/* Problem 550: NEW 9 */
3511
3535
expand(niceindices(integrate(z^n*a^(b*z^2)*h^(f*z+c*z^2),z))-
3512
-(c*log(h)+log(a)*b)^(-n-1)*%e^-(f^2*log(h)^2/(4*(c*log(h)+log(a)*b)))
3513
*('sum(f^(n-i)*log(h)^(n-i)
3514
*(-c*log(h)-log(a)*b)^((i+1)/2)
3515
*gamma_incomplete((i+1)/2,
3516
-(log(h)*(2*c*z+f)
3517
+2*log(a)*b*z)
3518
^2
3519
/(4
3520
*(c*log(h)
3521
+log(a)*b)))
3522
*(-1)^(i-n)*2^(i-n)*binomial(n,i)
3523
*(log(h)*(2*c*z+f)+2*log(a)*b*z)
3524
^(i+1)
3525
*abs(log(h)*(2*c*z+f)+2*log(a)*b*z)
3526
^(-i-1),i,0,n))
3527
/2);
3536
(-%e^-(f^2*log(h)^2/(4*(c*log(h)+log(a)*b)))
3537
*(c*log(h)+log(a)*b)^(-n-1)
3538
*('sum(f^(n-i)*log(h)^(n-i)
3539
*gamma_incomplete((i+1)/2,
3540
-(log(h)*(2*c*z+f)+2*log(a)*b*z)^2
3541
/(4*(c*log(h)+log(a)*b)))*(-1)^(i-n)*2^(i-n)
3542
*binomial(n,i)*(log(h)*(2*c*z+f)+2*log(a)*b*z)^(i+1)
3543
/(-(log(h)*(2*c*z+f)+2*log(a)*b*z)^2/(c*log(h)+log(a)*b))^((i+1)/2),
3544
i,0,n))
3545
/2));
3528
3546
0;
3529
3547
3530
/* Problem 551: NEW 11 */
3548
/* Problem 551: NEW 10 */
3531
3549
expand(niceindices(integrate(x^n*a^(b*sqrt(x))*h^(f*x+c*sqrt(x)),x))-
3532
%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))
3550
(%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))
3533
3551
*2^(-2*n-1)
3534
3552
*('sum(4^j*binomial(n,j)
3535
*'sum((-f*log(h))^((j+i+1)/2)*(c*log(h)+log(a)*b)^(2*n-j-i)
3536
*(-1)^(i-j)*binomial(j,i)
3537
*abs(
3538
2*f*log(h)*sqrt(x)+c*log(h)+log(a)*b)
3539
^(-j-i-1)
3553
*'sum((f*log(h))^((j+i+1)/2)*(c*log(h)+log(a)*b)^(2*n-j-i)
3554
*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
3555
*abs(
3556
2*f*log(h)*sqrt(x)+c*log(h)+log(a)*b)
3557
^(-j-i-1)
3558
*(2*f*log(h)*sqrt(x)+c*log(h)+log(a)*b)
3559
^(j+i)
3560
*((c*log(h)+log(a)*b)
3561
*gamma_incomplete((j+i+1)/2,
3562
-(2*f*log(h)*sqrt(x)
3563
+c*log(h)+log(a)*b)
3564
^2
3565
/(4*f*log(h)))
3540
3566
*(2*f*log(h)*sqrt(x)
3541
3567
+c*log(h)+log(a)*b)
3542
^(j+i)
3543
*((c*log(h)+log(a)*b)
3544
*gamma_incomplete((j+i+1)/2,
3545
-(2*f*log(h)
3546
*sqrt(x)
3547
+c*log(h)
3548
+log(a)*b)
3549
^2
3550
/(4*f*log(h)))
3551
*(2*f*log(h)*sqrt(x)
3552
+c*log(h)+log(a)*b)
3553
+2*f*log(h)
3554
*gamma_incomplete(
3555
(j+i+2)/2,
3556
-(2*f*log(h)*sqrt(x)
3557
+c*log(h)+log(a)*b)
3558
^2
3559
/(4*f*log(h)))
3560
*abs(
3561
2*f*log(h)*sqrt(x)
3562
+c*log(h)+log(a)*b)
3563
/sqrt(-f*log(h))),i,0,j),j,0,n))
3564
/(f^(2*(n+1))*log(h)^(2*(n+1))));
3568
+2*%i*f*log(h)
3569
*gamma_incomplete((j+i+2)/2,
3570
-(2*f*log(h)
3571
*sqrt(x)
3572
+c*log(h)
3573
+log(a)*b)
3574
^2
3575
/(4*f*log(h)))
3576
*abs(
3577
2*f*log(h)*sqrt(x)
3578
+c*log(h)+log(a)*b)
3579
/sqrt(f*log(h))),i,0,j),j,0,n))
3580
/(f^(2*(n+1))*log(h)^(2*(n+1)))));
3565
3581
0;
3566
3582
3567
/* Problem 552: NEW 10 */
3583
/* Problem 552: NEW 9 */
3568
3584
expand(niceindices(integrate(z^n*a^(e+b*z^2)*h^(f*z+c*z^2),z))-
3569
-a^e*(c*log(h)+log(a)*b)^(-n-1)*%e^-(f^2*log(h)^2/(4*(c*log(h)+log(a)*b)))
3570
*('sum(f^(n-i)*log(h)^(n-i)*(-c*log(h)-log(a)*b)^((i+1)/2)
3585
(-a^e*%e^-(f^2*log(h)^2/(4*(c*log(h)+log(a)*b)))*(c*log(h)+log(a)*b)^(-n-1)
3586
*('sum(f^(n-i)*log(h)^(n-i)
3571
3587
*gamma_incomplete((i+1)/2,
3572
3588
-(log(h)*(2*c*z+f)+2*log(a)*b*z)^2
3573
3589
/(4*(c*log(h)+log(a)*b)))*(-1)^(i-n)
3574
3590
*2^(i-n)*binomial(n,i)*(log(h)*(2*c*z+f)+2*log(a)*b*z)^(i+1)
3575
*abs(log(h)*(2*c*z+f)+2*log(a)*b*z)^(-i-1),i,0,n))
3576
/2);
3591
/(-(log(h)*(2*c*z+f)+2*log(a)*b*z)^2/(c*log(h)+log(a)*b))
3592
^((i+1)/2),i,0,n))
3593
/2));
3577
3594
0;
3578
3595
3579
/* Problem 553: NEW 11 */
3596
/* Problem 553: NEW 10 */
3580
3597
expand(niceindices(integrate(x^n*a^(e+b*sqrt(x))*h^(f*x+c*sqrt(x)),x))-
3581
a^e*%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))*2^(-2*n-1)
3598
(a^e*%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))*2^(-2*n-1)
3582
3599
*('sum(4^j*binomial(n,j)
3583
*'sum((-f*log(h))^((j+i+1)/2)*(c*log(h)+log(a)*b)^(2*n-j-i)
3584
*(-1)^(i-j)*binomial(j,i)
3585
*abs(
3586
2*f*log(h)*sqrt(x)
3587
+c*log(h)+log(a)*b)
3588
^(-j-i-1)
3600
*'sum((f*log(h))^((j+i+1)/2)*(c*log(h)+log(a)*b)^(2*n-j-i)
3601
*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
3602
*abs(
3603
2*f*log(h)*sqrt(x)+c*log(h)
3604
+log(a)*b)
3605
^(-j-i-1)
3606
*(2*f*log(h)*sqrt(x)
3607
+c*log(h)+log(a)*b)
3608
^(j+i)
3609
*((c*log(h)+log(a)*b)
3610
*gamma_incomplete((j+i+1)/2,
3611
-(2*f*log(h)
3612
*sqrt(x)
3613
+c*log(h)
3614
+log(a)*b)
3615
^2
3616
/(4*f*log(h)))
3589
3617
*(2*f*log(h)*sqrt(x)
3590
3618
+c*log(h)+log(a)*b)
3591
^(j+i)
3592
*((c*log(h)+log(a)*b)
3593
*gamma_incomplete(
3594
(j+i+1)/2,
3595
-(2*f*log(h)*sqrt(x)
3596
+c*log(h)+log(a)*b)
3597
^2
3598
/(4*f*log(h)))
3599
*(2*f*log(h)*sqrt(x)
3600
+c*log(h)+log(a)*b)
3601
+2*f*log(h)
3602
*gamma_incomplete(
3603
(j+i+2)/2,
3604
-(2*f*log(h)*sqrt(x)
3605
+c*log(h)+log(a)*b)
3606
^2
3607
/(4*f*log(h)))
3608
*abs(
3609
2*f*log(h)*sqrt(x)
3610
+c*log(h)+log(a)*b)
3611
/sqrt(-f*log(h))),i,0,j),j,0,n))
3612
/(f^(2*(n+1))*log(h)^(2*(n+1))));
3619
+2*%i*f*log(h)
3620
*gamma_incomplete(
3621
(j+i+2)/2,
3622
-(2*f*log(h)*sqrt(x)
3623
+c*log(h)+log(a)*b)
3624
^2
3625
/(4*f*log(h)))
3626
*abs(
3627
2*f*log(h)*sqrt(x)
3628
+c*log(h)+log(a)*b)
3629
/sqrt(f*log(h))),i,0,j),j,0,n))
3630
/(f^(2*(n+1))*log(h)^(2*(n+1)))));
3613
3631
0;
3614
3632
3615
/* Problem 554: NEW 10 */
3633
/* Problem 554: NEW 9 */
3616
3634
expand(niceindices(integrate(z^n*a^(d*z+b*z^2)*h^(f*z+c*z^2),z))-
3617
-(c*log(h)+log(a)*b)^(-n-1)*%e^-((f*log(h)+log(a)*d)^2
3618
/(4*(c*log(h)+log(a)*b)))
3619
*('sum((-c*log(h)-log(a)*b)^((i+1)/2)
3620
*(-f*log(h)-log(a)*d)^(n-i)
3621
*gamma_incomplete((i+1)/2,
3622
-(log(h)*(2*c*z+f)
3623
+log(a)*(2*b*z+d))
3624
^2
3625
/(4
3626
*(c*log(h)+log(a)*b)))
3627
*2^(i-n)*binomial(n,i)
3628
*(log(h)*(2*c*z+f)+log(a)*(2*b*z+d))^(i+1)
3629
*abs(log(h)*(2*c*z+f)+log(a)*(2*b*z+d))
3630
^(-i-1),i,0,n))
3631
/2);
3635
(-%e^-((f*log(h)+log(a)*d)^2/(4*(c*log(h)+log(a)*b)))
3636
*(c*log(h)+log(a)*b)^(-n-1)
3637
*('sum((-f*log(h)-log(a)*d)^(n-i)*gamma_incomplete((i+1)/2,
3638
-(log(h)*(2*c*z+f)
3639
+log(a)*(2*b*z+d))
3640
^2
3641
/(4*(c*log(h)+log(a)*b)))
3642
*2^(i-n)*binomial(n,i)
3643
*(log(h)*(2*c*z+f)+log(a)*(2*b*z+d))^(i+1)
3644
/(-(log(h)*(2*c*z+f)+log(a)*(2*b*z+d))^2/(c*log(h)+log(a)*b))
3645
^((i+1)/2),i,0,n))
3646
/2));
3632
3647
0;
3633
3648
3634
/* Problem 555: NEW 11 */
3649
/* Problem 555: NEW 10 */
3635
3650
expand(niceindices(integrate(x^n*a^(d*x+b*sqrt(x))*h^(f*x+c*sqrt(x)),x))-
3636
%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d)))
3651
(%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d)))
3637
3652
*2^(-2*n-1)
3638
3653
*('sum(4^j*binomial(n,j)
3639
*'sum((c*log(h)+log(a)*b)^(2*n-j-i)
3640
*(-f*log(h)-log(a)*d)^((j+i+1)/2)*(-1)^(i-j)*binomial(j,i)
3641
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h)+log(a)*b)
3642
^(-j-i-1)
3643
*(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h)+log(a)*b)^(j+i)
3644
*((c*log(h)+log(a)*b)*gamma_incomplete((j+i+1)/2,
3645
-(2
3646
*(f*log(h)+log(a)*d)
3647
*sqrt(x)
3648
+c*log(h)+log(a)*b)
3649
^2
3650
/(4
3651
*(f*log(h)
3652
+log(a)*d)))
3653
*(2*(f*log(h)+log(a)*d)*sqrt(x)
3654
+c*log(h)+log(a)*b)
3655
+2*(f*log(h)+log(a)*d)
3656
*gamma_incomplete((j+i+2)/2,
3657
-(2*(f*log(h)+log(a)*d)*sqrt(x)
3658
+c*log(h)+log(a)*b)
3659
^2
3660
/(4*(f*log(h)+log(a)*d)))
3661
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h)+log(a)*b)
3662
/sqrt(-f*log(h)-log(a)*d)),i,0,j),j,0,n))
3663
/(f*log(h)+log(a)*d)^(2*(n+1)));
3654
*'sum((c*log(h)+log(a)*b)^(2*n-j-i)*(f*log(h)+log(a)*d)^((j+i+1)/2)
3655
*(-1)^(-(j+i+1)/2+j-i)
3656
*binomial(j,i)
3657
*abs(
3658
2*(f*log(h)+log(a)*d)*sqrt(x)
3659
+c*log(h)+log(a)*b)
3660
^(-j-i-1)
3661
*(2*(f*log(h)+log(a)*d)*sqrt(x)
3662
+c*log(h)+log(a)*b)
3663
^(j+i)
3664
*((c*log(h)+log(a)*b)
3665
*gamma_incomplete(
3666
(j+i+1)/2,
3667
-(2*(f*log(h)+log(a)*d)
3668
*sqrt(x)
3669
+c*log(h)+log(a)*b)
3670
^2
3671
/(4*(f*log(h)+log(a)*d)))
3672
*(2*(f*log(h)+log(a)*d)*sqrt(x)
3673
+c*log(h)+log(a)*b)
3674
+2*%i*sqrt(f*log(h)+log(a)*d)
3675
*gamma_incomplete(
3676
(j+i+2)/2,
3677
-(2*(f*log(h)+log(a)*d)
3678
*sqrt(x)
3679
+c*log(h)+log(a)*b)
3680
^2
3681
/(4*(f*log(h)+log(a)*d)))
3682
*abs(
3683
2*(f*log(h)+log(a)*d)
3684
*sqrt(x)
3685
+c*log(h)+log(a)*b)),i,0,
3686
j),j,0,n))
3687
/(f*log(h)+log(a)*d)^(2*(n+1))));
3664
3688
0;
3665
3689
3666
/* Problem 556: NEW 10 */
3690
/* Problem 556: NEW 9 */
3667
3691
expand(niceindices(integrate(z^n*a^(d*z)*h^(g+f*z+c*z^2),z))-
3668
-c^(-n-1)*h^g*log(h)^(-n-1)*%e^-((f*log(h)+log(a)*d)^2/(4*c*log(h)))
3669
*('sum((-c*log(h))^((i+1)/2)*(-f*log(h)-log(a)*d)^(n-i)
3670
*gamma_incomplete((i+1)/2,
3671
-(log(h)*(2*c*z+f)
3672
+log(a)*d)
3673
^2
3674
/(4*c*log(h)))*2^(i-n)
3675
*binomial(n,i)
3676
*(log(h)*(2*c*z+f)+log(a)*d)^(i+1)
3677
*abs(log(h)*(2*c*z+f)+log(a)*d)^(-i-1),i,
3678
0,n))
3679
/2);
3692
(-c^(-n-1)*h^g*%e^-((f*log(h)+log(a)*d)^2/(4*c*log(h)))*log(h)^(-n-1)
3693
*('sum((-f*log(h)-log(a)*d)^(n-i)
3694
*gamma_incomplete((i+1)/2,
3695
-(log(h)*(2*c*z+f)+log(a)*d)^2
3696
/(4*c*log(h)))*2^(i-n)*binomial(n,i)
3697
*(log(h)*(2*c*z+f)+log(a)*d)^(i+1)
3698
/(-(log(h)*(2*c*z+f)+log(a)*d)^2/(c*log(h)))^((i+1)/2),i,0,
3699
n))
3700
/2));
3680
3701
0;
3681
3702
3682
/* Problem 557: NEW 11 */
3703
/* Problem 557: NEW 10 */
3683
3704
expand(niceindices(integrate(x^n*a^(d*x)*h^(g+f*x+c*sqrt(x)),x))-
3684
h^g*%e^-(c^2*log(h)^2/(4*(f*log(h)+log(a)*d)))*2^(-2*n-1)
3705
(h^g*%e^-(c^2*log(h)^2/(4*(f*log(h)+log(a)*d)))*2^(-2*n-1)
3685
3706
*('sum(4^j*binomial(n,j)
3686
3707
*'sum(c^(2*n-j-i)*log(h)^(2*n-j-i)
3687
*(-f*log(h)-log(a)*d)^((j+i+1)/2)*(-1)^(i-j)
3688
*binomial(j,i)
3708
*(f*log(h)+log(a)*d)^((j+i+1)/2)
3709
*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
3689
3710
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h))
3690
3711
^(-j-i-1)
3691
3712
*(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h))^(j+i)
3697
3718
^2
3698
3719
/(4*(f*log(h)+log(a)*d)))
3699
3720
*(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h))
3700
+2*(f*log(h)+log(a)*d)
3721
+2*%i*sqrt(f*log(h)+log(a)*d)
3701
3722
*gamma_incomplete((j+i+2)/2,
3702
3723
-(2*(f*log(h)+log(a)*d)
3703
3724
*sqrt(x)
3704
3725
+c*log(h))
3705
3726
^2
3706
3727
/(4*(f*log(h)+log(a)*d)))
3707
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h))
3708
/sqrt(-f*log(h)-log(a)*d)),i,0,j),j,0,n))
3709
/(f*log(h)+log(a)*d)^(2*(n+1)));
3728
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)
3729
+c*log(h))),i,0,j),j,0,n))
3730
/(f*log(h)+log(a)*d)^(2*(n+1))));
3710
3731
0;
3711
3732
3712
/* Problem 558: NEW 10 */
3733
/* Problem 558: NEW 9 */
3713
3734
expand(niceindices(integrate(z^n*a^(e+d*z)*h^(g+f*z+c*z^2),z))-
3714
-a^e*c^(-n-1)*h^g*log(h)^(-n-1)*%e^-((f*log(h)+log(a)*d)^2/(4*c*log(h)))
3715
*('sum((-c*log(h))^((i+1)/2)*(-f*log(h)-log(a)*d)^(n-i)
3716
*gamma_incomplete((i+1)/2,
3717
-(log(h)*(2*c*z+f)+log(a)*d)
3718
^2
3719
/(4*c*log(h)))*2^(i-n)
3720
*binomial(n,i)
3721
*(log(h)*(2*c*z+f)+log(a)*d)^(i+1)
3722
*abs(log(h)*(2*c*z+f)+log(a)*d)^(-i-1),i,0,n))
3723
/2);
3735
(-a^e*c^(-n-1)*h^g*%e^-((f*log(h)+log(a)*d)^2/(4*c*log(h)))*log(h)^(-n-1)
3736
*('sum((-f*log(h)-log(a)*d)^(n-i)*gamma_incomplete((i+1)/2,
3737
-(log(h)*(2*c*z+f)
3738
+log(a)*d)
3739
^2
3740
/(4*c*log(h)))*2^(i-n)
3741
*binomial(n,i)
3742
*(log(h)*(2*c*z+f)+log(a)*d)^(i+1)
3743
/(-(log(h)*(2*c*z+f)+log(a)*d)^2/(c*log(h)))^((i+1)/2),i,0,n))
3744
/2));
3724
3745
0;
3725
3746
3726
/* Problem 559: NEW 11 */
3747
/* Problem 559: NEW 10 */
3727
3748
expand(niceindices(integrate(x^n*a^(e+d*x)*h^(g+f*x+c*sqrt(x)),x))-
3728
a^e*h^g*%e^-(c^2*log(h)^2/(4*(f*log(h)+log(a)*d)))*2^(-2*n-1)
3749
(a^e*h^g*%e^-(c^2*log(h)^2/(4*(f*log(h)+log(a)*d)))*2^(-2*n-1)
3729
3750
*('sum(4^j*binomial(n,j)
3730
3751
*'sum(c^(2*n-j-i)*log(h)^(2*n-j-i)
3731
*(-f*log(h)-log(a)*d)^((j+i+1)/2)*(-1)^(i-j)
3732
*binomial(j,i)
3752
*(f*log(h)+log(a)*d)^((j+i+1)/2)
3753
*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
3733
3754
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h))
3734
3755
^(-j-i-1)
3735
3756
*(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h))^(j+i)
3741
3762
^2
3742
3763
/(4*(f*log(h)+log(a)*d)))
3743
3764
*(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h))
3744
+2*(f*log(h)+log(a)*d)
3765
+2*%i*sqrt(f*log(h)+log(a)*d)
3745
3766
*gamma_incomplete((j+i+2)/2,
3746
3767
-(2*(f*log(h)+log(a)*d)
3747
3768
*sqrt(x)
3748
3769
+c*log(h))
3749
3770
^2
3750
3771
/(4*(f*log(h)+log(a)*d)))
3751
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h))
3752
/sqrt(-f*log(h)-log(a)*d)),i,0,j),j,0,n))
3753
/(f*log(h)+log(a)*d)^(2*(n+1)));
3772
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)
3773
+c*log(h))),i,0,j),j,0,n))
3774
/(f*log(h)+log(a)*d)^(2*(n+1))));
3754
3775
0;
3755
3776
3756
/* Problem 560: NEW 10 */
3777
/* Problem 560: NEW 9 */
3757
3778
expand(niceindices(integrate(z^n*a^(b*z^2)*h^(g+f*z+c*z^2),z))-
3758
-h^g*(c*log(h)+log(a)*b)^(-n-1)*%e^-(f^2*log(h)^2/(4*(c*log(h)+log(a)*b)))
3759
*('sum(f^(n-i)*log(h)^(n-i)*(-c*log(h)-log(a)*b)^((i+1)/2)
3779
(-h^g*%e^-(f^2*log(h)^2/(4*(c*log(h)+log(a)*b)))*(c*log(h)+log(a)*b)^(-n-1)
3780
*('sum(f^(n-i)*log(h)^(n-i)
3760
3781
*gamma_incomplete((i+1)/2,
3761
3782
-(log(h)*(2*c*z+f)+2*log(a)*b*z)^2
3762
3783
/(4*(c*log(h)+log(a)*b)))*(-1)^(i-n)
3763
3784
*2^(i-n)*binomial(n,i)*(log(h)*(2*c*z+f)+2*log(a)*b*z)^(i+1)
3764
*abs(log(h)*(2*c*z+f)+2*log(a)*b*z)^(-i-1),i,0,n))
3765
/2);
3785
/(-(log(h)*(2*c*z+f)+2*log(a)*b*z)^2/(c*log(h)+log(a)*b))
3786
^((i+1)/2),i,0,n))
3787
/2));
3766
3788
0;
3767
3789
3768
/* Problem 561: NEW 11 */
3790
/* Problem 561: NEW 10 */
3769
3791
expand(niceindices(integrate(x^n*a^(b*sqrt(x))*h^(g+f*x+c*sqrt(x)),x))-
3770
h^g*%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))*2^(-2*n-1)
3792
(h^g*%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))*2^(-2*n-1)
3771
3793
*('sum(4^j*binomial(n,j)
3772
*'sum((-f*log(h))^((j+i+1)/2)*(c*log(h)+log(a)*b)^(2*n-j-i)
3773
*(-1)^(i-j)*binomial(j,i)
3774
*abs(
3775
2*f*log(h)*sqrt(x)
3776
+c*log(h)+log(a)*b)
3777
^(-j-i-1)
3794
*'sum((f*log(h))^((j+i+1)/2)*(c*log(h)+log(a)*b)^(2*n-j-i)
3795
*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
3796
*abs(
3797
2*f*log(h)*sqrt(x)+c*log(h)
3798
+log(a)*b)
3799
^(-j-i-1)
3800
*(2*f*log(h)*sqrt(x)
3801
+c*log(h)+log(a)*b)
3802
^(j+i)
3803
*((c*log(h)+log(a)*b)
3804
*gamma_incomplete((j+i+1)/2,
3805
-(2*f*log(h)
3806
*sqrt(x)
3807
+c*log(h)
3808
+log(a)*b)
3809
^2
3810
/(4*f*log(h)))
3778
3811
*(2*f*log(h)*sqrt(x)
3779
3812
+c*log(h)+log(a)*b)
3780
^(j+i)
3781
*((c*log(h)+log(a)*b)
3782
*gamma_incomplete(
3783
(j+i+1)/2,
3784
-(2*f*log(h)*sqrt(x)
3785
+c*log(h)+log(a)*b)
3786
^2
3787
/(4*f*log(h)))
3788
*(2*f*log(h)*sqrt(x)
3789
+c*log(h)+log(a)*b)
3790
+2*f*log(h)
3791
*gamma_incomplete(
3792
(j+i+2)/2,
3793
-(2*f*log(h)*sqrt(x)
3794
+c*log(h)+log(a)*b)
3795
^2
3796
/(4*f*log(h)))
3797
*abs(
3798
2*f*log(h)*sqrt(x)
3799
+c*log(h)+log(a)*b)
3800
/sqrt(-f*log(h))),i,0,j),j,0,n))
3801
/(f^(2*(n+1))*log(h)^(2*(n+1))));
3813
+2*%i*f*log(h)
3814
*gamma_incomplete(
3815
(j+i+2)/2,
3816
-(2*f*log(h)*sqrt(x)
3817
+c*log(h)+log(a)*b)
3818
^2
3819
/(4*f*log(h)))
3820
*abs(
3821
2*f*log(h)*sqrt(x)
3822
+c*log(h)+log(a)*b)
3823
/sqrt(f*log(h))),i,0,j),j,0,n))
3824
/(f^(2*(n+1))*log(h)^(2*(n+1)))));
3802
3825
0;
3803
3826
3804
/* Problem 562: NEW 10 */
3827
/* Problem 562: NEW 9 */
3805
3828
expand(niceindices(integrate(z^n*a^(e+b*z^2)*h^(g+f*z+c*z^2),z))-
3806
-a^e*h^g*(c*log(h)+log(a)*b)^(-n-1)*%e^-(f^2*log(h)^2/(4*(c*log(h)+log(a)*b)))
3807
*('sum(f^(n-i)*log(h)^(n-i)*(-c*log(h)-log(a)*b)^((i+1)/2)
3829
(-a^e*h^g*%e^-(f^2*log(h)^2/(4*(c*log(h)+log(a)*b)))*(c*log(h)+log(a)*b)^(-n-1)
3830
*('sum(f^(n-i)*log(h)^(n-i)
3808
3831
*gamma_incomplete((i+1)/2,
3809
3832
-(log(h)*(2*c*z+f)+2*log(a)*b*z)^2
3810
3833
/(4*(c*log(h)+log(a)*b)))*(-1)^(i-n)
3811
3834
*2^(i-n)*binomial(n,i)*(log(h)*(2*c*z+f)+2*log(a)*b*z)^(i+1)
3812
*abs(log(h)*(2*c*z+f)+2*log(a)*b*z)^(-i-1),i,0,n))
3813
/2);
3835
/(-(log(h)*(2*c*z+f)+2*log(a)*b*z)^2/(c*log(h)+log(a)*b))
3836
^((i+1)/2),i,0,n))
3837
/2));
3814
3838
0;
3815
3839
3816
/* Problem 563: NEW 11 */
3840
/* Problem 563: NEW 10 */
3817
3841
expand(niceindices(integrate(x^n*a^(e+b*sqrt(x))*h^(g+f*x+c*sqrt(x)),x))-
3818
a^e*h^g*%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))*2^(-2*n-1)
3842
(a^e*h^g*%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))*2^(-2*n-1)
3819
3843
*('sum(4^j*binomial(n,j)
3820
*'sum((-f*log(h))^((j+i+1)/2)*(c*log(h)+log(a)*b)^(2*n-j-i)
3821
*(-1)^(i-j)*binomial(j,i)
3822
*abs(
3823
2*f*log(h)*sqrt(x)
3824
+c*log(h)+log(a)*b)
3825
^(-j-i-1)
3844
*'sum((f*log(h))^((j+i+1)/2)*(c*log(h)+log(a)*b)^(2*n-j-i)
3845
*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
3846
*abs(
3847
2*f*log(h)*sqrt(x)+c*log(h)
3848
+log(a)*b)
3849
^(-j-i-1)
3850
*(2*f*log(h)*sqrt(x)
3851
+c*log(h)+log(a)*b)
3852
^(j+i)
3853
*((c*log(h)+log(a)*b)
3854
*gamma_incomplete((j+i+1)/2,
3855
-(2*f*log(h)
3856
*sqrt(x)
3857
+c*log(h)
3858
+log(a)*b)
3859
^2
3860
/(4*f*log(h)))
3826
3861
*(2*f*log(h)*sqrt(x)
3827
3862
+c*log(h)+log(a)*b)
3828
^(j+i)
3829
*((c*log(h)+log(a)*b)
3830
*gamma_incomplete(
3831
(j+i+1)/2,
3832
-(2*f*log(h)*sqrt(x)
3833
+c*log(h)+log(a)*b)
3834
^2
3835
/(4*f*log(h)))
3836
*(2*f*log(h)*sqrt(x)
3837
+c*log(h)+log(a)*b)
3838
+2*f*log(h)
3839
*gamma_incomplete(
3840
(j+i+2)/2,
3841
-(2*f*log(h)*sqrt(x)
3842
+c*log(h)+log(a)*b)
3843
^2
3844
/(4*f*log(h)))
3845
*abs(
3846
2*f*log(h)*sqrt(x)
3847
+c*log(h)+log(a)*b)
3848
/sqrt(-f*log(h))),i,0,j),j,0,n))
3849
/(f^(2*(n+1))*log(h)^(2*(n+1))));
3863
+2*%i*f*log(h)
3864
*gamma_incomplete(
3865
(j+i+2)/2,
3866
-(2*f*log(h)*sqrt(x)
3867
+c*log(h)+log(a)*b)
3868
^2
3869
/(4*f*log(h)))
3870
*abs(
3871
2*f*log(h)*sqrt(x)
3872
+c*log(h)+log(a)*b)
3873
/sqrt(f*log(h))),i,0,j),j,0,n))
3874
/(f^(2*(n+1))*log(h)^(2*(n+1)))));
3850
3875
0;
3851
3876
3852
/* Problem 564: NEW 10 */
3877
/* Problem 564: NEW 9 */
3853
3878
expand(niceindices(integrate(z^n*a^(d*z+b*z^2)*h^(g+f*z+c*z^2),z))-
3854
-h^g*(c*log(h)+log(a)*b)^(-n-1)
3855
*%e^-((f*log(h)+log(a)*d)^2/(4*(c*log(h)+log(a)*b)))
3856
*('sum((-c*log(h)-log(a)*b)^((i+1)/2)*(-f*log(h)-log(a)*d)^(n-i)
3857
*gamma_incomplete((i+1)/2,
3858
-(log(h)*(2*c*z+f)
3859
+log(a)*(2*b*z+d))
3860
^2
3861
/(4
3862
*(c*log(h)
3863
+log(a)*b)))
3864
*2^(i-n)*binomial(n,i)
3865
*(log(h)*(2*c*z+f)+log(a)*(2*b*z+d))
3866
^(i+1)
3867
*abs(
3868
log(h)*(2*c*z+f)+log(a)*(2*b*z+d))
3869
^(-i-1),i,0,n))
3870
/2);
3879
(-h^g*%e^-((f*log(h)+log(a)*d)^2/(4*(c*log(h)+log(a)*b)))
3880
*(c*log(h)+log(a)*b)^(-n-1)
3881
*('sum((-f*log(h)-log(a)*d)^(n-i)*gamma_incomplete((i+1)/2,
3882
-(log(h)*(2*c*z+f)
3883
+log(a)*(2*b*z+d))
3884
^2
3885
/(4
3886
*(c*log(h)
3887
+log(a)*b)))*2^(i-n)
3888
*binomial(n,i)
3889
*(log(h)*(2*c*z+f)+log(a)*(2*b*z+d))
3890
^(i+1)
3891
/(-(log(h)*(2*c*z+f)+log(a)*(2*b*z+d))^2/(c*log(h)+log(a)*b))
3892
^((i+1)/2),i,0,n))/2));
3871
3893
0;
3872
3894
3873
/* Problem 565: NEW 11 */
3895
/* Problem 565: NEW 10 */
3874
3896
expand(niceindices(integrate(x^n*a^(d*x+b*sqrt(x))*h^(g+f*x+c*sqrt(x)),x))-
3875
h^g*%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d)))*2^(-2*n-1)
3897
(h^g*%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d)))*2^(-2*n-1)
3876
3898
*('sum(4^j*binomial(n,j)
3877
3899
*'sum((c*log(h)+log(a)*b)^(2*n-j-i)
3878
*(-f*log(h)-log(a)*d)^((j+i+1)/2)*(-1)^(i-j)*binomial(j,i)
3900
*(f*log(h)+log(a)*d)^((j+i+1)/2)*(-1)^(-(j+i+1)/2+j-i)
3901
*binomial(j,i)
3879
3902
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h)+log(a)*b)
3880
3903
^(-j-i-1)
3881
3904
*(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h)+log(a)*b)^(j+i)
3886
3909
^2
3887
3910
/(4*(f*log(h)+log(a)*d)))
3888
3911
*(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h)+log(a)*b)
3889
+2*(f*log(h)+log(a)*d)
3912
+2*%i*sqrt(f*log(h)+log(a)*d)
3890
3913
*gamma_incomplete((j+i+2)/2,
3891
3914
-(2*(f*log(h)+log(a)*d)*sqrt(x)
3892
3915
+c*log(h)+log(a)*b)
3893
3916
^2
3894
3917
/(4*(f*log(h)+log(a)*d)))
3895
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h)+log(a)*b)
3896
/sqrt(-f*log(h)-log(a)*d)),i,0,j),j,0,n))
3897
/(f*log(h)+log(a)*d)^(2*(n+1)));
3918
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h)+log(a)*b)),
3919
i,0,j),j,0,n))
3920
/(f*log(h)+log(a)*d)^(2*(n+1))));
3898
3921
0;
3899
3922
3900
/* Problem 566: NEW 10 */
3923
/* Problem 566: NEW 9 */
3901
3924
expand(niceindices(integrate(z^n*a^(e+d*z+b*z^2)*h^(g+f*z+c*z^2),z))-
3902
-a^e*h^g*(c*log(h)+log(a)*b)^(-n-1)
3903
*%e^-((f*log(h)+log(a)*d)^2/(4*(c*log(h)+log(a)*b)))
3904
*('sum((-c*log(h)-log(a)*b)^((i+1)/2)*(-f*log(h)-log(a)*d)^(n-i)
3905
*gamma_incomplete((i+1)/2,
3906
-(log(h)*(2*c*z+f)
3907
+log(a)*(2*b*z+d))
3908
^2
3909
/(4
3910
*(c*log(h)
3911
+log(a)*b)))
3912
*2^(i-n)*binomial(n,i)
3913
*(log(h)*(2*c*z+f)+log(a)*(2*b*z+d))
3914
^(i+1)
3915
*abs(
3916
log(h)*(2*c*z+f)+log(a)*(2*b*z+d))
3917
^(-i-1),i,0,n))
3918
/2);
3925
(-a^e*h^g*%e^-((f*log(h)+log(a)*d)^2/(4*(c*log(h)+log(a)*b)))
3926
*(c*log(h)+log(a)*b)^(-n-1)
3927
*('sum((-f*log(h)-log(a)*d)^(n-i)*gamma_incomplete((i+1)/2,
3928
-(log(h)*(2*c*z+f)
3929
+log(a)*(2*b*z+d))
3930
^2
3931
/(4
3932
*(c*log(h)
3933
+log(a)*b)))*2^(i-n)
3934
*binomial(n,i)
3935
*(log(h)*(2*c*z+f)+log(a)*(2*b*z+d))
3936
^(i+1)
3937
/(-(log(h)*(2*c*z+f)+log(a)*(2*b*z+d))^2/(c*log(h)+log(a)*b))
3938
^((i+1)/2),i,0,n))
3939
/2));
3919
3940
0;
3920
3941
3921
/* Problem 567: NEW 11 */
3942
/* Problem 567: NEW 10 */
3922
3943
expand(niceindices(integrate(x^n*a^(e+d*x+b*sqrt(x))*h^(g+f*x+c*sqrt(x)),x))-
3923
a^e*h^g*%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d)))*2^(-2*n-1)
3944
(a^e*h^g*%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d)))*2^(-2*n-1)
3924
3945
*('sum(4^j*binomial(n,j)
3925
3946
*'sum((c*log(h)+log(a)*b)^(2*n-j-i)
3926
*(-f*log(h)-log(a)*d)^((j+i+1)/2)*(-1)^(i-j)*binomial(j,i)
3947
*(f*log(h)+log(a)*d)^((j+i+1)/2)*(-1)^(-(j+i+1)/2+j-i)
3948
*binomial(j,i)
3927
3949
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h)+log(a)*b)
3928
3950
^(-j-i-1)
3929
3951
*(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h)+log(a)*b)^(j+i)
3934
3956
^2
3935
3957
/(4*(f*log(h)+log(a)*d)))
3936
3958
*(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h)+log(a)*b)
3937
+2*(f*log(h)+log(a)*d)
3959
+2*%i*sqrt(f*log(h)+log(a)*d)
3938
3960
*gamma_incomplete((j+i+2)/2,
3939
3961
-(2*(f*log(h)+log(a)*d)*sqrt(x)
3940
3962
+c*log(h)+log(a)*b)
3941
3963
^2
3942
3964
/(4*(f*log(h)+log(a)*d)))
3943
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h)+log(a)*b)
3944
/sqrt(-f*log(h)-log(a)*d)),i,0,j),j,0,n))
3945
/(f*log(h)+log(a)*d)^(2*(n+1)));
3965
*abs(2*(f*log(h)+log(a)*d)*sqrt(x)+c*log(h)+log(a)*b)),
3966
i,0,j),j,0,n))
3967
/(f*log(h)+log(a)*d)^(2*(n+1))));
3946
3968
0;
3947
3969
3948
3970
/* Problem 568: NEW */
4014
4036
-%e^-(c^2*u^2/(4*b))*('sum(b^(-n+(i+1)/2-1/2)*c^(n-i)
4015
4037
*gamma_incomplete(
4016
4038
(i+1)/2,-(2*b*z+c*u)^2/(4*b))
4017
*2^(i-n)*(-1)^(-n-(i+1)/2+i)
4018
*binomial(n,i)*u^(n-i)
4019
*(2*b*z+c*u)^(i+1)
4020
*abs(2*b*z+c*u)^(-i-1),i,0,n))
4039
*(-1)^(i-n)*2^(i-n)*binomial(n,i)
4040
*u^(n-i)*(2*b*z+c*u)^(i+1)
4041
/(-(2*b*z+c*u)^2)^((i+1)/2),i,0,n))
4021
4042
/(2*sqrt(b)));
4022
4043
0;
4023
4044
4024
4045
/* Problem 581: NEW 6 */
4025
4046
expand(niceindices(integrate(x^n*%e^(b*sqrt(x))*(%e^(c*x))^u,x))-
4026
2^(-n-1)*%e^-(b^2/(4*c*u))
4047
(2^(-n-1)*%e^-(b^2/(4*c*u))
4027
4048
*('sum(4^j*binomial(n,j)
4028
*'sum(b^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)
4029
*(-c*u)^((j+i+1)/2)
4049
*'sum(b^(2*n-j-i)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
4050
*(c*u)^((j+i+1)/2)
4030
4051
*abs(2*c*u*sqrt(x)+b)^(-j-i-1)
4031
4052
*(2*c*u*sqrt(x)+b)^(j+i)
4032
4053
*(b*gamma_incomplete((j+i+1)/2,
4033
4054
-(2*c*u*sqrt(x)+b)^2
4034
4055
/(4*c*u))
4035
4056
*(2*c*u*sqrt(x)+b)
4036
+2*c
4057
+2*%i*c
4037
4058
*gamma_incomplete((j+i+2)/2,
4038
4059
-(2*c*u*sqrt(x)+b)^2
4039
4060
/(4*c*u))*u
4040
4061
*abs(2*c*u*sqrt(x)+b)
4041
/sqrt(-c*u)),i,0,j),j,0,n))
4042
/(c^(2*(n+1))*u^(2*(n+1))));
4062
/sqrt(c*u)),i,0,j),j,0,n))
4063
/(c^(2*(n+1))*u^(2*(n+1)))));
4043
4064
0;
4044
4065
4045
4066
/* Problem 582: NEW 5 */
4046
4067
expand(niceindices(integrate(z^n*%e^(e+b*z^2)*(%e^(c*z))^u,z))-
4047
-%e^(e-c^2*u^2/(4*b))*('sum(b^(-n+(i+1)/2-1/2)*c^(n-i)
4068
(-%e^(e-c^2*u^2/(4*b))*('sum(b^(-n+(i+1)/2-1/2)*c^(n-i)
4048
4069
*gamma_incomplete(
4049
4070
(i+1)/2,-(2*b*z+c*u)^2/(4*b))
4050
*2^(i-n)*(-1)^(-n-(i+1)/2+i)
4071
*(-1)^(i-n)*2^(i-n)
4051
4072
*binomial(n,i)*u^(n-i)
4052
4073
*(2*b*z+c*u)^(i+1)
4053
*abs(2*b*z+c*u)^(-i-1),i,0,n))
4054
/(2*sqrt(b)));
4074
/(-(2*b*z+c*u)^2)^((i+1)/2),i,0,n))
4075
/(2*sqrt(b))));
4055
4076
0;
4056
4077
4057
4078
/* Problem 583: NEW 6 */
4058
4079
expand(niceindices(integrate(x^n*%e^(e+b*sqrt(x))*(%e^(c*x))^u,x))-
4059
2^(-n-1)*%e^(e-b^2/(4*c*u))
4080
(2^(-n-1)*%e^(e-b^2/(4*c*u))
4060
4081
*('sum(4^j*binomial(n,j)
4061
*'sum(b^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)
4062
*(-c*u)^((j+i+1)/2)
4082
*'sum(b^(2*n-j-i)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
4083
*(c*u)^((j+i+1)/2)
4063
4084
*abs(2*c*u*sqrt(x)+b)^(-j-i-1)
4064
4085
*(2*c*u*sqrt(x)+b)^(j+i)
4065
4086
*(b*gamma_incomplete((j+i+1)/2,
4066
4087
-(2*c*u*sqrt(x)+b)^2
4067
4088
/(4*c*u))
4068
4089
*(2*c*u*sqrt(x)+b)
4069
+2*c
4090
+2*%i*c
4070
4091
*gamma_incomplete((j+i+2)/2,
4071
4092
-(2*c*u*sqrt(x)+b)^2
4072
4093
/(4*c*u))*u
4073
4094
*abs(2*c*u*sqrt(x)+b)
4074
/sqrt(-c*u)),i,0,j),j,0,n))
4075
/(c^(2*(n+1))*u^(2*(n+1))));
4095
/sqrt(c*u)),i,0,j),j,0,n))
4096
/(c^(2*(n+1))*u^(2*(n+1)))));
4076
4097
0;
4077
4098
4078
/* Problem 584: NEW 10 */
4099
/* Problem 584: NEW 5 */
4079
4100
expand(niceindices(integrate(z^n*%e^(d*z+b*z^2)*(%e^(c*z))^u,z))-
4080
-%e^-((c*u+d)^2/(4*b))*('sum(b^(-n+(i+1)/2-1/2)
4101
(-%e^-((c*u+d)^2/(4*b))*('sum(b^(-n+(i+1)/2-1/2)
4081
4102
*gamma_incomplete((i+1)/2,
4082
4103
-(2*b*z+c*u+d)^2/(4*b))
4083
4104
*2^(i-n)*binomial(n,i)*(-c*u-d)^(n-i)
4084
*(2*b*z+c*u+d)^(i+1)*abs(2*b*z+c*u+d)^(-i-1)
4085
/(-1)^((i+1)/2),i,0,n))
4086
/(2*sqrt(b)));
4105
*(2*b*z+c*u+d)^(i+1)
4106
/(-(2*b*z+c*u+d)^2)^((i+1)/2),i,0,n))
4107
/(2*sqrt(b))));
4087
4108
0;
4088
4109
4089
/* Problem 585: NEW 11 */
4110
/* Problem 585: NEW 6 */
4090
4111
expand(niceindices(integrate(x^n*%e^(d*x+b*sqrt(x))*(%e^(c*x))^u,x))-
4091
2^(-n-1)*%e^-(b^2/(4*(c*u+d)))
4112
(2^(-n-1)*%e^-(b^2/(4*(c*u+d)))
4092
4113
*('sum(4^j*binomial(n,j)
4093
*'sum(b^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)
4094
*(-c*u-d)^((j+i+1)/2)
4114
*'sum(b^(2*n-j-i)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
4115
*(c*u+d)^((j+i+1)/2)
4095
4116
*abs(2*(c*u+d)*sqrt(x)+b)^(-j-i-1)
4096
4117
*(2*(c*u+d)*sqrt(x)+b)^(j+i)
4097
4118
*(b*gamma_incomplete((j+i+1)/2,
4099
4120
^2
4100
4121
/(4*(c*u+d)))
4101
4122
*(2*(c*u+d)*sqrt(x)+b)
4102
+2*gamma_incomplete((j+i+2)/2,
4123
+2*%i
4124
*gamma_incomplete((j+i+2)/2,
4103
4125
-(2*(c*u+d)*sqrt(x)+b)
4104
4126
^2
4105
/(4*(c*u+d)))*(c*u+d)
4106
*abs(2*(c*u+d)*sqrt(x)+b)
4107
/sqrt(-c*u-d)),i,0,j),j,0,n))
4108
/(c*u+d)^(2*(n+1)));
4127
/(4*(c*u+d)))
4128
*sqrt(c*u+d)*abs(2*(c*u+d)*sqrt(x)+b)),
4129
i,0,j),j,0,n))
4130
/(c*u+d)^(2*(n+1))));
4109
4131
0;
4110
4132
4111
/* Problem 586: NEW 10 */
4133
/* Problem 586: NEW 5 */
4112
4134
expand(niceindices(integrate(z^n*%e^(e+d*z+b*z^2)*(%e^(c*z))^u,z))-
4113
-%e^(e-(c*u+d)^2/(4*b))*('sum(b^(-n+(i+1)/2-1/2)
4135
(-%e^(e-(c*u+d)^2/(4*b))*('sum(b^(-n+(i+1)/2-1/2)
4114
4136
*gamma_incomplete((i+1)/2,
4115
4137
-(2*b*z+c*u+d)^2/(4*b))
4116
4138
*2^(i-n)*binomial(n,i)*(-c*u-d)^(n-i)
4117
*(2*b*z+c*u+d)^(i+1)*abs(2*b*z+c*u+d)^(-i-1)
4118
/(-1)^((i+1)/2),i,0,n))
4119
/(2*sqrt(b)));
4139
*(2*b*z+c*u+d)^(i+1)
4140
/(-(2*b*z+c*u+d)^2)^((i+1)/2),i,0,n))
4141
/(2*sqrt(b))));
4120
4142
0;
4121
4143
4122
/* Problem 587: NEW 11 */
4144
/* Problem 587: NEW 6 */
4123
4145
expand(niceindices(integrate(x^n*%e^(e+d*x+b*sqrt(x))*(%e^(c*x))^u,x))-
4124
2^(-n-1)*%e^(e-b^2/(4*(c*u+d)))
4146
(2^(-n-1)*%e^(e-b^2/(4*(c*u+d)))
4125
4147
*('sum(4^j*binomial(n,j)
4126
*'sum(b^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)
4127
*(-c*u-d)^((j+i+1)/2)
4148
*'sum(b^(2*n-j-i)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
4149
*(c*u+d)^((j+i+1)/2)
4128
4150
*abs(2*(c*u+d)*sqrt(x)+b)^(-j-i-1)
4129
4151
*(2*(c*u+d)*sqrt(x)+b)^(j+i)
4130
4152
*(b*gamma_incomplete((j+i+1)/2,
4132
4154
^2
4133
4155
/(4*(c*u+d)))
4134
4156
*(2*(c*u+d)*sqrt(x)+b)
4135
+2*gamma_incomplete((j+i+2)/2,
4157
+2*%i
4158
*gamma_incomplete((j+i+2)/2,
4136
4159
-(2*(c*u+d)*sqrt(x)+b)
4137
4160
^2
4138
/(4*(c*u+d)))*(c*u+d)
4139
*abs(2*(c*u+d)*sqrt(x)+b)
4140
/sqrt(-c*u-d)),i,0,j),j,0,n))
4141
/(c*u+d)^(2*(n+1)));
4161
/(4*(c*u+d)))
4162
*sqrt(c*u+d)*abs(2*(c*u+d)*sqrt(x)+b)),
4163
i,0,j),j,0,n))
4164
/(c*u+d)^(2*(n+1))));
4142
4165
0;
4143
4166
4144
4167
/* Problem 588: NEW 5 */
4145
4168
expand(niceindices(integrate(z^n*%e^(b*z^2)*(%e^(g+f*z))^u,z))-
4146
-%e^(g*u-f^2*u^2/(4*b))*('sum(b^(-n+(i+1)/2-1/2)*f^(n-i)
4169
(-%e^(g*u-f^2*u^2/(4*b))*('sum(b^(-n+(i+1)/2-1/2)*f^(n-i)
4147
4170
*gamma_incomplete(
4148
4171
(i+1)/2,-(2*b*z+f*u)^2/(4*b))
4149
*2^(i-n)*(-1)^(-n-(i+1)/2+i)
4172
*(-1)^(i-n)*2^(i-n)
4150
4173
*binomial(n,i)*u^(n-i)
4151
4174
*(2*b*z+f*u)^(i+1)
4152
*abs(2*b*z+f*u)^(-i-1),i,0,n))
4153
/(2*sqrt(b)));
4175
/(-(2*b*z+f*u)^2)^((i+1)/2),i,0,n))
4176
/(2*sqrt(b))));
4154
4177
0;
4155
4178
4156
4179
/* Problem 589: NEW 6 */
4157
4180
expand(niceindices(integrate(x^n*%e^(b*sqrt(x))*(%e^(g+f*x))^u,x))-
4158
2^(-n-1)*%e^(g*u-b^2/(4*f*u))
4181
(2^(-n-1)*%e^(g*u-b^2/(4*f*u))
4159
4182
*('sum(4^j*binomial(n,j)
4160
*'sum(b^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)
4161
*(-f*u)^((j+i+1)/2)
4183
*'sum(b^(2*n-j-i)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
4184
*(f*u)^((j+i+1)/2)
4162
4185
*abs(2*f*u*sqrt(x)+b)^(-j-i-1)
4163
4186
*(2*f*u*sqrt(x)+b)^(j+i)
4164
4187
*(b*gamma_incomplete((j+i+1)/2,
4165
4188
-(2*f*u*sqrt(x)+b)^2
4166
4189
/(4*f*u))
4167
4190
*(2*f*u*sqrt(x)+b)
4168
+2*f
4191
+2*%i*f
4169
4192
*gamma_incomplete((j+i+2)/2,
4170
4193
-(2*f*u*sqrt(x)+b)^2
4171
4194
/(4*f*u))*u
4172
4195
*abs(2*f*u*sqrt(x)+b)
4173
/sqrt(-f*u)),i,0,j),j,0,n))
4174
/(f^(2*(n+1))*u^(2*(n+1))));
4196
/sqrt(f*u)),i,0,j),j,0,n))
4197
/(f^(2*(n+1))*u^(2*(n+1)))));
4175
4198
0;
4176
4199
4177
4200
/* Problem 590: NEW 5 */
4178
4201
expand(niceindices(integrate(z^n*%e^(e+b*z^2)*(%e^(g+f*z))^u,z))-
4179
-%e^(-f^2*u^2/(4*b)+g*u+e)*('sum(b^(-n+(i+1)/2-1/2)*f^(n-i)
4202
(-%e^(-f^2*u^2/(4*b)+g*u+e)*('sum(b^(-n+(i+1)/2-1/2)*f^(n-i)
4180
4203
*gamma_incomplete(
4181
4204
(i+1)/2,
4182
4205
-(2*b*z+f*u)^2/(4*b))
4183
*2^(i-n)
4184
*(-1)^(-n-(i+1)/2+i)
4206
*(-1)^(i-n)*2^(i-n)
4185
4207
*binomial(n,i)*u^(n-i)
4186
4208
*(2*b*z+f*u)^(i+1)
4187
*abs(2*b*z+f*u)^(-i-1),i,0,
4188
n))
4189
/(2*sqrt(b)));
4209
/(-(2*b*z+f*u)^2)^((i+1)/2),i,0,n))
4210
/(2*sqrt(b))));
4190
4211
0;
4191
4212
4192
4213
/* Problem 591: NEW 6 */
4193
4214
expand(niceindices(integrate(x^n*%e^(e+b*sqrt(x))*(%e^(g+f*x))^u,x))-
4194
2^(-n-1)*%e^(g*u-b^2/(4*f*u)+e)
4215
(2^(-n-1)*%e^(g*u-b^2/(4*f*u)+e)
4195
4216
*('sum(4^j*binomial(n,j)
4196
*'sum(b^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)
4197
*(-f*u)^((j+i+1)/2)
4217
*'sum(b^(2*n-j-i)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
4218
*(f*u)^((j+i+1)/2)
4198
4219
*abs(2*f*u*sqrt(x)+b)^(-j-i-1)
4199
4220
*(2*f*u*sqrt(x)+b)^(j+i)
4200
4221
*(b*gamma_incomplete((j+i+1)/2,
4201
4222
-(2*f*u*sqrt(x)+b)^2
4202
4223
/(4*f*u))
4203
4224
*(2*f*u*sqrt(x)+b)
4204
+2*f
4225
+2*%i*f
4205
4226
*gamma_incomplete((j+i+2)/2,
4206
4227
-(2*f*u*sqrt(x)+b)^2
4207
4228
/(4*f*u))*u
4208
4229
*abs(2*f*u*sqrt(x)+b)
4209
/sqrt(-f*u)),i,0,j),j,0,n))
4210
/(f^(2*(n+1))*u^(2*(n+1))));
4230
/sqrt(f*u)),i,0,j),j,0,n))
4231
/(f^(2*(n+1))*u^(2*(n+1)))));
4211
4232
0;
4212
4233
4213
/* Problem 592: NEW 10 */
4234
/* Problem 592: NEW 5 */
4214
4235
expand(niceindices(integrate(z^n*%e^(d*z+b*z^2)*(%e^(g+f*z))^u,z))-
4215
-%e^(g*u-(f*u+d)^2/(4*b))*('sum(b^(-n+(i+1)/2-1/2)
4236
(-%e^(g*u-(f*u+d)^2/(4*b))*('sum(b^(-n+(i+1)/2-1/2)
4216
4237
*gamma_incomplete((i+1)/2,
4217
4238
-(2*b*z+f*u+d)^2/(4*b))
4218
4239
*2^(i-n)*binomial(n,i)*(-f*u-d)^(n-i)
4219
*(2*b*z+f*u+d)^(i+1)*abs(2*b*z+f*u+d)^(-i-1)
4220
/(-1)^((i+1)/2),i,0,n))
4221
/(2*sqrt(b)));
4240
*(2*b*z+f*u+d)^(i+1)
4241
/(-(2*b*z+f*u+d)^2)^((i+1)/2),i,0,n))
4242
/(2*sqrt(b))));
4222
4243
0;
4223
4244
4224
/* Problem 593: NEW 11 */
4245
/* Problem 593: NEW 6 */
4225
4246
expand(niceindices(integrate(x^n*%e^(d*x+b*sqrt(x))*(%e^(g+f*x))^u,x))-
4226
2^(-n-1)*%e^(g*u-b^2/(4*(f*u+d)))
4247
(2^(-n-1)*%e^(g*u-b^2/(4*(f*u+d)))
4227
4248
*('sum(4^j*binomial(n,j)
4228
*'sum(b^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)
4229
*(-f*u-d)^((j+i+1)/2)
4249
*'sum(b^(2*n-j-i)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
4250
*(f*u+d)^((j+i+1)/2)
4230
4251
*abs(2*(f*u+d)*sqrt(x)+b)^(-j-i-1)
4231
4252
*(2*(f*u+d)*sqrt(x)+b)^(j+i)
4232
4253
*(b*gamma_incomplete((j+i+1)/2,
4234
4255
^2
4235
4256
/(4*(f*u+d)))
4236
4257
*(2*(f*u+d)*sqrt(x)+b)
4237
+2*gamma_incomplete((j+i+2)/2,
4258
+2*%i
4259
*gamma_incomplete((j+i+2)/2,
4238
4260
-(2*(f*u+d)*sqrt(x)+b)
4239
4261
^2
4240
/(4*(f*u+d)))*(f*u+d)
4241
*abs(2*(f*u+d)*sqrt(x)+b)
4242
/sqrt(-f*u-d)),i,0,j),j,0,n))
4243
/(f*u+d)^(2*(n+1)));
4262
/(4*(f*u+d)))
4263
*sqrt(f*u+d)*abs(2*(f*u+d)*sqrt(x)+b)),
4264
i,0,j),j,0,n))
4265
/(f*u+d)^(2*(n+1))));
4244
4266
0;
4245
4267
4246
/* Problem 594: NEW 10 */
4268
/* Problem 594: NEW 5 */
4247
4269
expand(niceindices(integrate(z^n*%e^(e+d*z+b*z^2)*(%e^(g+f*z))^u,z))-
4248
-%e^(-(f*u+d)^2/(4*b)+g*u+e)*('sum(b^(-n+(i+1)/2-1/2)
4270
(-%e^(-(f*u+d)^2/(4*b)+g*u+e)*('sum(b^(-n+(i+1)/2-1/2)
4249
4271
*gamma_incomplete((i+1)/2,
4250
4272
-(2*b*z+f*u+d)^2/(4*b))
4251
4273
*2^(i-n)*binomial(n,i)*(-f*u-d)^(n-i)
4252
4274
*(2*b*z+f*u+d)^(i+1)
4253
*abs(2*b*z+f*u+d)^(-i-1)
4254
/(-1)^((i+1)/2),i,0,n))
4255
/(2*sqrt(b)));
4275
/(-(2*b*z+f*u+d)^2)^((i+1)/2),i,0,n))
4276
/(2*sqrt(b))));
4256
4277
0;
4257
4278
4258
/* Problem 595: NEW 11 */
4279
/* Problem 595: NEW 6 */
4259
4280
expand(niceindices(integrate(x^n*%e^(e+d*x+b*sqrt(x))*(%e^(g+f*x))^u,x))-
4260
2^(-n-1)*%e^(-b^2/(4*(f*u+d))+g*u+e)
4281
(2^(-n-1)*%e^(-b^2/(4*(f*u+d))+g*u+e)
4261
4282
*('sum(4^j*binomial(n,j)
4262
*'sum(b^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)
4263
*(-f*u-d)^((j+i+1)/2)
4283
*'sum(b^(2*n-j-i)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
4284
*(f*u+d)^((j+i+1)/2)
4264
4285
*abs(2*(f*u+d)*sqrt(x)+b)^(-j-i-1)
4265
4286
*(2*(f*u+d)*sqrt(x)+b)^(j+i)
4266
4287
*(b*gamma_incomplete((j+i+1)/2,
4268
4289
^2
4269
4290
/(4*(f*u+d)))
4270
4291
*(2*(f*u+d)*sqrt(x)+b)
4271
+2*gamma_incomplete((j+i+2)/2,
4292
+2*%i
4293
*gamma_incomplete((j+i+2)/2,
4272
4294
-(2*(f*u+d)*sqrt(x)+b)
4273
4295
^2
4274
/(4*(f*u+d)))*(f*u+d)
4275
*abs(2*(f*u+d)*sqrt(x)+b)
4276
/sqrt(-f*u-d)),i,0,j),j,0,n))
4277
/(f*u+d)^(2*(n+1)));
4296
/(4*(f*u+d)))
4297
*sqrt(f*u+d)*abs(2*(f*u+d)*sqrt(x)+b)),
4298
i,0,j),j,0,n))
4299
/(f*u+d)^(2*(n+1))));
4278
4300
0;
4279
4301
4280
4302
/* Problem 596: NEW 5 */
4281
4303
expand(niceindices(integrate(z^n*%e^(d*z)*(%e^(c*z^2))^u,z))-
4282
-c^(-n-1)*u^(-n-1)*%e^-(d^2/(4*c*u))
4304
(-c^(-n-1)*u^(-n-1)*%e^-(d^2/(4*c*u))
4283
4305
*('sum(d^(n-i)*gamma_incomplete((i+1)/2,-(2*c*u*z+d)^2/(4*c*u))
4284
*(-1)^(i-n)*2^(i-n)*binomial(n,i)*(-c*u)^((i+1)/2)
4285
*(2*c*u*z+d)^(i+1)*abs(2*c*u*z+d)^(-i-1),i,0,n))
4286
/2);
4306
*(-1)^(i-n)*2^(i-n)*binomial(n,i)*(2*c*u*z+d)^(i+1)
4307
/(-(2*c*u*z+d)^2/(c*u))^((i+1)/2),i,0,n))
4308
/2));
4287
4309
0;
4288
4310
4289
4311
/* Problem 597: NEW 6 */
4290
4312
expand(niceindices(integrate(x^n*%e^(d*x)*(%e^(c*sqrt(x)))^u,x))-
4291
2^(-n-1)*%e^-(c^2*u^2/(4*d))
4313
(2^(-n-1)*%e^-(c^2*u^2/(4*d))
4292
4314
*('sum(4^j*binomial(n,j)
4293
*'sum(c^(2*n-j-i)*(-d)^((j+i+1)/2)*(-1)^(i-j)*binomial(j,i)
4294
*u^(2*n-j-i)*abs(2*d*sqrt(x)+c*u)^(-j-i-1)
4315
*'sum(c^(2*n-j-i)*d^((j+i+1)/2)*(-1)^(-(j+i+1)/2+j-i)
4316
*binomial(j,i)*u^(2*n-j-i)
4317
*abs(2*d*sqrt(x)+c*u)^(-j-i-1)
4295
4318
*(2*d*sqrt(x)+c*u)^(j+i)
4296
4319
*(c*gamma_incomplete((j+i+1)/2,
4297
4320
-(2*d*sqrt(x)+c*u)^2
4298
4321
/(4*d))*u
4299
4322
*(2*d*sqrt(x)+c*u)
4300
+2*d
4323
+2*%i*sqrt(d)
4301
4324
*gamma_incomplete((j+i+2)/2,
4302
4325
-(2*d*sqrt(x)+c*u)^2
4303
4326
/(4*d))
4304
*abs(2*d*sqrt(x)+c*u)
4305
/sqrt(-d)),i,0,j),j,0,n))
4306
/d^(2*(n+1)));
4327
*abs(2*d*sqrt(x)+c*u)),i,0,j),j,0,n))
4328
/d^(2*(n+1))));
4307
4329
0;
4308
4330
4309
4331
/* Problem 598: NEW 5 */
4310
4332
expand(niceindices(integrate(z^n*%e^(e+d*z)*(%e^(c*z^2))^u,z))-
4311
-c^(-n-1)*u^(-n-1)*%e^(e-d^2/(4*c*u))
4333
(-c^(-n-1)*u^(-n-1)*%e^(e-d^2/(4*c*u))
4312
4334
*('sum(d^(n-i)*gamma_incomplete((i+1)/2,-(2*c*u*z+d)^2/(4*c*u))
4313
*(-1)^(i-n)*2^(i-n)*binomial(n,i)*(-c*u)^((i+1)/2)
4314
*(2*c*u*z+d)^(i+1)*abs(2*c*u*z+d)^(-i-1),i,0,n))
4315
/2);
4335
*(-1)^(i-n)*2^(i-n)*binomial(n,i)*(2*c*u*z+d)^(i+1)
4336
/(-(2*c*u*z+d)^2/(c*u))^((i+1)/2),i,0,n))
4337
/2));
4316
4338
0;
4317
4339
4318
4340
/* Problem 599: NEW 6 */
4319
4341
expand(niceindices(integrate(x^n*%e^(e+d*x)*(%e^(c*sqrt(x)))^u,x))-
4320
2^(-n-1)*%e^(e-c^2*u^2/(4*d))
4342
(2^(-n-1)*%e^(e-c^2*u^2/(4*d))
4321
4343
*('sum(4^j*binomial(n,j)
4322
*'sum(c^(2*n-j-i)*(-d)^((j+i+1)/2)*(-1)^(i-j)*binomial(j,i)
4323
*u^(2*n-j-i)*abs(2*d*sqrt(x)+c*u)^(-j-i-1)
4344
*'sum(c^(2*n-j-i)*d^((j+i+1)/2)*(-1)^(-(j+i+1)/2+j-i)
4345
*binomial(j,i)*u^(2*n-j-i)
4346
*abs(2*d*sqrt(x)+c*u)^(-j-i-1)
4324
4347
*(2*d*sqrt(x)+c*u)^(j+i)
4325
4348
*(c*gamma_incomplete((j+i+1)/2,
4326
4349
-(2*d*sqrt(x)+c*u)^2
4327
4350
/(4*d))*u
4328
4351
*(2*d*sqrt(x)+c*u)
4329
+2*d
4352
+2*%i*sqrt(d)
4330
4353
*gamma_incomplete((j+i+2)/2,
4331
4354
-(2*d*sqrt(x)+c*u)^2
4332
4355
/(4*d))
4333
*abs(2*d*sqrt(x)+c*u)
4334
/sqrt(-d)),i,0,j),j,0,n))
4335
/d^(2*(n+1)));
4356
*abs(2*d*sqrt(x)+c*u)),i,0,j),j,0,n))
4357
/d^(2*(n+1))));
4336
4358
0;
4337
4359
4338
4360
/* Problem 600: NEW */
4340
4362
-gamma_incomplete(v/r,-(c*u+b)*z^r)*z^v/(r*(-(c*u+b)*z^r)^(v/r)));
4341
4363
0;
4342
4364
4343
/* Problem 601: NEW */
4365
/* Problem 601: NEW 2 */
4344
4366
expand(integrate(z^n*%e^(b*z^2)*(%e^(c*z^2))^u,z)-
4345
-gamma_incomplete((n+1)/2,-(c*u+b)*z^2)*z^(n+1)
4346
/(2*(-c*u-b)^((n+1)/2)*(z^2)^((n+1)/2)));
4367
(-gamma_incomplete((n+1)/2,-(c*u+b)*z^2)*z^(n+1)/(2*(-(c*u+b)*z^2)^((n+1)/2))));
4347
4368
0;
4348
4369
4349
/* Problem 602: NEW */
4370
/* Problem 602: NEW 2 */
4350
4371
expand(integrate(z^(2*n)*%e^(b*z^2)*(%e^(c*z^2))^u,z)-
4351
-gamma_incomplete((2*n+1)/2,-(c*u+b)*z^2)*z^(2*n+1)
4352
/(2*(-c*u-b)^((2*n+1)/2)*(z^2)^((2*n+1)/2)));
4372
(-gamma_incomplete((2*n+1)/2,-(c*u+b)*z^2)*z^(2*n+1)
4373
/(2*(-(c*u+b)*z^2)^((2*n+1)/2))));
4353
4374
0;
4354
4375
4355
4376
/* Problem 603: NEW */
4368
4389
-%e^e*gamma_incomplete(v/r,-(c*u+b)*z^r)*z^v/(r*(-(c*u+b)*z^r)^(v/r)));
4369
4390
0;
4370
4391
4371
/* Problem 606: NEW */
4392
/* Problem 606: NEW 2 */
4372
4393
expand(integrate(z^n*%e^(e+b*z^2)*(%e^(c*z^2))^u,z)-
4373
-%e^e*gamma_incomplete((n+1)/2,-(c*u+b)*z^2)*z^(n+1)
4374
/(2*(-c*u-b)^((n+1)/2)*(z^2)^((n+1)/2)));
4394
(-%e^e*gamma_incomplete((n+1)/2,-(c*u+b)*z^2)*z^(n+1)
4395
/(2*(-(c*u+b)*z^2)^((n+1)/2))));
4375
4396
0;
4376
4397
4377
/* Problem 607: NEW */
4398
/* Problem 607: NEW 2 */
4378
4399
expand(integrate(z^(2*n)*%e^(e+b*z^2)*(%e^(c*z^2))^u,z)-
4379
-%e^e*gamma_incomplete((2*n+1)/2,-(c*u+b)*z^2)*z^(2*n+1)
4380
/(2*(-c*u-b)^((2*n+1)/2)*(z^2)^((2*n+1)/2)));
4400
(-%e^e*gamma_incomplete((2*n+1)/2,-(c*u+b)*z^2)*z^(2*n+1)
4401
/(2*(-(c*u+b)*z^2)^((2*n+1)/2))));
4381
4402
0;
4382
4403
4383
4404
/* Problem 608: NEW */
4391
4412
-2*%e^e*gamma_incomplete(2*(n+1),-(c*u+b)*sqrt(x))/(c*u+b)^(2*(n+1)));
4392
4413
0;
4393
4414
4394
/* Problem 610: NEW 10 */
4415
/* Problem 610: NEW 5 */
4395
4416
expand(niceindices(integrate(z^n*%e^(d*z+b*z^2)*(%e^(c*z^2))^u,z))-
4396
-(c*u+b)^(-n-1)*%e^-(d^2/(4*(c*u+b)))
4417
(-(c*u+b)^(-n-1)*%e^-(d^2/(4*(c*u+b)))
4397
4418
*('sum(d^(n-i)*gamma_incomplete((i+1)/2,
4398
4419
-(2*(c*u+b)*z+d)^2/(4*(c*u+b)))
4399
4420
*(-1)^(i-n)*2^(i-n)*binomial(n,i)
4400
*(-c*u-b)^((i+1)/2)*(2*(c*u+b)*z+d)^(i+1)
4401
*abs(2*(c*u+b)*z+d)^(-i-1),i,0,n))
4402
/2);
4421
*(2*(c*u+b)*z+d)^(i+1)
4422
/(-(2*(c*u+b)*z+d)^2/(c*u+b))^((i+1)/2),i,0,n))
4423
/2));
4403
4424
0;
4404
4425
4405
/* Problem 611: NEW 11 */
4426
/* Problem 611: NEW 6 */
4406
4427
expand(niceindices(integrate(x^n*%e^(d*x+b*sqrt(x))*(%e^(c*sqrt(x)))^u,x))-
4407
2^(-n-1)*%e^-((c*u+b)^2/(4*d))
4428
(2^(-n-1)*%e^-((c*u+b)^2/(4*d))
4408
4429
*('sum(4^j*binomial(n,j)
4409
*'sum((-d)^((j+i+1)/2)*(-1)^(i-j)*binomial(j,i)
4410
*(c*u+b)^(2*n-j-i)
4411
*abs(2*d*sqrt(x)+c*u+b)^(-j-i-1)
4412
*(2*d*sqrt(x)+c*u+b)^(j+i)
4413
*(gamma_incomplete((j+i+1)/2,
4414
-(2*d*sqrt(x)
4415
+c*u+b)
4416
^2
4417
/(4*d))
4418
*(c*u+b)*(2*d*sqrt(x)+c*u+b)
4419
+2*d
4420
*gamma_incomplete(
4421
(j+i+2)/2,
4422
-(2*d*sqrt(x)+c*u+b)^2/(4*d))
4423
*abs(2*d*sqrt(x)+c*u+b)
4424
/sqrt(-d)),i,0,j),j,0,n))
4425
/d^(2*(n+1)));
4430
*'sum(d^((j+i+1)/2)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
4431
*(c*u+b)^(2*n-j-i)
4432
*abs(2*d*sqrt(x)+c*u+b)^(-j-i-1)
4433
*(2*d*sqrt(x)+c*u+b)^(j+i)
4434
*(gamma_incomplete((j+i+1)/2,
4435
-(2*d*sqrt(x)+c*u+b)^2
4436
/(4*d))
4437
*(c*u+b)*(2*d*sqrt(x)+c*u+b)
4438
+2*%i*sqrt(d)
4439
*gamma_incomplete((j+i+2)/2,
4440
-(2*d*sqrt(x)+c*u+b)
4441
^2
4442
/(4*d))
4443
*abs(2*d*sqrt(x)+c*u+b)),i,0,j),j,0,
4444
n))
4445
/d^(2*(n+1))));
4426
4446
0;
4427
4447
4428
/* Problem 612: NEW 10 */
4448
/* Problem 612: NEW 5 */
4429
4449
expand(niceindices(integrate(z^n*%e^(e+d*z+b*z^2)*(%e^(c*z^2))^u,z))-
4430
-(c*u+b)^(-n-1)*%e^(e-d^2/(4*(c*u+b)))
4450
(-(c*u+b)^(-n-1)*%e^(e-d^2/(4*(c*u+b)))
4431
4451
*('sum(d^(n-i)*gamma_incomplete((i+1)/2,
4432
4452
-(2*(c*u+b)*z+d)^2/(4*(c*u+b)))
4433
4453
*(-1)^(i-n)*2^(i-n)*binomial(n,i)
4434
*(-c*u-b)^((i+1)/2)*(2*(c*u+b)*z+d)^(i+1)
4435
*abs(2*(c*u+b)*z+d)^(-i-1),i,0,n))
4436
/2);
4454
*(2*(c*u+b)*z+d)^(i+1)
4455
/(-(2*(c*u+b)*z+d)^2/(c*u+b))^((i+1)/2),i,0,n))
4456
/2));
4437
4457
0;
4438
4458
4439
/* Problem 613: NEW */
4459
/* Problem 613: NEW 6 */
4440
4460
expand(niceindices(integrate(x^n*%e^(e+d*x+b*sqrt(x))*(%e^(c*sqrt(x)))^u,x))-
4441
2^(-n-1)*%e^(e-(c*u+b)^2/(4*d))
4461
(2^(-n-1)*%e^(e-(c*u+b)^2/(4*d))
4442
4462
*('sum(4^j*binomial(n,j)
4443
*'sum((-d)^((j+i+1)/2)*(-1)^(i-j)*binomial(j,i)
4444
*(c*u+b)^(2*n-j-i)
4445
*abs(2*d*sqrt(x)+c*u+b)^(-j-i-1)
4446
*(2*d*sqrt(x)+c*u+b)^(j+i)
4447
*(gamma_incomplete((j+i+1)/2,
4448
-(2*d*sqrt(x)
4449
+c*u+b)
4450
^2
4451
/(4*d))
4452
*(c*u+b)*(2*d*sqrt(x)+c*u+b)
4453
+2*d
4454
*gamma_incomplete(
4455
(j+i+2)/2,
4456
-(2*d*sqrt(x)+c*u+b)^2/(4*d))
4457
*abs(2*d*sqrt(x)+c*u+b)
4458
/sqrt(-d)),i,0,j),j,0,n))
4459
/d^(2*(n+1)));
4463
*'sum(d^((j+i+1)/2)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
4464
*(c*u+b)^(2*n-j-i)
4465
*abs(2*d*sqrt(x)+c*u+b)^(-j-i-1)
4466
*(2*d*sqrt(x)+c*u+b)^(j+i)
4467
*(gamma_incomplete((j+i+1)/2,
4468
-(2*d*sqrt(x)+c*u+b)^2
4469
/(4*d))
4470
*(c*u+b)*(2*d*sqrt(x)+c*u+b)
4471
+2*%i*sqrt(d)
4472
*gamma_incomplete((j+i+2)/2,
4473
-(2*d*sqrt(x)+c*u+b)
4474
^2
4475
/(4*d))
4476
*abs(2*d*sqrt(x)+c*u+b)),i,0,j),j,0,
4477
n))
4478
/d^(2*(n+1))));
4460
4479
0;
4461
4480
4462
4481
/* Problem 614: NEW 5 */
4463
4482
expand(niceindices(integrate(z^n*%e^(d*z)*(%e^(g+c*z^2))^u,z))-
4464
-c^(-n-1)*u^(-n-1)*%e^(g*u-d^2/(4*c*u))
4483
(-c^(-n-1)*u^(-n-1)*%e^(g*u-d^2/(4*c*u))
4465
4484
*('sum(d^(n-i)*gamma_incomplete((i+1)/2,-(2*c*u*z+d)^2/(4*c*u))
4466
*(-1)^(i-n)*2^(i-n)*binomial(n,i)*(-c*u)^((i+1)/2)
4467
*(2*c*u*z+d)^(i+1)*abs(2*c*u*z+d)^(-i-1),i,0,n))
4468
/2);
4485
*(-1)^(i-n)*2^(i-n)*binomial(n,i)*(2*c*u*z+d)^(i+1)
4486
/(-(2*c*u*z+d)^2/(c*u))^((i+1)/2),i,0,n))
4487
/2));
4469
4488
0;
4470
4489
4471
4490
/* Problem 615: NEW 6 */
4472
4491
expand(niceindices(integrate(x^n*%e^(d*x)*(%e^(g+c*sqrt(x)))^u,x))-
4473
2^(-n-1)*%e^(g*u-c^2*u^2/(4*d))
4492
(2^(-n-1)*%e^(g*u-c^2*u^2/(4*d))
4474
4493
*('sum(4^j*binomial(n,j)
4475
*'sum(c^(2*n-j-i)*(-d)^((j+i+1)/2)*(-1)^(i-j)*binomial(j,i)
4476
*u^(2*n-j-i)*abs(2*d*sqrt(x)+c*u)^(-j-i-1)
4494
*'sum(c^(2*n-j-i)*d^((j+i+1)/2)*(-1)^(-(j+i+1)/2+j-i)
4495
*binomial(j,i)*u^(2*n-j-i)
4496
*abs(2*d*sqrt(x)+c*u)^(-j-i-1)
4477
4497
*(2*d*sqrt(x)+c*u)^(j+i)
4478
4498
*(c*gamma_incomplete((j+i+1)/2,
4479
4499
-(2*d*sqrt(x)+c*u)^2
4480
4500
/(4*d))*u
4481
4501
*(2*d*sqrt(x)+c*u)
4482
+2*d
4502
+2*%i*sqrt(d)
4483
4503
*gamma_incomplete((j+i+2)/2,
4484
4504
-(2*d*sqrt(x)+c*u)^2
4485
4505
/(4*d))
4486
*abs(2*d*sqrt(x)+c*u)
4487
/sqrt(-d)),i,0,j),j,0,n))
4488
/d^(2*(n+1)));
4506
*abs(2*d*sqrt(x)+c*u)),i,0,j),j,0,n))
4507
/d^(2*(n+1))));
4489
4508
0;
4490
4509
4491
4510
/* Problem 616: NEW 5 */
4492
4511
expand(niceindices(integrate(z^n*%e^(e+d*z)*(%e^(g+c*z^2))^u,z))-
4493
-c^(-n-1)*u^(-n-1)*%e^(g*u-d^2/(4*c*u)+e)
4512
(-c^(-n-1)*u^(-n-1)*%e^(g*u-d^2/(4*c*u)+e)
4494
4513
*('sum(d^(n-i)*gamma_incomplete((i+1)/2,-(2*c*u*z+d)^2/(4*c*u))
4495
*(-1)^(i-n)*2^(i-n)*binomial(n,i)*(-c*u)^((i+1)/2)
4496
*(2*c*u*z+d)^(i+1)*abs(2*c*u*z+d)^(-i-1),i,0,n))
4497
/2);
4514
*(-1)^(i-n)*2^(i-n)*binomial(n,i)*(2*c*u*z+d)^(i+1)
4515
/(-(2*c*u*z+d)^2/(c*u))^((i+1)/2),i,0,n))
4516
/2));
4498
4517
0;
4499
4518
4500
4519
/* Problem 617: NEW 6 */
4501
4520
expand(niceindices(integrate(x^n*%e^(e+d*x)*(%e^(g+c*sqrt(x)))^u,x))-
4502
2^(-n-1)*%e^(-c^2*u^2/(4*d)+g*u+e)
4521
(2^(-n-1)*%e^(-c^2*u^2/(4*d)+g*u+e)
4503
4522
*('sum(4^j*binomial(n,j)
4504
*'sum(c^(2*n-j-i)*(-d)^((j+i+1)/2)*(-1)^(i-j)*binomial(j,i)
4505
*u^(2*n-j-i)*abs(2*d*sqrt(x)+c*u)^(-j-i-1)
4523
*'sum(c^(2*n-j-i)*d^((j+i+1)/2)*(-1)^(-(j+i+1)/2+j-i)
4524
*binomial(j,i)*u^(2*n-j-i)
4525
*abs(2*d*sqrt(x)+c*u)^(-j-i-1)
4506
4526
*(2*d*sqrt(x)+c*u)^(j+i)
4507
4527
*(c*gamma_incomplete((j+i+1)/2,
4508
4528
-(2*d*sqrt(x)+c*u)^2
4509
4529
/(4*d))*u
4510
4530
*(2*d*sqrt(x)+c*u)
4511
+2*d
4531
+2*%i*sqrt(d)
4512
4532
*gamma_incomplete((j+i+2)/2,
4513
4533
-(2*d*sqrt(x)+c*u)^2
4514
4534
/(4*d))
4515
*abs(2*d*sqrt(x)+c*u)
4516
/sqrt(-d)),i,0,j),j,0,n))
4517
/d^(2*(n+1)));
4535
*abs(2*d*sqrt(x)+c*u)),i,0,j),j,0,n))
4536
/d^(2*(n+1))));
4518
4537
0;
4519
4538
4520
4539
/* Problem 618: NEW */
4522
4541
-%e^(g*u)*gamma_incomplete(v/r,-(c*u+b)*z^r)*z^v/(r*(-(c*u+b)*z^r)^(v/r)));
4523
4542
0;
4524
4543
4525
/* Problem 619: NEW */
4544
/* Problem 619: NEW 2 */
4526
4545
expand(integrate(z^n*%e^(b*z^2)*(%e^(g+c*z^2))^u,z)-
4527
-gamma_incomplete((n+1)/2,-(c*u+b)*z^2)*%e^(g*u)*z^(n+1)
4528
/(2*(-c*u-b)^((n+1)/2)*(z^2)^((n+1)/2)));
4546
(-gamma_incomplete((n+1)/2,-(c*u+b)*z^2)*%e^(g*u)*z^(n+1)
4547
/(2*(-(c*u+b)*z^2)^((n+1)/2))));
4529
4548
0;
4530
4549
4531
/* Problem 620: NEW */
4550
/* Problem 620: NEW 2 */
4532
4551
expand(integrate(z^(2*n)*%e^(b*z^2)*(%e^(g+c*z^2))^u,z)-
4533
-gamma_incomplete((2*n+1)/2,-(c*u+b)*z^2)*%e^(g*u)*z^(2*n+1)
4534
/(2*(-c*u-b)^((2*n+1)/2)*(z^2)^((2*n+1)/2)));
4552
(-gamma_incomplete((2*n+1)/2,-(c*u+b)*z^2)*%e^(g*u)*z^(2*n+1)
4553
/(2*(-(c*u+b)*z^2)^((2*n+1)/2))));
4535
4554
0;
4536
4555
4537
4556
/* Problem 621: NEW */
4550
4569
-%e^(g*u+e)*gamma_incomplete(v/r,-(c*u+b)*z^r)*z^v/(r*(-(c*u+b)*z^r)^(v/r)));
4551
4570
0;
4552
4571
4553
/* Problem 624: NEW */
4572
/* Problem 624: NEW 2 */
4554
4573
expand(integrate(z^n*%e^(e+b*z^2)*(%e^(g+c*z^2))^u,z)-
4555
-gamma_incomplete((n+1)/2,-(c*u+b)*z^2)*%e^(g*u+e)*z^(n+1)
4556
/(2*(-c*u-b)^((n+1)/2)*(z^2)^((n+1)/2)));
4574
(-gamma_incomplete((n+1)/2,-(c*u+b)*z^2)*%e^(g*u+e)*z^(n+1)
4575
/(2*(-(c*u+b)*z^2)^((n+1)/2))));
4557
4576
0;
4558
4577
4559
/* Problem 625: NEW */
4578
/* Problem 625: NEW 2 */
4560
4579
expand(integrate(z^(2*n)*%e^(e+b*z^2)*(%e^(g+c*z^2))^u,z)-
4561
-gamma_incomplete((2*n+1)/2,-(c*u+b)*z^2)*%e^(g*u+e)*z^(2*n+1)
4562
/(2*(-c*u-b)^((2*n+1)/2)*(z^2)^((2*n+1)/2)));
4580
(-gamma_incomplete((2*n+1)/2,-(c*u+b)*z^2)*%e^(g*u+e)*z^(2*n+1)
4581
/(2*(-(c*u+b)*z^2)^((2*n+1)/2))));
4563
4582
0;
4564
4583
4565
4584
/* Problem 626: NEW */
4573
4592
-2*gamma_incomplete(2*(n+1),-(c*u+b)*sqrt(x))*%e^(g*u+e)/(c*u+b)^(2*(n+1)));
4574
4593
0;
4575
4594
4576
/* Problem 628: NEW 10 */
4595
/* Problem 628: NEW 5 */
4577
4596
expand(niceindices(integrate(z^n*%e^(d*z+b*z^2)*(%e^(g+c*z^2))^u,z))-
4578
-(c*u+b)^(-n-1)*%e^(g*u-d^2/(4*(c*u+b)))
4597
(-(c*u+b)^(-n-1)*%e^(g*u-d^2/(4*(c*u+b)))
4579
4598
*('sum(d^(n-i)*gamma_incomplete((i+1)/2,
4580
4599
-(2*(c*u+b)*z+d)^2/(4*(c*u+b)))
4581
4600
*(-1)^(i-n)*2^(i-n)*binomial(n,i)
4582
*(-c*u-b)^((i+1)/2)*(2*(c*u+b)*z+d)^(i+1)
4583
*abs(2*(c*u+b)*z+d)^(-i-1),i,0,n))
4584
/2);
4601
*(2*(c*u+b)*z+d)^(i+1)
4602
/(-(2*(c*u+b)*z+d)^2/(c*u+b))^((i+1)/2),i,0,n))
4603
/2));
4585
4604
0;
4586
4605
4587
/* Problem 629: NEW 11 */
4606
/* Problem 629: NEW 6 */
4588
4607
expand(niceindices(integrate(x^n*%e^(d*x+b*sqrt(x))*(%e^(g+c*sqrt(x)))^u,x))-
4589
2^(-n-1)*%e^(g*u-(c*u+b)^2/(4*d))
4608
(2^(-n-1)*%e^(g*u-(c*u+b)^2/(4*d))
4590
4609
*('sum(4^j*binomial(n,j)
4591
*'sum((-d)^((j+i+1)/2)*(-1)^(i-j)*binomial(j,i)
4592
*(c*u+b)^(2*n-j-i)
4593
*abs(2*d*sqrt(x)+c*u+b)^(-j-i-1)
4594
*(2*d*sqrt(x)+c*u+b)^(j+i)
4595
*(gamma_incomplete((j+i+1)/2,
4596
-(2*d*sqrt(x)
4597
+c*u+b)
4598
^2
4599
/(4*d))
4600
*(c*u+b)*(2*d*sqrt(x)+c*u+b)
4601
+2*d
4602
*gamma_incomplete(
4603
(j+i+2)/2,
4604
-(2*d*sqrt(x)+c*u+b)^2/(4*d))
4605
*abs(2*d*sqrt(x)+c*u+b)
4606
/sqrt(-d)),i,0,j),j,0,n))
4607
/d^(2*(n+1)));
4610
*'sum(d^((j+i+1)/2)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
4611
*(c*u+b)^(2*n-j-i)
4612
*abs(2*d*sqrt(x)+c*u+b)^(-j-i-1)
4613
*(2*d*sqrt(x)+c*u+b)^(j+i)
4614
*(gamma_incomplete((j+i+1)/2,
4615
-(2*d*sqrt(x)+c*u+b)^2
4616
/(4*d))
4617
*(c*u+b)*(2*d*sqrt(x)+c*u+b)
4618
+2*%i*sqrt(d)
4619
*gamma_incomplete((j+i+2)/2,
4620
-(2*d*sqrt(x)+c*u+b)
4621
^2
4622
/(4*d))
4623
*abs(2*d*sqrt(x)+c*u+b)),i,0,j),j,0,
4624
n))
4625
/d^(2*(n+1))));
4608
4626
0;
4609
4627
4610
/* Problem 630: NEW 10 */
4628
/* Problem 630: NEW 5 */
4611
4629
expand(niceindices(integrate(z^n*%e^(e+d*z+b*z^2)*(%e^(g+c*z^2))^u,z))-
4612
-(c*u+b)^(-n-1)*%e^(-d^2/(4*(c*u+b))+g*u+e)
4630
(-(c*u+b)^(-n-1)*%e^(-d^2/(4*(c*u+b))+g*u+e)
4613
4631
*('sum(d^(n-i)*gamma_incomplete((i+1)/2,
4614
4632
-(2*(c*u+b)*z+d)^2/(4*(c*u+b)))
4615
4633
*(-1)^(i-n)*2^(i-n)*binomial(n,i)
4616
*(-c*u-b)^((i+1)/2)*(2*(c*u+b)*z+d)^(i+1)
4617
*abs(2*(c*u+b)*z+d)^(-i-1),i,0,n))
4618
/2);
4634
*(2*(c*u+b)*z+d)^(i+1)
4635
/(-(2*(c*u+b)*z+d)^2/(c*u+b))^((i+1)/2),i,0,n))
4636
/2));
4619
4637
0;
4620
4638
4621
/* Problem 631: NEW 11 */
4639
/* Problem 631: NEW 6 */
4622
4640
expand(niceindices(integrate(x^n*%e^(e+d*x+b*sqrt(x))*(%e^(g+c*sqrt(x)))^u,x))-
4623
4641
2^(-n-1)*%e^(-(c*u+b)^2/(4*d)+g*u+e)
4624
4642
*('sum(4^j*binomial(n,j)
4625
*'sum((-d)^((j+i+1)/2)*(-1)^(i-j)*binomial(j,i)
4626
*(c*u+b)^(2*n-j-i)
4627
*abs(2*d*sqrt(x)+c*u+b)^(-j-i-1)
4628
*(2*d*sqrt(x)+c*u+b)^(j+i)
4629
*(gamma_incomplete((j+i+1)/2,
4630
-(2*d*sqrt(x)
4631
+c*u+b)
4632
^2
4633
/(4*d))
4634
*(c*u+b)*(2*d*sqrt(x)+c*u+b)
4635
+2*d
4636
*gamma_incomplete(
4637
(j+i+2)/2,
4638
-(2*d*sqrt(x)+c*u+b)^2/(4*d))
4639
*abs(2*d*sqrt(x)+c*u+b)
4640
/sqrt(-d)),i,0,j),j,0,n))
4643
*'sum(d^((j+i+1)/2)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
4644
*(c*u+b)^(2*n-j-i)
4645
*abs(2*d*sqrt(x)+c*u+b)^(-j-i-1)
4646
*(2*d*sqrt(x)+c*u+b)^(j+i)
4647
*(gamma_incomplete((j+i+1)/2,
4648
-(2*d*sqrt(x)+c*u+b)^2
4649
/(4*d))
4650
*(c*u+b)*(2*d*sqrt(x)+c*u+b)
4651
+2*%i*sqrt(d)
4652
*gamma_incomplete((j+i+2)/2,
4653
-(2*d*sqrt(x)+c*u+b)
4654
^2
4655
/(4*d))
4656
*abs(2*d*sqrt(x)+c*u+b)),i,0,j),j,0,
4657
n))
4641
4658
/d^(2*(n+1)));
4642
4659
0;
4643
4660
4644
/* Problem 632: NEW 10 */
4661
/* Problem 632: NEW 5 */
4645
4662
expand(niceindices(integrate(z^n*%e^(d*z)*(%e^(f*z+c*z^2))^u,z))-
4646
-c^(-n-1)*u^(-n-1)*%e^-((f*u+d)^2/(4*c*u))
4663
(-c^(-n-1)*u^(-n-1)*%e^-((f*u+d)^2/(4*c*u))
4647
4664
*('sum(gamma_incomplete((i+1)/2,-(2*c*u*z+f*u+d)^2/(4*c*u))
4648
*2^(i-n)*binomial(n,i)*(-c*u)^((i+1)/2)*(-f*u-d)^(n-i)
4649
*(2*c*u*z+f*u+d)^(i+1)*abs(2*c*u*z+f*u+d)^(-i-1),i,0,n))
4650
/2);
4665
*2^(i-n)*binomial(n,i)*(-f*u-d)^(n-i)*(2*c*u*z+f*u+d)^(i+1)
4666
/(-(2*c*u*z+f*u+d)^2/(c*u))^((i+1)/2),i,0,n))
4667
/2));
4651
4668
0;
4652
4669
4653
/* Problem 633: NEW 11 */
4670
/* Problem 633: NEW 6 */
4654
4671
expand(niceindices(integrate(x^n*%e^(d*x)*(%e^(f*x+c*sqrt(x)))^u,x))-
4655
2^(-n-1)*%e^-(c^2*u^2/(4*(f*u+d)))
4672
(2^(-n-1)*%e^-(c^2*u^2/(4*(f*u+d)))
4656
4673
*('sum(4^j*binomial(n,j)
4657
*'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i)
4658
*(-f*u-d)^((j+i+1)/2)
4674
*'sum(c^(2*n-j-i)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
4675
*u^(2*n-j-i)*(f*u+d)^((j+i+1)/2)
4659
4676
*abs(2*(f*u+d)*sqrt(x)+c*u)^(-j-i-1)
4660
4677
*(2*(f*u+d)*sqrt(x)+c*u)^(j+i)
4661
4678
*(c*gamma_incomplete((j+i+1)/2,
4664
4681
^2
4665
4682
/(4*(f*u+d)))*u
4666
4683
*(2*(f*u+d)*sqrt(x)+c*u)
4667
+2*gamma_incomplete((j+i+2)/2,
4684
+2*%i
4685
*gamma_incomplete((j+i+2)/2,
4668
4686
-(2*(f*u+d)*sqrt(x)
4669
4687
+c*u)
4670
4688
^2
4671
/(4*(f*u+d)))*(f*u+d)
4672
*abs(2*(f*u+d)*sqrt(x)+c*u)
4673
/sqrt(-f*u-d)),i,0,j),j,0,n))
4674
/(f*u+d)^(2*(n+1)));
4689
/(4*(f*u+d)))
4690
*sqrt(f*u+d)
4691
*abs(2*(f*u+d)*sqrt(x)+c*u)),i,0,j),j,0,
4692
n))
4693
/(f*u+d)^(2*(n+1))));
4675
4694
0;
4676
4695
4677
/* Problem 634: NEW 10 */
4696
/* Problem 634: NEW 5 */
4678
4697
expand(niceindices(integrate(z^n*%e^(e+d*z)*(%e^(f*z+c*z^2))^u,z))-
4679
-c^(-n-1)*u^(-n-1)*%e^(e-(f*u+d)^2/(4*c*u))
4698
(-c^(-n-1)*u^(-n-1)*%e^(e-(f*u+d)^2/(4*c*u))
4680
4699
*('sum(gamma_incomplete((i+1)/2,-(2*c*u*z+f*u+d)^2/(4*c*u))
4681
*2^(i-n)*binomial(n,i)*(-c*u)^((i+1)/2)*(-f*u-d)^(n-i)
4682
*(2*c*u*z+f*u+d)^(i+1)*abs(2*c*u*z+f*u+d)^(-i-1),i,0,n))
4683
/2);
4700
*2^(i-n)*binomial(n,i)*(-f*u-d)^(n-i)*(2*c*u*z+f*u+d)^(i+1)
4701
/(-(2*c*u*z+f*u+d)^2/(c*u))^((i+1)/2),i,0,n))
4702
/2));
4684
4703
0;
4685
4704
4686
/* Problem 635: NEW 11 */
4705
/* Problem 635: NEW 6 */
4687
4706
expand(niceindices(integrate(x^n*%e^(e+d*x)*(%e^(f*x+c*sqrt(x)))^u,x))-
4688
2^(-n-1)*%e^(e-c^2*u^2/(4*(f*u+d)))
4707
(2^(-n-1)*%e^(e-c^2*u^2/(4*(f*u+d)))
4689
4708
*('sum(4^j*binomial(n,j)
4690
*'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i)
4691
*(-f*u-d)^((j+i+1)/2)
4709
*'sum(c^(2*n-j-i)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
4710
*u^(2*n-j-i)*(f*u+d)^((j+i+1)/2)
4692
4711
*abs(2*(f*u+d)*sqrt(x)+c*u)^(-j-i-1)
4693
4712
*(2*(f*u+d)*sqrt(x)+c*u)^(j+i)
4694
4713
*(c*gamma_incomplete((j+i+1)/2,
4697
4716
^2
4698
4717
/(4*(f*u+d)))*u
4699
4718
*(2*(f*u+d)*sqrt(x)+c*u)
4700
+2*gamma_incomplete((j+i+2)/2,
4719
+2*%i
4720
*gamma_incomplete((j+i+2)/2,
4701
4721
-(2*(f*u+d)*sqrt(x)
4702
4722
+c*u)
4703
4723
^2
4704
/(4*(f*u+d)))*(f*u+d)
4705
*abs(2*(f*u+d)*sqrt(x)+c*u)
4706
/sqrt(-f*u-d)),i,0,j),j,0,n))
4707
/(f*u+d)^(2*(n+1)));
4724
/(4*(f*u+d)))
4725
*sqrt(f*u+d)
4726
*abs(2*(f*u+d)*sqrt(x)+c*u)),i,0,j),j,0,
4727
n))
4728
/(f*u+d)^(2*(n+1))));
4708
4729
0;
4709
4730
4710
/* Problem 636: NEW 10 */
4731
/* Problem 636: NEW 5 */
4711
4732
expand(niceindices(integrate(z^n*%e^(b*z^2)*(%e^(f*z+c*z^2))^u,z))-
4712
-(c*u+b)^(-n-1)*%e^-(f^2*u^2/(4*(c*u+b)))
4733
(-(c*u+b)^(-n-1)*%e^-(f^2*u^2/(4*(c*u+b)))
4713
4734
*('sum(f^(n-i)*gamma_incomplete((i+1)/2,
4714
4735
-(2*(c*u+b)*z+f*u)^2
4715
4736
/(4*(c*u+b)))*(-1)^(i-n)
4716
*2^(i-n)*binomial(n,i)*u^(n-i)*(-c*u-b)^((i+1)/2)
4737
*2^(i-n)*binomial(n,i)*u^(n-i)
4717
4738
*(2*(c*u+b)*z+f*u)^(i+1)
4718
*abs(2*(c*u+b)*z+f*u)^(-i-1),i,0,n))
4719
/2);
4739
/(-(2*(c*u+b)*z+f*u)^2/(c*u+b))^((i+1)/2),i,0,n))
4740
/2));
4720
4741
0;
4721
4742
4722
/* Problem 637: NEW 11 */
4743
/* Problem 637: NEW 6 */
4723
4744
expand(niceindices(integrate(x^n*%e^(b*sqrt(x))*(%e^(f*x+c*sqrt(x)))^u,x))-
4724
2^(-n-1)*%e^-((c*u+b)^2/(4*f*u))
4745
(2^(-n-1)*%e^-((c*u+b)^2/(4*f*u))
4725
4746
*('sum(4^j*binomial(n,j)
4726
*'sum((-1)^(i-j)*binomial(j,i)*(-f*u)^((j+i+1)/2)
4727
*(c*u+b)^(2*n-j-i)
4728
*abs(2*f*u*sqrt(x)+c*u+b)^(-j-i-1)
4729
*(2*f*u*sqrt(x)+c*u+b)^(j+i)
4730
*(gamma_incomplete((j+i+1)/2,
4731
-(2*f*u*sqrt(x)+c*u+b)^2
4732
/(4*f*u))
4733
*(c*u+b)*(2*f*u*sqrt(x)+c*u+b)
4734
+2*f
4735
*gamma_incomplete((j+i+2)/2,
4736
-(2*f*u*sqrt(x)+c*u+b)
4737
^2
4738
/(4*f*u))*u
4739
*abs(2*f*u*sqrt(x)+c*u+b)
4740
/sqrt(-f*u)),i,0,j),j,0,n))
4741
/(f^(2*(n+1))*u^(2*(n+1))));
4747
*'sum((-1)^(-(j+i+1)/2+j-i)*binomial(j,i)*(f*u)^((j+i+1)/2)
4748
*(c*u+b)^(2*n-j-i)
4749
*abs(2*f*u*sqrt(x)+c*u+b)
4750
^(-j-i-1)
4751
*(2*f*u*sqrt(x)+c*u+b)^(j+i)
4752
*(gamma_incomplete(
4753
(j+i+1)/2,
4754
-(2*f*u*sqrt(x)+c*u+b)^2
4755
/(4*f*u))
4756
*(c*u+b)*(2*f*u*sqrt(x)+c*u+b)
4757
+2*%i*f
4758
*gamma_incomplete(
4759
(j+i+2)/2,
4760
-(2*f*u*sqrt(x)+c*u+b)^2
4761
/(4*f*u))*u
4762
*abs(2*f*u*sqrt(x)+c*u+b)
4763
/sqrt(f*u)),i,0,j),j,0,n))
4764
/(f^(2*(n+1))*u^(2*(n+1)))));
4742
4765
0;
4743
4766
4744
/* Problem 638: NEW 10 */
4767
/* Problem 638: NEW 5 */
4745
4768
expand(niceindices(integrate(z^n*%e^(e+b*z^2)*(%e^(f*z+c*z^2))^u,z))-
4746
-(c*u+b)^(-n-1)*%e^(e-f^2*u^2/(4*(c*u+b)))
4769
(-(c*u+b)^(-n-1)*%e^(e-f^2*u^2/(4*(c*u+b)))
4747
4770
*('sum(f^(n-i)*gamma_incomplete((i+1)/2,
4748
4771
-(2*(c*u+b)*z+f*u)^2
4749
4772
/(4*(c*u+b)))*(-1)^(i-n)
4750
*2^(i-n)*binomial(n,i)*u^(n-i)*(-c*u-b)^((i+1)/2)
4773
*2^(i-n)*binomial(n,i)*u^(n-i)
4751
4774
*(2*(c*u+b)*z+f*u)^(i+1)
4752
*abs(2*(c*u+b)*z+f*u)^(-i-1),i,0,n))
4753
/2);
4775
/(-(2*(c*u+b)*z+f*u)^2/(c*u+b))^((i+1)/2),i,0,n))
4776
/2));
4754
4777
0;
4755
4778
4756
/* Problem 639: NEW 11 */
4779
/* Problem 639: NEW 6 */
4757
4780
expand(niceindices(integrate(x^n*%e^(e+b*sqrt(x))*(%e^(f*x+c*sqrt(x)))^u,x))-
4758
2^(-n-1)*%e^(e-(c*u+b)^2/(4*f*u))
4781
(2^(-n-1)*%e^(e-(c*u+b)^2/(4*f*u))
4759
4782
*('sum(4^j*binomial(n,j)
4760
*'sum((-1)^(i-j)*binomial(j,i)*(-f*u)^((j+i+1)/2)
4761
*(c*u+b)^(2*n-j-i)
4762
*abs(2*f*u*sqrt(x)+c*u+b)^(-j-i-1)
4763
*(2*f*u*sqrt(x)+c*u+b)^(j+i)
4764
*(gamma_incomplete((j+i+1)/2,
4765
-(2*f*u*sqrt(x)+c*u+b)^2
4766
/(4*f*u))
4767
*(c*u+b)*(2*f*u*sqrt(x)+c*u+b)
4768
+2*f
4769
*gamma_incomplete((j+i+2)/2,
4770
-(2*f*u*sqrt(x)+c*u+b)
4771
^2
4772
/(4*f*u))*u
4773
*abs(2*f*u*sqrt(x)+c*u+b)
4774
/sqrt(-f*u)),i,0,j),j,0,n))
4775
/(f^(2*(n+1))*u^(2*(n+1))));
4776
0;
4777
4778
/* Problem 640: NEW 10 */
4779
expand(niceindices(integrate(z^n*%e^(d*z+b*z^2)*(%e^(f*z+c*z^2))^u,z))-
4780
-(c*u+b)^(-n-1)*%e^-((f*u+d)^2/(4*(c*u+b)))
4781
*('sum(gamma_incomplete((i+1)/2,
4782
-(2*(c*u+b)*z+f*u+d)^2/(4*(c*u+b)))
4783
*2^(i-n)*binomial(n,i)*(-c*u-b)^((i+1)/2)
4784
*(-f*u-d)^(n-i)*(2*(c*u+b)*z+f*u+d)^(i+1)
4785
*abs(2*(c*u+b)*z+f*u+d)^(-i-1),i,0,n))
4786
/2);
4787
0;
4788
4789
/* Problem 641: NEW 11 */
4783
*'sum((-1)^(-(j+i+1)/2+j-i)*binomial(j,i)*(f*u)^((j+i+1)/2)
4784
*(c*u+b)^(2*n-j-i)
4785
*abs(2*f*u*sqrt(x)+c*u+b)
4786
^(-j-i-1)
4787
*(2*f*u*sqrt(x)+c*u+b)^(j+i)
4788
*(gamma_incomplete(
4789
(j+i+1)/2,
4790
-(2*f*u*sqrt(x)+c*u+b)^2
4791
/(4*f*u))
4792
*(c*u+b)*(2*f*u*sqrt(x)+c*u+b)
4793
+2*%i*f
4794
*gamma_incomplete(
4795
(j+i+2)/2,
4796
-(2*f*u*sqrt(x)+c*u+b)^2
4797
/(4*f*u))*u
4798
*abs(2*f*u*sqrt(x)+c*u+b)
4799
/sqrt(f*u)),i,0,j),j,0,n))
4800
/(f^(2*(n+1))*u^(2*(n+1)))));
4801
0;
4802
4803
/* Problem 640: NEW 5 */
4804
expand(niceindices(integrate(x^n*%e^(d*x+b*x^2)*(%e^(f*x+c*x^2))^u,x))-
4805
(-%e^-((f*u+d)^2/(4*(c*u+b)))*('sum(gamma_incomplete((i+1)/2,
4806
-(2*(c*u+b)*x+f*u+d)^2
4807
/(4*(c*u+b)))
4808
*2^(i-n)*binomial(n,i)
4809
*(c*u+b)^(-n+(i+1)/2-1/2)*(-f*u-d)^(n-i)
4810
*(2*(c*u+b)*x+f*u+d)^(i+1)
4811
*abs(2*(c*u+b)*x+f*u+d)^(-i-1)
4812
/(-1)^((i+1)/2),i,0,n))
4813
/(2*sqrt(c*u+b))));
4814
0;
4815
4816
/* Problem 641: NEW 6 */
4790
4817
expand(niceindices(integrate(x^n*%e^(d*x+b*sqrt(x))*(%e^(f*x+c*sqrt(x)))^u,x))-
4791
2^(-n-1)*%e^-((c*u+b)^2/(4*(f*u+d)))
4818
(2^(-n-1)*%e^-((c*u+b)^2/(4*(f*u+d)))
4792
4819
*('sum(4^j*binomial(n,j)
4793
*'sum((-1)^(i-j)*binomial(j,i)*(c*u+b)^(2*n-j-i)
4794
*(-f*u-d)^((j+i+1)/2)
4795
*abs(2*(f*u+d)*sqrt(x)+c*u+b)^(-j-i-1)
4796
*(2*(f*u+d)*sqrt(x)+c*u+b)^(j+i)
4797
*(gamma_incomplete((j+i+1)/2,
4798
-(2*(f*u+d)*sqrt(x)
4799
+c*u+b)
4800
^2
4801
/(4*(f*u+d)))
4802
*(c*u+b)*(2*(f*u+d)*sqrt(x)+c*u+b)
4803
+2*gamma_incomplete((j+i+2)/2,
4804
-(2*(f*u+d)*sqrt(x)
4805
+c*u+b)
4806
^2
4807
/(4*(f*u+d)))*(f*u+d)
4808
*abs(2*(f*u+d)*sqrt(x)+c*u+b)
4809
/sqrt(-f*u-d)),i,0,j),j,0,n))
4810
/(f*u+d)^(2*(n+1)));
4811
0;
4812
4813
/* Problem 642: NEW 10 */
4814
expand(niceindices(integrate(z^n*%e^(e+d*z+b*z^2)*(%e^(f*z+c*z^2))^u,z))-
4815
-(c*u+b)^(-n-1)*%e^(e-(f*u+d)^2/(4*(c*u+b)))
4816
*('sum(gamma_incomplete((i+1)/2,
4817
-(2*(c*u+b)*z+f*u+d)^2/(4*(c*u+b)))
4818
*2^(i-n)*binomial(n,i)*(-c*u-b)^((i+1)/2)
4819
*(-f*u-d)^(n-i)*(2*(c*u+b)*z+f*u+d)^(i+1)
4820
*abs(2*(c*u+b)*z+f*u+d)^(-i-1),i,0,n))
4821
/2);
4822
0;
4823
4824
/* Problem 643: NEW 11 */
4820
*'sum((-1)^(-(j+i+1)/2+j-i)*binomial(j,i)*(c*u+b)^(2*n-j-i)
4821
*(f*u+d)^((j+i+1)/2)
4822
*abs(2*(f*u+d)*sqrt(x)+c*u+b)
4823
^(-j-i-1)
4824
*(2*(f*u+d)*sqrt(x)+c*u+b)^(j+i)
4825
*(gamma_incomplete(
4826
(j+i+1)/2,
4827
-(2*(f*u+d)*sqrt(x)+c*u+b)^2
4828
/(4*(f*u+d)))
4829
*(c*u+b)
4830
*(2*(f*u+d)*sqrt(x)+c*u+b)
4831
+2*%i
4832
*gamma_incomplete(
4833
(j+i+2)/2,
4834
-(2*(f*u+d)*sqrt(x)+c*u+b)^2
4835
/(4*(f*u+d)))*sqrt(f*u+d)
4836
*abs(
4837
2*(f*u+d)*sqrt(x)+c*u+b)),i,
4838
0,j),j,0,n))
4839
/(f*u+d)^(2*(n+1))));
4840
0;
4841
4842
/* Problem 642: NEW 5 */
4843
expand(niceindices(integrate(x^n*%e^(e+d*x+b*x^2)*(%e^(f*x+c*x^2))^u,x))-
4844
(-%e^(e-(f*u+d)^2/(4*(c*u+b)))*('sum(gamma_incomplete((i+1)/2,
4845
-(2*(c*u+b)*x+f*u+d)^2
4846
/(4*(c*u+b)))
4847
*2^(i-n)*binomial(n,i)
4848
*(c*u+b)^(-n+(i+1)/2-1/2)*(-f*u-d)^(n-i)
4849
*(2*(c*u+b)*x+f*u+d)^(i+1)
4850
*abs(2*(c*u+b)*x+f*u+d)^(-i-1)
4851
/(-1)^((i+1)/2),i,0,n))
4852
/(2*sqrt(c*u+b))));
4853
0;
4854
4855
/* Problem 643: NEW 6 */
4825
4856
expand(niceindices(integrate(x^n*%e^(e+d*x+b*sqrt(x))*(%e^(f*x+c*sqrt(x)))^u,x))-
4826
2^(-n-1)*%e^(e-(c*u+b)^2/(4*(f*u+d)))
4857
(2^(-n-1)*%e^(e-(c*u+b)^2/(4*(f*u+d)))
4827
4858
*('sum(4^j*binomial(n,j)
4828
*'sum((-1)^(i-j)*binomial(j,i)*(c*u+b)^(2*n-j-i)
4829
*(-f*u-d)^((j+i+1)/2)
4830
*abs(2*(f*u+d)*sqrt(x)+c*u+b)^(-j-i-1)
4831
*(2*(f*u+d)*sqrt(x)+c*u+b)^(j+i)
4832
*(gamma_incomplete((j+i+1)/2,
4833
-(2*(f*u+d)*sqrt(x)
4834
+c*u+b)
4835
^2
4836
/(4*(f*u+d)))
4837
*(c*u+b)*(2*(f*u+d)*sqrt(x)+c*u+b)
4838
+2*gamma_incomplete((j+i+2)/2,
4839
-(2*(f*u+d)*sqrt(x)
4840
+c*u+b)
4841
^2
4842
/(4*(f*u+d)))*(f*u+d)
4843
*abs(2*(f*u+d)*sqrt(x)+c*u+b)
4844
/sqrt(-f*u-d)),i,0,j),j,0,n))
4845
/(f*u+d)^(2*(n+1)));
4846
0;
4847
4848
/* Problem 644: NEW 10 */
4849
expand(niceindices(integrate(z^n*%e^(d*z)*(%e^(g+f*z+c*z^2))^u,z))-
4850
-c^(-n-1)*u^(-n-1)*%e^(g*u-(f*u+d)^2/(4*c*u))
4851
*('sum(gamma_incomplete((i+1)/2,-(2*c*u*z+f*u+d)^2/(4*c*u))
4852
*2^(i-n)*binomial(n,i)*(-c*u)^((i+1)/2)*(-f*u-d)^(n-i)
4853
*(2*c*u*z+f*u+d)^(i+1)*abs(2*c*u*z+f*u+d)^(-i-1),i,0,n))
4854
/2);
4855
0;
4856
4857
/* Problem 645: NEW 11 */
4859
*'sum((-1)^(-(j+i+1)/2+j-i)*binomial(j,i)*(c*u+b)^(2*n-j-i)
4860
*(f*u+d)^((j+i+1)/2)
4861
*abs(2*(f*u+d)*sqrt(x)+c*u+b)
4862
^(-j-i-1)
4863
*(2*(f*u+d)*sqrt(x)+c*u+b)^(j+i)
4864
*(gamma_incomplete(
4865
(j+i+1)/2,
4866
-(2*(f*u+d)*sqrt(x)+c*u+b)^2
4867
/(4*(f*u+d)))
4868
*(c*u+b)
4869
*(2*(f*u+d)*sqrt(x)+c*u+b)
4870
+2*%i
4871
*gamma_incomplete(
4872
(j+i+2)/2,
4873
-(2*(f*u+d)*sqrt(x)+c*u+b)^2
4874
/(4*(f*u+d)))*sqrt(f*u+d)
4875
*abs(
4876
2*(f*u+d)*sqrt(x)+c*u+b)),i,
4877
0,j),j,0,n))
4878
/(f*u+d)^(2*(n+1))));
4879
0;
4880
4881
/* Problem 644: NEW 5 */
4882
expand(niceindices(integrate(x^n*%e^(d*x)*(%e^(g+f*x+c*x^2))^u,x))-
4883
(-%e^(g*u-(f*u+d)^2/(4*c*u))*('sum(gamma_incomplete((i+1)/2,
4884
-(2*c*u*x+f*u+d)^2/(4*c*u))
4885
*2^(i-n)*binomial(n,i)
4886
*(c*u)^(-n+(i+1)/2-1/2)*(-f*u-d)^(n-i)
4887
*(2*c*u*x+f*u+d)^(i+1)
4888
*abs(2*c*u*x+f*u+d)^(-i-1)
4889
/(-1)^((i+1)/2),i,0,n))
4890
/(2*sqrt(c*u))));
4891
0;
4892
4893
/* Problem 645: NEW 6 */
4858
4894
expand(niceindices(integrate(x^n*%e^(d*x)*(%e^(g+f*x+c*sqrt(x)))^u,x))-
4859
2^(-n-1)*%e^(g*u-c^2*u^2/(4*(f*u+d)))
4895
(2^(-n-1)*%e^(g*u-c^2*u^2/(4*(f*u+d)))
4860
4896
*('sum(4^j*binomial(n,j)
4861
*'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i)
4862
*(-f*u-d)^((j+i+1)/2)
4897
*'sum(c^(2*n-j-i)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
4898
*u^(2*n-j-i)*(f*u+d)^((j+i+1)/2)
4863
4899
*abs(2*(f*u+d)*sqrt(x)+c*u)^(-j-i-1)
4864
4900
*(2*(f*u+d)*sqrt(x)+c*u)^(j+i)
4865
4901
*(c*gamma_incomplete((j+i+1)/2,
4868
4904
^2
4869
4905
/(4*(f*u+d)))*u
4870
4906
*(2*(f*u+d)*sqrt(x)+c*u)
4871
+2*gamma_incomplete((j+i+2)/2,
4907
+2*%i
4908
*gamma_incomplete((j+i+2)/2,
4872
4909
-(2*(f*u+d)*sqrt(x)
4873
4910
+c*u)
4874
4911
^2
4875
/(4*(f*u+d)))*(f*u+d)
4876
*abs(2*(f*u+d)*sqrt(x)+c*u)
4877
/sqrt(-f*u-d)),i,0,j),j,0,n))
4878
/(f*u+d)^(2*(n+1)));
4879
0;
4880
4881
/* Problem 646: NEW 10 */
4882
expand(niceindices(integrate(z^n*%e^(e+d*z)*(%e^(g+f*z+c*z^2))^u,z))-
4883
-c^(-n-1)*u^(-n-1)*%e^(-(f*u+d)^2/(4*c*u)+g*u+e)
4884
*('sum(gamma_incomplete((i+1)/2,-(2*c*u*z+f*u+d)^2/(4*c*u))
4885
*2^(i-n)*binomial(n,i)*(-c*u)^((i+1)/2)*(-f*u-d)^(n-i)
4886
*(2*c*u*z+f*u+d)^(i+1)*abs(2*c*u*z+f*u+d)^(-i-1),i,0,n))
4887
/2);
4888
0;
4889
4890
/* Problem 647: NEW 11 */
4912
/(4*(f*u+d)))
4913
*sqrt(f*u+d)
4914
*abs(2*(f*u+d)*sqrt(x)+c*u)),i,0,j),j,0,
4915
n))
4916
/(f*u+d)^(2*(n+1))));
4917
0;
4918
4919
/* Problem 646: NEW 5 */
4920
expand(niceindices(integrate(x^n*%e^(e+d*x)*(%e^(g+f*x+c*x^2))^u,x))-
4921
(-%e^(-(f*u+d)^2/(4*c*u)+g*u+e)*('sum(gamma_incomplete((i+1)/2,
4922
-(2*c*u*x+f*u+d)^2
4923
/(4*c*u))
4924
*2^(i-n)*binomial(n,i)
4925
*(c*u)^(-n+(i+1)/2-1/2)*(-f*u-d)^(n-i)
4926
*(2*c*u*x+f*u+d)^(i+1)
4927
*abs(2*c*u*x+f*u+d)^(-i-1)
4928
/(-1)^((i+1)/2),i,0,n))
4929
/(2*sqrt(c*u))));
4930
0;
4931
4932
/* Problem 647: NEW 6 */
4891
4933
expand(niceindices(integrate(x^n*%e^(e+d*x)*(%e^(g+f*x+c*sqrt(x)))^u,x))-
4892
2^(-n-1)*%e^(-c^2*u^2/(4*(f*u+d))+g*u+e)
4934
(2^(-n-1)*%e^(-c^2*u^2/(4*(f*u+d))+g*u+e)
4893
4935
*('sum(4^j*binomial(n,j)
4894
*'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i)
4895
*(-f*u-d)^((j+i+1)/2)
4936
*'sum(c^(2*n-j-i)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
4937
*u^(2*n-j-i)*(f*u+d)^((j+i+1)/2)
4896
4938
*abs(2*(f*u+d)*sqrt(x)+c*u)^(-j-i-1)
4897
4939
*(2*(f*u+d)*sqrt(x)+c*u)^(j+i)
4898
4940
*(c*gamma_incomplete((j+i+1)/2,
4901
4943
^2
4902
4944
/(4*(f*u+d)))*u
4903
4945
*(2*(f*u+d)*sqrt(x)+c*u)
4904
+2*gamma_incomplete((j+i+2)/2,
4946
+2*%i
4947
*gamma_incomplete((j+i+2)/2,
4905
4948
-(2*(f*u+d)*sqrt(x)
4906
4949
+c*u)
4907
4950
^2
4908
/(4*(f*u+d)))*(f*u+d)
4909
*abs(2*(f*u+d)*sqrt(x)+c*u)
4910
/sqrt(-f*u-d)),i,0,j),j,0,n))
4911
/(f*u+d)^(2*(n+1)));
4912
0;
4913
4914
/* Problem 648: NEW 10 */
4915
expand(niceindices(integrate(z^n*%e^(b*z^2)*(%e^(g+f*z+c*z^2))^u,z))-
4916
-(c*u+b)^(-n-1)*%e^(g*u-f^2*u^2/(4*(c*u+b)))
4917
*('sum(f^(n-i)*gamma_incomplete((i+1)/2,
4918
-(2*(c*u+b)*z+f*u)^2
4919
/(4*(c*u+b)))*(-1)^(i-n)
4920
*2^(i-n)*binomial(n,i)*u^(n-i)*(-c*u-b)^((i+1)/2)
4921
*(2*(c*u+b)*z+f*u)^(i+1)
4922
*abs(2*(c*u+b)*z+f*u)^(-i-1),i,0,n))
4923
/2);
4924
0;
4925
4926
/* Problem 649: NEW 11 */
4951
/(4*(f*u+d)))
4952
*sqrt(f*u+d)
4953
*abs(2*(f*u+d)*sqrt(x)+c*u)),i,0,j),j,0,
4954
n))
4955
/(f*u+d)^(2*(n+1))));
4956
0;
4957
4958
/* Problem 648: NEW 5 */
4959
expand(niceindices(integrate(x^n*%e^(b*x^2)*(%e^(g+f*x+c*x^2))^u,x))-
4960
(-%e^(g*u-f^2*u^2/(4*(c*u+b)))*('sum(f^(n-i)
4961
*gamma_incomplete((i+1)/2,
4962
-(2*(c*u+b)*x+f*u)^2
4963
/(4*(c*u+b)))*2^(i-n)
4964
*(-1)^(-n-(i+1)/2+i)*binomial(n,i)
4965
*u^(n-i)*(c*u+b)^(-n+(i+1)/2-1/2)
4966
*(2*(c*u+b)*x+f*u)^(i+1)
4967
*abs(2*(c*u+b)*x+f*u)^(-i-1),i,0,n))
4968
/(2*sqrt(c*u+b))));
4969
0;
4970
4971
/* Problem 649: NEW 6 */
4927
4972
expand(niceindices(integrate(x^n*%e^(b*sqrt(x))*(%e^(g+f*x+c*sqrt(x)))^u,x))-
4928
2^(-n-1)*%e^(g*u-(c*u+b)^2/(4*f*u))
4973
(2^(-n-1)*%e^(g*u-(c*u+b)^2/(4*f*u))
4929
4974
*('sum(4^j*binomial(n,j)
4930
*'sum((-1)^(i-j)*binomial(j,i)*(-f*u)^((j+i+1)/2)
4931
*(c*u+b)^(2*n-j-i)
4932
*abs(2*f*u*sqrt(x)+c*u+b)^(-j-i-1)
4933
*(2*f*u*sqrt(x)+c*u+b)^(j+i)
4934
*(gamma_incomplete((j+i+1)/2,
4935
-(2*f*u*sqrt(x)+c*u+b)^2
4936
/(4*f*u))
4937
*(c*u+b)*(2*f*u*sqrt(x)+c*u+b)
4938
+2*f
4939
*gamma_incomplete((j+i+2)/2,
4940
-(2*f*u*sqrt(x)+c*u+b)
4941
^2
4942
/(4*f*u))*u
4943
*abs(2*f*u*sqrt(x)+c*u+b)
4944
/sqrt(-f*u)),i,0,j),j,0,n))
4945
/(f^(2*(n+1))*u^(2*(n+1))));
4946
0;
4947
4948
/* Problem 650: NEW 10 */
4949
expand(niceindices(integrate(z^n*%e^(e+b*z^2)*(%e^(g+f*z+c*z^2))^u,z))-
4950
-(c*u+b)^(-n-1)*%e^(-f^2*u^2/(4*(c*u+b))+g*u+e)
4951
*('sum(f^(n-i)*gamma_incomplete((i+1)/2,
4952
-(2*(c*u+b)*z+f*u)^2
4953
/(4*(c*u+b)))*(-1)^(i-n)
4954
*2^(i-n)*binomial(n,i)*u^(n-i)*(-c*u-b)^((i+1)/2)
4955
*(2*(c*u+b)*z+f*u)^(i+1)
4956
*abs(2*(c*u+b)*z+f*u)^(-i-1),i,0,n))
4957
/2);
4958
0;
4959
4960
/* Problem 651: NEW 11 */
4975
*'sum((-1)^(-(j+i+1)/2+j-i)*binomial(j,i)*(f*u)^((j+i+1)/2)
4976
*(c*u+b)^(2*n-j-i)
4977
*abs(2*f*u*sqrt(x)+c*u+b)
4978
^(-j-i-1)
4979
*(2*f*u*sqrt(x)+c*u+b)^(j+i)
4980
*(gamma_incomplete(
4981
(j+i+1)/2,
4982
-(2*f*u*sqrt(x)+c*u+b)^2
4983
/(4*f*u))
4984
*(c*u+b)*(2*f*u*sqrt(x)+c*u+b)
4985
+2*%i*f
4986
*gamma_incomplete(
4987
(j+i+2)/2,
4988
-(2*f*u*sqrt(x)+c*u+b)^2
4989
/(4*f*u))*u
4990
*abs(2*f*u*sqrt(x)+c*u+b)
4991
/sqrt(f*u)),i,0,j),j,0,n))
4992
/(f^(2*(n+1))*u^(2*(n+1)))));
4993
0;
4994
4995
/* Problem 650: NEW 5 */
4996
expand(niceindices(integrate(x^n*%e^(e+b*x^2)*(%e^(g+f*x+c*x^2))^u,x))-
4997
(-%e^(-f^2*u^2/(4*(c*u+b))+g*u+e)*('sum(f^(n-i)
4998
*gamma_incomplete((i+1)/2,
4999
-(2*(c*u+b)*x+f*u)^2
5000
/(4*(c*u+b)))
5001
*2^(i-n)*(-1)^(-n-(i+1)/2+i)
5002
*binomial(n,i)*u^(n-i)
5003
*(c*u+b)^(-n+(i+1)/2-1/2)
5004
*(2*(c*u+b)*x+f*u)^(i+1)
5005
*abs(2*(c*u+b)*x+f*u)^(-i-1),i,0,n))
5006
/(2*sqrt(c*u+b))));
5007
0;
5008
5009
/* Problem 651: NEW 6 */
4961
5010
expand(niceindices(integrate(x^n*%e^(e+b*sqrt(x))*(%e^(g+f*x+c*sqrt(x)))^u,x))-
4962
2^(-n-1)*%e^(-(c*u+b)^2/(4*f*u)+g*u+e)
5011
(2^(-n-1)*%e^(-(c*u+b)^2/(4*f*u)+g*u+e)
4963
5012
*('sum(4^j*binomial(n,j)
4964
*'sum((-1)^(i-j)*binomial(j,i)*(-f*u)^((j+i+1)/2)
4965
*(c*u+b)^(2*n-j-i)
4966
*abs(2*f*u*sqrt(x)+c*u+b)^(-j-i-1)
4967
*(2*f*u*sqrt(x)+c*u+b)^(j+i)
4968
*(gamma_incomplete((j+i+1)/2,
4969
-(2*f*u*sqrt(x)+c*u+b)^2
4970
/(4*f*u))
4971
*(c*u+b)*(2*f*u*sqrt(x)+c*u+b)
4972
+2*f
4973
*gamma_incomplete((j+i+2)/2,
4974
-(2*f*u*sqrt(x)+c*u+b)
4975
^2
4976
/(4*f*u))*u
4977
*abs(2*f*u*sqrt(x)+c*u+b)
4978
/sqrt(-f*u)),i,0,j),j,0,n))
4979
/(f^(2*(n+1))*u^(2*(n+1))));
4980
0;
4981
4982
/* Problem 652: NEW 10 */
4983
expand(niceindices(integrate(z^n*%e^(d*z+b*z^2)*(%e^(g+f*z+c*z^2))^u,z))-
4984
-(c*u+b)^(-n-1)*%e^(g*u-(f*u+d)^2/(4*(c*u+b)))
4985
*('sum(gamma_incomplete((i+1)/2,
4986
-(2*(c*u+b)*z+f*u+d)^2/(4*(c*u+b)))
4987
*2^(i-n)*binomial(n,i)*(-c*u-b)^((i+1)/2)
4988
*(-f*u-d)^(n-i)*(2*(c*u+b)*z+f*u+d)^(i+1)
4989
*abs(2*(c*u+b)*z+f*u+d)^(-i-1),i,0,n))
4990
/2);
4991
0;
4992
4993
/* Problem 653: NEW 11 */
5013
*'sum((-1)^(-(j+i+1)/2+j-i)*binomial(j,i)*(f*u)^((j+i+1)/2)
5014
*(c*u+b)^(2*n-j-i)
5015
*abs(2*f*u*sqrt(x)+c*u+b)
5016
^(-j-i-1)
5017
*(2*f*u*sqrt(x)+c*u+b)^(j+i)
5018
*(gamma_incomplete(
5019
(j+i+1)/2,
5020
-(2*f*u*sqrt(x)+c*u+b)^2
5021
/(4*f*u))
5022
*(c*u+b)*(2*f*u*sqrt(x)+c*u+b)
5023
+2*%i*f
5024
*gamma_incomplete(
5025
(j+i+2)/2,
5026
-(2*f*u*sqrt(x)+c*u+b)^2
5027
/(4*f*u))*u
5028
*abs(2*f*u*sqrt(x)+c*u+b)
5029
/sqrt(f*u)),i,0,j),j,0,n))
5030
/(f^(2*(n+1))*u^(2*(n+1)))));
5031
0;
5032
5033
/* Problem 652: NEW 5 */
5034
expand(niceindices(integrate(x^n*%e^(d*x+b*x^2)*(%e^(g+f*x+c*x^2))^u,x))-
5035
(-%e^(g*u-(f*u+d)^2/(4*(c*u+b)))*('sum(gamma_incomplete((i+1)/2,
5036
-(2*(c*u+b)*x+f*u+d)^2
5037
/(4*(c*u+b)))
5038
*2^(i-n)*binomial(n,i)
5039
*(c*u+b)^(-n+(i+1)/2-1/2)
5040
*(-f*u-d)^(n-i)
5041
*(2*(c*u+b)*x+f*u+d)^(i+1)
5042
*abs(2*(c*u+b)*x+f*u+d)^(-i-1)
5043
/(-1)^((i+1)/2),i,0,n))
5044
/(2*sqrt(c*u+b))));
5045
0;
5046
5047
/* Problem 653: NEW 6 */
4994
5048
expand(niceindices(integrate(x^n*%e^(d*x+b*sqrt(x))*(%e^(g+f*x+c*sqrt(x)))^u,x))-
4995
2^(-n-1)*%e^(g*u-(c*u+b)^2/(4*(f*u+d)))
5049
(2^(-n-1)*%e^(g*u-(c*u+b)^2/(4*(f*u+d)))
4996
5050
*('sum(4^j*binomial(n,j)
4997
*'sum((-1)^(i-j)*binomial(j,i)*(c*u+b)^(2*n-j-i)
4998
*(-f*u-d)^((j+i+1)/2)
4999
*abs(2*(f*u+d)*sqrt(x)+c*u+b)^(-j-i-1)
5000
*(2*(f*u+d)*sqrt(x)+c*u+b)^(j+i)
5001
*(gamma_incomplete((j+i+1)/2,
5002
-(2*(f*u+d)*sqrt(x)
5003
+c*u+b)
5004
^2
5005
/(4*(f*u+d)))
5006
*(c*u+b)*(2*(f*u+d)*sqrt(x)+c*u+b)
5007
+2*gamma_incomplete((j+i+2)/2,
5008
-(2*(f*u+d)*sqrt(x)
5009
+c*u+b)
5010
^2
5011
/(4*(f*u+d)))*(f*u+d)
5012
*abs(2*(f*u+d)*sqrt(x)+c*u+b)
5013
/sqrt(-f*u-d)),i,0,j),j,0,n))
5014
/(f*u+d)^(2*(n+1)));
5015
0;
5016
5017
/* Problem 654: NEW 10 */
5018
expand(niceindices(integrate(z^n*%e^(e+d*z+b*z^2)*(%e^(g+f*z+c*z^2))^u,z))-
5019
-(c*u+b)^(-n-1)*%e^(-(f*u+d)^2/(4*(c*u+b))+g*u+e)
5020
*('sum(gamma_incomplete((i+1)/2,
5021
-(2*(c*u+b)*z+f*u+d)^2/(4*(c*u+b)))
5022
*2^(i-n)*binomial(n,i)*(-c*u-b)^((i+1)/2)
5023
*(-f*u-d)^(n-i)*(2*(c*u+b)*z+f*u+d)^(i+1)
5024
*abs(2*(c*u+b)*z+f*u+d)^(-i-1),i,0,n))
5025
/2);
5026
0;
5027
5028
/* Problem 655: NEW 11 */
5051
*'sum((-1)^(-(j+i+1)/2+j-i)*binomial(j,i)*(c*u+b)^(2*n-j-i)
5052
*(f*u+d)^((j+i+1)/2)
5053
*abs(2*(f*u+d)*sqrt(x)+c*u+b)
5054
^(-j-i-1)
5055
*(2*(f*u+d)*sqrt(x)+c*u+b)^(j+i)
5056
*(gamma_incomplete(
5057
(j+i+1)/2,
5058
-(2*(f*u+d)*sqrt(x)+c*u+b)^2
5059
/(4*(f*u+d)))
5060
*(c*u+b)
5061
*(2*(f*u+d)*sqrt(x)+c*u+b)
5062
+2*%i
5063
*gamma_incomplete(
5064
(j+i+2)/2,
5065
-(2*(f*u+d)*sqrt(x)+c*u+b)^2
5066
/(4*(f*u+d)))*sqrt(f*u+d)
5067
*abs(
5068
2*(f*u+d)*sqrt(x)+c*u+b)),i,
5069
0,j),j,0,n))
5070
/(f*u+d)^(2*(n+1))));
5071
0;
5072
5073
/* Problem 654: NEW 5 */
5074
expand(niceindices(integrate(x^n*%e^(e+d*x+b*x^2)*(%e^(g+f*x+c*x^2))^u,x))-
5075
(-%e^(-(f*u+d)^2/(4*(c*u+b))+g*u+e)*('sum(
5076
gamma_incomplete((i+1)/2,
5077
-(2*(c*u+b)*x+f*u+d)^2
5078
/(4*(c*u+b)))
5079
*2^(i-n)*binomial(n,i)
5080
*(c*u+b)^(-n+(i+1)/2-1/2)*(-f*u-d)^(n-i)
5081
*(2*(c*u+b)*x+f*u+d)^(i+1)
5082
*abs(2*(c*u+b)*x+f*u+d)^(-i-1)
5083
/(-1)^((i+1)/2),i,0,n))
5084
/(2*sqrt(c*u+b))));
5085
0;
5086
5087
/* Problem 655: NEW 6 */
5029
5088
expand(niceindices(integrate(x^n*%e^(e+d*x+b*sqrt(x))*(%e^(g+f*x+c*sqrt(x)))^u,x))-
5030
2^(-n-1)*%e^(-(c*u+b)^2/(4*(f*u+d))+g*u+e)
5089
(2^(-n-1)*%e^(-(c*u+b)^2/(4*(f*u+d))+g*u+e)
5031
5090
*('sum(4^j*binomial(n,j)
5032
*'sum((-1)^(i-j)*binomial(j,i)*(c*u+b)^(2*n-j-i)
5033
*(-f*u-d)^((j+i+1)/2)
5034
*abs(2*(f*u+d)*sqrt(x)+c*u+b)^(-j-i-1)
5035
*(2*(f*u+d)*sqrt(x)+c*u+b)^(j+i)
5036
*(gamma_incomplete((j+i+1)/2,
5037
-(2*(f*u+d)*sqrt(x)
5038
+c*u+b)
5039
^2
5040
/(4*(f*u+d)))
5041
*(c*u+b)*(2*(f*u+d)*sqrt(x)+c*u+b)
5042
+2*gamma_incomplete((j+i+2)/2,
5043
-(2*(f*u+d)*sqrt(x)
5044
+c*u+b)
5045
^2
5046
/(4*(f*u+d)))*(f*u+d)
5047
*abs(2*(f*u+d)*sqrt(x)+c*u+b)
5048
/sqrt(-f*u-d)),i,0,j),j,0,n))
5049
/(f*u+d)^(2*(n+1)));
5091
*'sum((-1)^(-(j+i+1)/2+j-i)*binomial(j,i)*(c*u+b)^(2*n-j-i)
5092
*(f*u+d)^((j+i+1)/2)
5093
*abs(2*(f*u+d)*sqrt(x)+c*u+b)
5094
^(-j-i-1)
5095
*(2*(f*u+d)*sqrt(x)+c*u+b)^(j+i)
5096
*(gamma_incomplete(
5097
(j+i+1)/2,
5098
-(2*(f*u+d)*sqrt(x)+c*u+b)^2
5099
/(4*(f*u+d)))
5100
*(c*u+b)
5101
*(2*(f*u+d)*sqrt(x)+c*u+b)
5102
+2*%i
5103
*gamma_incomplete(
5104
(j+i+2)/2,
5105
-(2*(f*u+d)*sqrt(x)+c*u+b)^2
5106
/(4*(f*u+d)))*sqrt(f*u+d)
5107
*abs(
5108
2*(f*u+d)*sqrt(x)+c*u+b)),i,
5109
0,j),j,0,n))
5110
/(f*u+d)^(2*(n+1))));
5050
5111
0;
5051
5112
5052
5113
/* Problem 656: NEW */
5098
5159
0;
5099
5160
5100
5161
/* Problem 665: NEW 5 */
5101
expand(niceindices(integrate(z^n*(%e^(d*z))^q*(%e^(c*z^2))^u,z))-
5102
-c^(-n-1)*u^(-n-1)*%e^-(d^2*q^2/(4*c*u))
5103
*('sum(d^(n-i)*gamma_incomplete((i+1)/2,-(2*c*u*z+d*q)^2/(4*c*u))
5104
*(-1)^(i-n)*2^(i-n)*binomial(n,i)*q^(n-i)
5105
*(-c*u)^((i+1)/2)*(2*c*u*z+d*q)^(i+1)
5106
*abs(2*c*u*z+d*q)^(-i-1),i,0,n))
5107
/2);
5162
expand(niceindices(integrate(x^n*(%e^(d*x))^q*(%e^(c*x^2))^u,x))-
5163
(-%e^-(d^2*q^2/(4*c*u))*('sum(d^(n-i)*gamma_incomplete((i+1)/2,
5164
-(2*c*u*x+d*q)^2
5165
/(4*c*u))*2^(i-n)
5166
*(-1)^(-n-(i+1)/2+i)*binomial(n,i)*q^(n-i)
5167
*(c*u)^(-n+(i+1)/2-1/2)
5168
*(2*c*u*x+d*q)^(i+1)
5169
*abs(2*c*u*x+d*q)^(-i-1),i,0,n))
5170
/(2*sqrt(c*u))));
5108
5171
0;
5109
5172
5110
5173
/* Problem 666: NEW 6 */
5111
5174
expand(niceindices(integrate(x^n*(%e^(d*x))^q*(%e^(c*sqrt(x)))^u,x))-
5112
2^(-n-1)*%e^-(c^2*u^2/(4*d*q))
5175
(2^(-n-1)*%e^-(c^2*u^2/(4*d*q))
5113
5176
*('sum(4^j*binomial(n,j)
5114
*'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)
5115
*(-d*q)^((j+i+1)/2)*u^(2*n-j-i)
5177
*'sum(c^(2*n-j-i)*d^((j+i+1)/2)*(-1)^(-(j+i+1)/2+j-i)
5178
*binomial(j,i)*q^((j+i+1)/2)*u^(2*n-j-i)
5116
5179
*abs(2*d*q*sqrt(x)+c*u)^(-j-i-1)
5117
5180
*(2*d*q*sqrt(x)+c*u)^(j+i)
5118
5181
*(c*gamma_incomplete((j+i+1)/2,
5119
5182
-(2*d*q*sqrt(x)+c*u)^2
5120
5183
/(4*d*q))*u
5121
5184
*(2*d*q*sqrt(x)+c*u)
5122
+2*d
5185
+2*%i*sqrt(d)
5123
5186
*gamma_incomplete((j+i+2)/2,
5124
5187
-(2*d*q*sqrt(x)+c*u)^2
5125
/(4*d*q))*q
5126
*abs(2*d*q*sqrt(x)+c*u)
5127
/sqrt(-d*q)),i,0,j),j,0,n))
5128
/(d^(2*(n+1))*q^(2*(n+1))));
5188
/(4*d*q))*sqrt(q)
5189
*abs(2*d*q*sqrt(x)+c*u)),i,0,j),j,0,n))
5190
/(d^(2*(n+1))*q^(2*(n+1)))));
5129
5191
0;
5130
5192
5131
5193
/* Problem 667: NEW 5 */
5132
expand(niceindices(integrate(z^n*(%e^(e+d*z))^q*(%e^(c*z^2))^u,z)) -
5133
-('sum(d^(n+(-1)*i)*binomial(n,i)*q^(n+(-1)*i)*(-c*u)^(1/2*(1+i))
5134
*(c*u)^(-n-1/2)*(2*c*u*z+d*q)^(1+i)
5135
*abs(2*c*u*z+d*q)^(-1+(-1)*i)*(-1)^(-n+i)*2^(-n+i)
5136
*gamma_incomplete(1/2*(1+i),-(2*c*u*z+d*q)^2/(4*c*u)),
5137
i,0,n))
5138
*%e^(e*q-d^2*q^2/(4*c*u))
5139
/(2*sqrt(c*u)));
5194
expand(niceindices(integrate(x^n*(%e^(e+d*x))^q*(%e^(c*x^2))^u,x)) -
5195
(-%e^(e*q-d^2*q^2/(4*c*u))*('sum(d^(n-i)*gamma_incomplete((i+1)/2,
5196
-(2*c*u*x+d*q)^2
5197
/(4*c*u))*2^(i-n)
5198
*(-1)^(-n-(i+1)/2+i)*binomial(n,i)
5199
*q^(n-i)*(c*u)^(-n+(i+1)/2-1/2)
5200
*(2*c*u*x+d*q)^(i+1)
5201
*abs(2*c*u*x+d*q)^(-i-1),i,0,n))
5202
/(2*sqrt(c*u))));
5140
5203
0;
5141
5204
5142
5205
/* Problem 668: NEW 6 */
5143
5206
expand(niceindices(integrate(x^n*(%e^(e+d*x))^q*(%e^(c*sqrt(x)))^u,x))-
5144
2^(-n-1)*%e^(e*q-c^2*u^2/(4*d*q))
5207
(2^(-n-1)*%e^(e*q-c^2*u^2/(4*d*q))
5145
5208
*('sum(4^j*binomial(n,j)
5146
*'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)
5147
*(-d*q)^((j+i+1)/2)*u^(2*n-j-i)
5209
*'sum(c^(2*n-j-i)*d^((j+i+1)/2)*(-1)^(-(j+i+1)/2+j-i)
5210
*binomial(j,i)*q^((j+i+1)/2)*u^(2*n-j-i)
5148
5211
*abs(2*d*q*sqrt(x)+c*u)^(-j-i-1)
5149
5212
*(2*d*q*sqrt(x)+c*u)^(j+i)
5150
5213
*(c*gamma_incomplete((j+i+1)/2,
5151
5214
-(2*d*q*sqrt(x)+c*u)^2
5152
5215
/(4*d*q))*u
5153
5216
*(2*d*q*sqrt(x)+c*u)
5154
+2*d
5217
+2*%i*sqrt(d)
5155
5218
*gamma_incomplete((j+i+2)/2,
5156
5219
-(2*d*q*sqrt(x)+c*u)^2
5157
/(4*d*q))*q
5158
*abs(2*d*q*sqrt(x)+c*u)
5159
/sqrt(-d*q)),i,0,j),j,0,n))
5160
/(d^(2*(n+1))*q^(2*(n+1))));
5220
/(4*d*q))*sqrt(q)
5221
*abs(2*d*q*sqrt(x)+c*u)),i,0,j),j,0,n))
5222
/(d^(2*(n+1))*q^(2*(n+1)))));
5161
5223
0;
5162
5224
5163
5225
/* Problem 669: NEW */
5165
5227
-gamma_incomplete(v/r,-(c*u+b*q)*z^r)*z^v/(r*(-(c*u+b*q)*z^r)^(v/r)));
5166
5228
0;
5167
5229
5168
/* Problem 670: NEW */
5230
/* Problem 670: NEW 2 */
5169
5231
expand(integrate(z^n*(%e^(b*z^2))^q*(%e^(c*z^2))^u,z)-
5170
-gamma_incomplete((n+1)/2,-(c*u+b*q)*z^2)*z^(n+1)
5171
/(2*(-c*u-b*q)^((n+1)/2)*(z^2)^((n+1)/2)));
5232
(-gamma_incomplete((n+1)/2,-(c*u+b*q)*z^2)*z^(n+1)
5233
/(2*(-(c*u+b*q)*z^2)^((n+1)/2))));
5172
5234
0;
5173
5235
5174
/* Problem 671: NEW */
5236
/* Problem 671: NEW 2 */
5175
5237
expand(integrate(z^(2*n)*(%e^(b*z^2))^q*(%e^(c*z^2))^u,z)-
5176
-gamma_incomplete((2*n+1)/2,-(c*u+b*q)*z^2)*z^(2*n+1)
5177
/(2*(-c*u-b*q)^((2*n+1)/2)*(z^2)^((2*n+1)/2)));
5238
(-gamma_incomplete((2*n+1)/2,-(c*u+b*q)*z^2)*z^(2*n+1)
5239
/(2*(-(c*u+b*q)*z^2)^((2*n+1)/2))));
5178
5240
0;
5179
5241
5180
5242
/* Problem 672: NEW */
5190
5252
5191
5253
/* Problem 674: NEW 5 */
5192
5254
expand(niceindices(integrate(z^n*(%e^(d*z))^q*(%e^(g+c*z^2))^u,z))-
5193
-c^(-n-1)*u^(-n-1)*%e^(g*u-d^2*q^2/(4*c*u))
5255
(-c^(-n-1)*u^(-n-1)*%e^(g*u-d^2*q^2/(4*c*u))
5194
5256
*('sum(d^(n-i)*gamma_incomplete((i+1)/2,-(2*c*u*z+d*q)^2/(4*c*u))
5195
5257
*(-1)^(i-n)*2^(i-n)*binomial(n,i)*q^(n-i)
5196
*(-c*u)^((i+1)/2)*(2*c*u*z+d*q)^(i+1)
5197
*abs(2*c*u*z+d*q)^(-i-1),i,0,n))
5198
/2);
5258
*(2*c*u*z+d*q)^(i+1)
5259
/(-(2*c*u*z+d*q)^2/(c*u))^((i+1)/2),i,0,n))
5260
/2));
5199
5261
0;
5200
5262
5201
5263
/* Problem 675: NEW 6 */
5202
5264
expand(niceindices(integrate(x^n*(%e^(d*x))^q*(%e^(g+c*sqrt(x)))^u,x))-
5203
2^(-n-1)*%e^(g*u-c^2*u^2/(4*d*q))
5265
(2^(-n-1)*%e^(g*u-c^2*u^2/(4*d*q))
5204
5266
*('sum(4^j*binomial(n,j)
5205
*'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)
5206
*(-d*q)^((j+i+1)/2)*u^(2*n-j-i)
5267
*'sum(c^(2*n-j-i)*d^((j+i+1)/2)*(-1)^(-(j+i+1)/2+j-i)
5268
*binomial(j,i)*q^((j+i+1)/2)*u^(2*n-j-i)
5207
5269
*abs(2*d*q*sqrt(x)+c*u)^(-j-i-1)
5208
5270
*(2*d*q*sqrt(x)+c*u)^(j+i)
5209
5271
*(c*gamma_incomplete((j+i+1)/2,
5210
5272
-(2*d*q*sqrt(x)+c*u)^2
5211
5273
/(4*d*q))*u
5212
5274
*(2*d*q*sqrt(x)+c*u)
5213
+2*d
5275
+2*%i*sqrt(d)
5214
5276
*gamma_incomplete((j+i+2)/2,
5215
5277
-(2*d*q*sqrt(x)+c*u)^2
5216
/(4*d*q))*q
5217
*abs(2*d*q*sqrt(x)+c*u)
5218
/sqrt(-d*q)),i,0,j),j,0,n))
5219
/(d^(2*(n+1))*q^(2*(n+1))));
5278
/(4*d*q))*sqrt(q)
5279
*abs(2*d*q*sqrt(x)+c*u)),i,0,j),j,0,n))
5280
/(d^(2*(n+1))*q^(2*(n+1)))));
5220
5281
0;
5221
5282
5222
5283
/* Problem 676: NEW 5 */
5223
5284
expand(niceindices(integrate(z^n*(%e^(e+d*z))^q*(%e^(g+c*z^2))^u,z))-
5224
-c^(-n-1)*u^(-n-1)*%e^(g*u-d^2*q^2/(4*c*u)+e*q)
5285
(-c^(-n-1)*u^(-n-1)*%e^(g*u-d^2*q^2/(4*c*u)+e*q)
5225
5286
*('sum(d^(n-i)*gamma_incomplete((i+1)/2,-(2*c*u*z+d*q)^2/(4*c*u))
5226
5287
*(-1)^(i-n)*2^(i-n)*binomial(n,i)*q^(n-i)
5227
*(-c*u)^((i+1)/2)*(2*c*u*z+d*q)^(i+1)
5228
*abs(2*c*u*z+d*q)^(-i-1),i,0,n))
5229
/2);
5288
*(2*c*u*z+d*q)^(i+1)
5289
/(-(2*c*u*z+d*q)^2/(c*u))^((i+1)/2),i,0,n))
5290
/2));
5230
5291
0;
5231
5292
5232
5293
/* Problem 677: NEW 6 */
5233
5294
expand(niceindices(integrate(x^n*(%e^(e+d*x))^q*(%e^(g+c*sqrt(x)))^u,x))-
5234
2^(-n-1)*%e^(-c^2*u^2/(4*d*q)+g*u+e*q)
5295
(2^(-n-1)*%e^(-c^2*u^2/(4*d*q)+g*u+e*q)
5235
5296
*('sum(4^j*binomial(n,j)
5236
*'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)
5237
*(-d*q)^((j+i+1)/2)*u^(2*n-j-i)
5297
*'sum(c^(2*n-j-i)*d^((j+i+1)/2)*(-1)^(-(j+i+1)/2+j-i)
5298
*binomial(j,i)*q^((j+i+1)/2)*u^(2*n-j-i)
5238
5299
*abs(2*d*q*sqrt(x)+c*u)^(-j-i-1)
5239
5300
*(2*d*q*sqrt(x)+c*u)^(j+i)
5240
5301
*(c*gamma_incomplete((j+i+1)/2,
5241
5302
-(2*d*q*sqrt(x)+c*u)^2
5242
5303
/(4*d*q))*u
5243
5304
*(2*d*q*sqrt(x)+c*u)
5244
+2*d
5305
+2*%i*sqrt(d)
5245
5306
*gamma_incomplete((j+i+2)/2,
5246
5307
-(2*d*q*sqrt(x)+c*u)^2
5247
/(4*d*q))*q
5248
*abs(2*d*q*sqrt(x)+c*u)
5249
/sqrt(-d*q)),i,0,j),j,0,n))
5250
/(d^(2*(n+1))*q^(2*(n+1))));
5308
/(4*d*q))*sqrt(q)
5309
*abs(2*d*q*sqrt(x)+c*u)),i,0,j),j,0,n))
5310
/(d^(2*(n+1))*q^(2*(n+1)))));
5251
5311
0;
5252
5312
5253
5313
/* Problem 678: NEW */
5255
5315
-%e^(g*u)*gamma_incomplete(v/r,-(c*u+b*q)*z^r)*z^v/(r*(-(c*u+b*q)*z^r)^(v/r)));
5256
5316
0;
5257
5317
5258
/* Problem 679: NEW */
5318
/* Problem 679: NEW 2 */
5259
5319
expand(integrate(z^n*(%e^(b*z^2))^q*(%e^(g+c*z^2))^u,z)-
5260
-gamma_incomplete((n+1)/2,-(c*u+b*q)*z^2)*%e^(g*u)*z^(n+1)
5261
/(2*(-c*u-b*q)^((n+1)/2)*(z^2)^((n+1)/2)));
5320
(-gamma_incomplete((n+1)/2,-(c*u+b*q)*z^2)*%e^(g*u)*z^(n+1)
5321
/(2*(-(c*u+b*q)*z^2)^((n+1)/2))));
5262
5322
0;
5263
5323
5264
/* Problem 680: NEW */
5324
/* Problem 680: NEW 2 */
5265
5325
expand(integrate(z^(2*n)*(%e^(b*z^2))^q*(%e^(g+c*z^2))^u,z)-
5266
-gamma_incomplete((2*n+1)/2,-(c*u+b*q)*z^2)*%e^(g*u)*z^(2*n+1)
5267
/(2*(-c*u-b*q)^((2*n+1)/2)*(z^2)^((2*n+1)/2)));
5326
(-gamma_incomplete((2*n+1)/2,-(c*u+b*q)*z^2)*%e^(g*u)*z^(2*n+1)
5327
/(2*(-(c*u+b*q)*z^2)^((2*n+1)/2))));
5268
5328
0;
5269
5329
5270
5330
/* Problem 681: NEW */
5284
5344
/(r*(-(c*u+b*q)*z^r)^(v/r)));
5285
5345
0;
5286
5346
5287
/* Problem 684: NEW */
5347
/* Problem 684: NEW 2 */
5288
5348
expand(integrate(z^n*(%e^(e+b*z^2))^q*(%e^(g+c*z^2))^u,z)-
5289
-gamma_incomplete((n+1)/2,-(c*u+b*q)*z^2)*%e^(g*u+e*q)*z^(n+1)
5290
/(2*(-c*u-b*q)^((n+1)/2)*(z^2)^((n+1)/2)));
5349
(-gamma_incomplete((n+1)/2,-(c*u+b*q)*z^2)*%e^(g*u+e*q)*z^(n+1)
5350
/(2*(-(c*u+b*q)*z^2)^((n+1)/2))));
5291
5351
0;
5292
5352
5293
/* Problem 685: NEW */
5353
/* Problem 685: NEW 2 */
5294
5354
expand(integrate(z^(2*n)*(%e^(e+b*z^2))^q*(%e^(g+c*z^2))^u,z)-
5295
-gamma_incomplete((2*n+1)/2,-(c*u+b*q)*z^2)*%e^(g*u+e*q)*z^(2*n+1)
5296
/(2*(-c*u-b*q)^((2*n+1)/2)*(z^2)^((2*n+1)/2)));
5355
(-gamma_incomplete((2*n+1)/2,-(c*u+b*q)*z^2)*%e^(g*u+e*q)*z^(2*n+1)
5356
/(2*(-(c*u+b*q)*z^2)^((2*n+1)/2))));
5297
5357
0;
5298
5358
5299
5359
/* Problem 686: NEW */
5308
5368
/(c*u+b*q)^(2*(n+1)));
5309
5369
0;
5310
5370
5311
/* Problem 688: NEW 10 */
5371
/* Problem 688: NEW 5 */
5312
5372
expand(niceindices(integrate(z^n*(%e^(d*z))^q*(%e^(f*z+c*z^2))^u,z))-
5313
-c^(-n-1)*u^(-n-1)*%e^-((f*u+d*q)^2/(4*c*u))
5373
(-c^(-n-1)*u^(-n-1)*%e^-((f*u+d*q)^2/(4*c*u))
5314
5374
*('sum(gamma_incomplete((i+1)/2,-(2*c*u*z+f*u+d*q)^2/(4*c*u))
5315
*2^(i-n)*binomial(n,i)*(-c*u)^((i+1)/2)*(-f*u-d*q)^(n-i)
5316
*(2*c*u*z+f*u+d*q)^(i+1)*abs(2*c*u*z+f*u+d*q)^(-i-1),i,0,n))
5317
/2);
5375
*2^(i-n)*binomial(n,i)*(-f*u-d*q)^(n-i)
5376
*(2*c*u*z+f*u+d*q)^(i+1)
5377
/(-(2*c*u*z+f*u+d*q)^2/(c*u))^((i+1)/2),i,0,n))
5378
/2));
5318
5379
0;
5319
5380
5320
/* Problem 689: NEW 11 */
5381
/* Problem 689: NEW 6 */
5321
5382
expand(niceindices(integrate(x^n*(%e^(d*x))^q*(%e^(f*x+c*sqrt(x)))^u,x))-
5322
2^(-n-1)*%e^-(c^2*u^2/(4*(f*u+d*q)))
5383
(2^(-n-1)*%e^-(c^2*u^2/(4*(f*u+d*q)))
5323
5384
*('sum(4^j*binomial(n,j)
5324
*'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i)
5325
*(-f*u-d*q)^((j+i+1)/2)
5385
*'sum(c^(2*n-j-i)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
5386
*u^(2*n-j-i)*(f*u+d*q)^((j+i+1)/2)
5326
5387
*abs(2*(f*u+d*q)*sqrt(x)+c*u)^(-j-i-1)
5327
5388
*(2*(f*u+d*q)*sqrt(x)+c*u)^(j+i)
5328
5389
*(c*gamma_incomplete((j+i+1)/2,
5331
5392
^2
5332
5393
/(4*(f*u+d*q)))*u
5333
5394
*(2*(f*u+d*q)*sqrt(x)+c*u)
5334
+2*gamma_incomplete((j+i+2)/2,
5395
+2*%i
5396
*gamma_incomplete((j+i+2)/2,
5335
5397
-(2*(f*u+d*q)*sqrt(x)
5336
5398
+c*u)
5337
5399
^2
5338
5400
/(4*(f*u+d*q)))
5339
*(f*u+d*q)*abs(2*(f*u+d*q)*sqrt(x)+c*u)
5340
/sqrt(-f*u-d*q)),i,0,j),j,0,n))
5341
/(f*u+d*q)^(2*(n+1)));
5401
*sqrt(f*u+d*q)
5402
*abs(2*(f*u+d*q)*sqrt(x)+c*u)),i,0,j),j,
5403
0,n))
5404
/(f*u+d*q)^(2*(n+1))));
5342
5405
0;
5343
5406
5344
/* Problem 690: NEW 10 */
5407
/* Problem 690: NEW 5 */
5345
5408
expand(niceindices(integrate(z^n*(%e^(e+d*z))^q*(%e^(f*z+c*z^2))^u,z))-
5346
-c^(-n-1)*u^(-n-1)*%e^(e*q-(f*u+d*q)^2/(4*c*u))
5409
(-c^(-n-1)*u^(-n-1)*%e^(e*q-(f*u+d*q)^2/(4*c*u))
5347
5410
*('sum(gamma_incomplete((i+1)/2,-(2*c*u*z+f*u+d*q)^2/(4*c*u))
5348
*2^(i-n)*binomial(n,i)*(-c*u)^((i+1)/2)*(-f*u-d*q)^(n-i)
5349
*(2*c*u*z+f*u+d*q)^(i+1)*abs(2*c*u*z+f*u+d*q)^(-i-1),i,0,n))
5350
/2);
5411
*2^(i-n)*binomial(n,i)*(-f*u-d*q)^(n-i)
5412
*(2*c*u*z+f*u+d*q)^(i+1)
5413
/(-(2*c*u*z+f*u+d*q)^2/(c*u))^((i+1)/2),i,0,n))
5414
/2));
5351
5415
0;
5352
5416
5353
/* Problem 691: NEW 11 */
5417
/* Problem 691: NEW 6 */
5354
5418
expand(niceindices(integrate(x^n*(%e^(e+d*x))^q*(%e^(f*x+c*sqrt(x)))^u,x))-
5355
2^(-n-1)*%e^(e*q-c^2*u^2/(4*(f*u+d*q)))
5419
(2^(-n-1)*%e^(e*q-c^2*u^2/(4*(f*u+d*q)))
5356
5420
*('sum(4^j*binomial(n,j)
5357
*'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i)
5358
*(-f*u-d*q)^((j+i+1)/2)
5421
*'sum(c^(2*n-j-i)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
5422
*u^(2*n-j-i)*(f*u+d*q)^((j+i+1)/2)
5359
5423
*abs(2*(f*u+d*q)*sqrt(x)+c*u)^(-j-i-1)
5360
5424
*(2*(f*u+d*q)*sqrt(x)+c*u)^(j+i)
5361
5425
*(c*gamma_incomplete((j+i+1)/2,
5364
5428
^2
5365
5429
/(4*(f*u+d*q)))*u
5366
5430
*(2*(f*u+d*q)*sqrt(x)+c*u)
5367
+2*gamma_incomplete((j+i+2)/2,
5431
+2*%i
5432
*gamma_incomplete((j+i+2)/2,
5368
5433
-(2*(f*u+d*q)*sqrt(x)
5369
5434
+c*u)
5370
5435
^2
5371
5436
/(4*(f*u+d*q)))
5372
*(f*u+d*q)*abs(2*(f*u+d*q)*sqrt(x)+c*u)
5373
/sqrt(-f*u-d*q)),i,0,j),j,0,n))
5374
/(f*u+d*q)^(2*(n+1)));
5437
*sqrt(f*u+d*q)
5438
*abs(2*(f*u+d*q)*sqrt(x)+c*u)),i,0,j),j,
5439
0,n))
5440
/(f*u+d*q)^(2*(n+1))));
5375
5441
0;
5376
5442
5377
/* Problem 692: NEW 10 */
5443
/* Problem 692: NEW 5 */
5378
5444
expand(niceindices(integrate(z^n*(%e^(b*z^2))^q*(%e^(f*z+c*z^2))^u,z))-
5379
-(c*u+b*q)^(-n-1)*%e^-(f^2*u^2/(4*(c*u+b*q)))
5445
(-(c*u+b*q)^(-n-1)*%e^-(f^2*u^2/(4*(c*u+b*q)))
5380
5446
*('sum(f^(n-i)*gamma_incomplete((i+1)/2,
5381
5447
-(2*(c*u+b*q)*z+f*u)^2
5382
5448
/(4*(c*u+b*q)))*(-1)^(i-n)
5383
5449
*2^(i-n)*binomial(n,i)*u^(n-i)
5384
*(-c*u-b*q)^((i+1)/2)*(2*(c*u+b*q)*z+f*u)^(i+1)
5385
*abs(2*(c*u+b*q)*z+f*u)^(-i-1),i,0,n))
5386
/2);
5450
*(2*(c*u+b*q)*z+f*u)^(i+1)
5451
/(-(2*(c*u+b*q)*z+f*u)^2/(c*u+b*q))^((i+1)/2),i,0,n))
5452
/2));
5387
5453
0;
5388
5454
5389
/* Problem 693: NEW 11 */
5455
/* Problem 693: NEW 6 */
5390
5456
expand(niceindices(integrate(x^n*(%e^(b*sqrt(x)))^q*(%e^(f*x+c*sqrt(x)))^u,x))-
5391
2^(-n-1)*%e^-((c*u+b*q)^2/(4*f*u))
5457
(2^(-n-1)*%e^-((c*u+b*q)^2/(4*f*u))
5392
5458
*('sum(4^j*binomial(n,j)
5393
*'sum((-1)^(i-j)*binomial(j,i)*(-f*u)^((j+i+1)/2)
5394
*(c*u+b*q)^(2*n-j-i)
5395
*abs(2*f*u*sqrt(x)+c*u+b*q)^(-j-i-1)
5396
*(2*f*u*sqrt(x)+c*u+b*q)^(j+i)
5397
*(gamma_incomplete((j+i+1)/2,
5398
-(2*f*u*sqrt(x)+c*u+b*q)
5399
^2
5400
/(4*f*u))
5401
*(c*u+b*q)*(2*f*u*sqrt(x)+c*u+b*q)
5402
+2*f
5403
*gamma_incomplete((j+i+2)/2,
5404
-(2*f*u*sqrt(x)
5405
+c*u+b*q)
5406
^2
5407
/(4*f*u))*u
5408
*abs(2*f*u*sqrt(x)+c*u+b*q)
5409
/sqrt(-f*u)),i,0,j),j,0,n))
5410
/(f^(2*(n+1))*u^(2*(n+1))));
5459
*'sum((-1)^(-(j+i+1)/2+j-i)*binomial(j,i)*(f*u)^((j+i+1)/2)
5460
*(c*u+b*q)^(2*n-j-i)
5461
*abs(2*f*u*sqrt(x)+c*u+b*q)
5462
^(-j-i-1)
5463
*(2*f*u*sqrt(x)+c*u+b*q)^(j+i)
5464
*(gamma_incomplete(
5465
(j+i+1)/2,
5466
-(2*f*u*sqrt(x)+c*u+b*q)^2
5467
/(4*f*u))
5468
*(c*u+b*q)
5469
*(2*f*u*sqrt(x)+c*u+b*q)
5470
+2*%i*f
5471
*gamma_incomplete(
5472
(j+i+2)/2,
5473
-(2*f*u*sqrt(x)+c*u+b*q)^2
5474
/(4*f*u))*u
5475
*abs(2*f*u*sqrt(x)+c*u+b*q)
5476
/sqrt(f*u)),i,0,j),j,0,n))
5477
/(f^(2*(n+1))*u^(2*(n+1)))));
5411
5478
0;
5412
5479
5413
/* Problem 694: NEW 10 */
5480
/* Problem 694: NEW 5 */
5414
5481
expand(niceindices(integrate(z^n*(%e^(e+b*z^2))^q*(%e^(f*z+c*z^2))^u,z))-
5415
-(c*u+b*q)^(-n-1)*%e^(e*q-f^2*u^2/(4*(c*u+b*q)))
5482
(-(c*u+b*q)^(-n-1)*%e^(e*q-f^2*u^2/(4*(c*u+b*q)))
5416
5483
*('sum(f^(n-i)*gamma_incomplete((i+1)/2,
5417
5484
-(2*(c*u+b*q)*z+f*u)^2
5418
5485
/(4*(c*u+b*q)))*(-1)^(i-n)
5419
5486
*2^(i-n)*binomial(n,i)*u^(n-i)
5420
*(-c*u-b*q)^((i+1)/2)*(2*(c*u+b*q)*z+f*u)^(i+1)
5421
*abs(2*(c*u+b*q)*z+f*u)^(-i-1),i,0,n))
5422
/2);
5487
*(2*(c*u+b*q)*z+f*u)^(i+1)
5488
/(-(2*(c*u+b*q)*z+f*u)^2/(c*u+b*q))^((i+1)/2),i,0,n))
5489
/2));
5423
5490
0;
5424
5491
5425
/* Problem 695: NEW 11 */
5492
/* Problem 695: NEW 6 */
5426
5493
expand(niceindices(integrate(x^n*(%e^(e+b*sqrt(x)))^q*(%e^(f*x+c*sqrt(x)))^u,x))-
5427
2^(-n-1)*%e^(e*q-(c*u+b*q)^2/(4*f*u))
5494
(2^(-n-1)*%e^(e*q-(c*u+b*q)^2/(4*f*u))
5428
5495
*('sum(4^j*binomial(n,j)
5429
*'sum((-1)^(i-j)*binomial(j,i)*(-f*u)^((j+i+1)/2)
5430
*(c*u+b*q)^(2*n-j-i)
5431
*abs(2*f*u*sqrt(x)+c*u+b*q)^(-j-i-1)
5432
*(2*f*u*sqrt(x)+c*u+b*q)^(j+i)
5433
*(gamma_incomplete((j+i+1)/2,
5434
-(2*f*u*sqrt(x)+c*u+b*q)
5435
^2
5436
/(4*f*u))
5437
*(c*u+b*q)*(2*f*u*sqrt(x)+c*u+b*q)
5438
+2*f
5439
*gamma_incomplete((j+i+2)/2,
5440
-(2*f*u*sqrt(x)
5441
+c*u+b*q)
5442
^2
5443
/(4*f*u))*u
5444
*abs(2*f*u*sqrt(x)+c*u+b*q)
5445
/sqrt(-f*u)),i,0,j),j,0,n))
5446
/(f^(2*(n+1))*u^(2*(n+1))));
5496
*'sum((-1)^(-(j+i+1)/2+j-i)*binomial(j,i)*(f*u)^((j+i+1)/2)
5497
*(c*u+b*q)^(2*n-j-i)
5498
*abs(2*f*u*sqrt(x)+c*u+b*q)
5499
^(-j-i-1)
5500
*(2*f*u*sqrt(x)+c*u+b*q)^(j+i)
5501
*(gamma_incomplete(
5502
(j+i+1)/2,
5503
-(2*f*u*sqrt(x)+c*u+b*q)^2
5504
/(4*f*u))
5505
*(c*u+b*q)
5506
*(2*f*u*sqrt(x)+c*u+b*q)
5507
+2*%i*f
5508
*gamma_incomplete(
5509
(j+i+2)/2,
5510
-(2*f*u*sqrt(x)+c*u+b*q)^2
5511
/(4*f*u))*u
5512
*abs(2*f*u*sqrt(x)+c*u+b*q)
5513
/sqrt(f*u)),i,0,j),j,0,n))
5514
/(f^(2*(n+1))*u^(2*(n+1)))));
5447
5515
0;
5448
5516
5449
/* Problem 696: NEW 10 */
5517
/* Problem 696: NEW 5 */
5450
5518
expand(niceindices(integrate(z^n*(%e^(d*z+b*z^2))^q*(%e^(f*z+c*z^2))^u,z))-
5451
-(c*u+b*q)^(-n-1)*%e^-((f*u+d*q)^2/(4*(c*u+b*q)))
5519
(-(c*u+b*q)^(-n-1)*%e^-((f*u+d*q)^2/(4*(c*u+b*q)))
5452
5520
*('sum(gamma_incomplete((i+1)/2,
5453
5521
-(2*(c*u+b*q)*z+f*u+d*q)^2
5454
5522
/(4*(c*u+b*q)))
5455
*2^(i-n)*binomial(n,i)*(-c*u-b*q)^((i+1)/2)
5456
*(-f*u-d*q)^(n-i)*(2*(c*u+b*q)*z+f*u+d*q)^(i+1)
5457
*abs(2*(c*u+b*q)*z+f*u+d*q)^(-i-1),i,0,n))
5458
/2);
5523
*2^(i-n)*binomial(n,i)*(-f*u-d*q)^(n-i)
5524
*(2*(c*u+b*q)*z+f*u+d*q)^(i+1)
5525
/(-(2*(c*u+b*q)*z+f*u+d*q)^2/(c*u+b*q))^((i+1)/2),i,
5526
0,n))
5527
/2));
5459
5528
0;
5460
5529
5461
/* Problem 697: NEW 11 */
5530
/* Problem 697: NEW 6 */
5462
5531
expand(niceindices(integrate(x^n*(%e^(d*x+b*sqrt(x)))^q*(%e^(f*x+c*sqrt(x)))^u,x))-
5463
2^(-n-1)*%e^-((c*u+b*q)^2/(4*(f*u+d*q)))
5532
(2^(-n-1)*%e^-((c*u+b*q)^2/(4*(f*u+d*q)))
5464
5533
*('sum(4^j*binomial(n,j)
5465
*'sum((-1)^(i-j)*binomial(j,i)*(c*u+b*q)^(2*n-j-i)
5466
*(-f*u-d*q)^((j+i+1)/2)
5467
*abs(2*(f*u+d*q)*sqrt(x)+c*u+b*q)^(-j-i-1)
5468
*(2*(f*u+d*q)*sqrt(x)+c*u+b*q)^(j+i)
5469
*(gamma_incomplete((j+i+1)/2,
5470
-(2*(f*u+d*q)*sqrt(x)
5471
+c*u+b*q)
5472
^2
5473
/(4*(f*u+d*q)))
5474
*(c*u+b*q)*(2*(f*u+d*q)*sqrt(x)+c*u+b*q)
5475
+2*gamma_incomplete((j+i+2)/2,
5476
-(2*(f*u+d*q)*sqrt(x)
5477
+c*u+b*q)
5478
^2
5479
/(4*(f*u+d*q)))
5480
*(f*u+d*q)
5481
*abs(2*(f*u+d*q)*sqrt(x)+c*u+b*q)
5482
/sqrt(-f*u-d*q)),i,0,j),j,0,n))
5483
/(f*u+d*q)^(2*(n+1)));
5534
*'sum((-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
5535
*(c*u+b*q)^(2*n-j-i)
5536
*(f*u+d*q)^((j+i+1)/2)
5537
*abs(2*(f*u+d*q)*sqrt(x)+c*u+b*q)
5538
^(-j-i-1)
5539
*(2*(f*u+d*q)*sqrt(x)+c*u+b*q)
5540
^(j+i)
5541
*(gamma_incomplete(
5542
(j+i+1)/2,
5543
-(2*(f*u+d*q)*sqrt(x)+c*u+b*q)^2
5544
/(4*(f*u+d*q)))
5545
*(c*u+b*q)
5546
*(2*(f*u+d*q)*sqrt(x)+c*u+b*q)
5547
+2*%i
5548
*gamma_incomplete(
5549
(j+i+2)/2,
5550
-(2*(f*u+d*q)*sqrt(x)
5551
+c*u+b*q)
5552
^2
5553
/(4*(f*u+d*q)))
5554
*sqrt(f*u+d*q)
5555
*abs(
5556
2*(f*u+d*q)*sqrt(x)
5557
+c*u+b*q)),i,0,j),j,0,n))
5558
/(f*u+d*q)^(2*(n+1))));
5484
5559
0;
5485
5560
5486
/* Problem 698: NEW 10 */
5561
/* Problem 698: NEW 5 */
5487
5562
expand(niceindices(integrate(z^n*(%e^(d*z))^q*(%e^(g+f*z+c*z^2))^u,z))-
5488
-c^(-n-1)*u^(-n-1)*%e^(g*u-(f*u+d*q)^2/(4*c*u))
5563
(-c^(-n-1)*u^(-n-1)*%e^(g*u-(f*u+d*q)^2/(4*c*u))
5489
5564
*('sum(gamma_incomplete((i+1)/2,-(2*c*u*z+f*u+d*q)^2/(4*c*u))
5490
*2^(i-n)*binomial(n,i)*(-c*u)^((i+1)/2)*(-f*u-d*q)^(n-i)
5491
*(2*c*u*z+f*u+d*q)^(i+1)*abs(2*c*u*z+f*u+d*q)^(-i-1),i,0,n))
5492
/2);
5565
*2^(i-n)*binomial(n,i)*(-f*u-d*q)^(n-i)
5566
*(2*c*u*z+f*u+d*q)^(i+1)
5567
/(-(2*c*u*z+f*u+d*q)^2/(c*u))^((i+1)/2),i,0,n))
5568
/2));
5493
5569
0;
5494
5570
5495
/* Problem 699: NEW 11 */
5571
/* Problem 699: NEW 6 */
5496
5572
expand(niceindices(integrate(x^n*(%e^(d*x))^q*(%e^(g+f*x+c*sqrt(x)))^u,x))-
5497
2^(-n-1)*%e^(g*u-c^2*u^2/(4*(f*u+d*q)))
5573
(2^(-n-1)*%e^(g*u-c^2*u^2/(4*(f*u+d*q)))
5498
5574
*('sum(4^j*binomial(n,j)
5499
*'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i)
5500
*(-f*u-d*q)^((j+i+1)/2)
5575
*'sum(c^(2*n-j-i)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
5576
*u^(2*n-j-i)*(f*u+d*q)^((j+i+1)/2)
5501
5577
*abs(2*(f*u+d*q)*sqrt(x)+c*u)^(-j-i-1)
5502
5578
*(2*(f*u+d*q)*sqrt(x)+c*u)^(j+i)
5503
5579
*(c*gamma_incomplete((j+i+1)/2,
5506
5582
^2
5507
5583
/(4*(f*u+d*q)))*u
5508
5584
*(2*(f*u+d*q)*sqrt(x)+c*u)
5509
+2*gamma_incomplete((j+i+2)/2,
5585
+2*%i
5586
*gamma_incomplete((j+i+2)/2,
5510
5587
-(2*(f*u+d*q)*sqrt(x)
5511
5588
+c*u)
5512
5589
^2
5513
5590
/(4*(f*u+d*q)))
5514
*(f*u+d*q)*abs(2*(f*u+d*q)*sqrt(x)+c*u)
5515
/sqrt(-f*u-d*q)),i,0,j),j,0,n))
5516
/(f*u+d*q)^(2*(n+1)));
5591
*sqrt(f*u+d*q)
5592
*abs(2*(f*u+d*q)*sqrt(x)+c*u)),i,0,j),j,
5593
0,n))
5594
/(f*u+d*q)^(2*(n+1))));
5517
5595
0;
5518
5596
5519
/* Problem 700: NEW 10 */
5597
/* Problem 700: NEW 5 */
5520
5598
expand(niceindices(integrate(z^n*(%e^(e+d*z))^q*(%e^(g+f*z+c*z^2))^u,z))-
5521
-c^(-n-1)*u^(-n-1)*%e^(-(f*u+d*q)^2/(4*c*u)+g*u+e*q)
5599
(-c^(-n-1)*u^(-n-1)*%e^(-(f*u+d*q)^2/(4*c*u)+g*u+e*q)
5522
5600
*('sum(gamma_incomplete((i+1)/2,-(2*c*u*z+f*u+d*q)^2/(4*c*u))
5523
*2^(i-n)*binomial(n,i)*(-c*u)^((i+1)/2)*(-f*u-d*q)^(n-i)
5524
*(2*c*u*z+f*u+d*q)^(i+1)*abs(2*c*u*z+f*u+d*q)^(-i-1),i,0,n))
5525
/2);
5601
*2^(i-n)*binomial(n,i)*(-f*u-d*q)^(n-i)
5602
*(2*c*u*z+f*u+d*q)^(i+1)
5603
/(-(2*c*u*z+f*u+d*q)^2/(c*u))^((i+1)/2),i,0,n))
5604
/2));
5526
5605
0;
5527
5606
5528
/* Problem 701: NEW 11 */
5607
/* Problem 701: NEW 6 */
5529
5608
expand(niceindices(integrate(x^n*(%e^(e+d*x))^q*(%e^(g+f*x+c*sqrt(x)))^u,x))-
5530
2^(-n-1)*%e^(-c^2*u^2/(4*(f*u+d*q))+g*u+e*q)
5609
(2^(-n-1)*%e^(-c^2*u^2/(4*(f*u+d*q))+g*u+e*q)
5531
5610
*('sum(4^j*binomial(n,j)
5532
*'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i)
5533
*(-f*u-d*q)^((j+i+1)/2)
5611
*'sum(c^(2*n-j-i)*(-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
5612
*u^(2*n-j-i)*(f*u+d*q)^((j+i+1)/2)
5534
5613
*abs(2*(f*u+d*q)*sqrt(x)+c*u)^(-j-i-1)
5535
5614
*(2*(f*u+d*q)*sqrt(x)+c*u)^(j+i)
5536
5615
*(c*gamma_incomplete((j+i+1)/2,
5539
5618
^2
5540
5619
/(4*(f*u+d*q)))*u
5541
5620
*(2*(f*u+d*q)*sqrt(x)+c*u)
5542
+2*gamma_incomplete((j+i+2)/2,
5621
+2*%i
5622
*gamma_incomplete((j+i+2)/2,
5543
5623
-(2*(f*u+d*q)*sqrt(x)
5544
5624
+c*u)
5545
5625
^2
5546
5626
/(4*(f*u+d*q)))
5547
*(f*u+d*q)*abs(2*(f*u+d*q)*sqrt(x)+c*u)
5548
/sqrt(-f*u-d*q)),i,0,j),j,0,n))
5549
/(f*u+d*q)^(2*(n+1)));
5627
*sqrt(f*u+d*q)
5628
*abs(2*(f*u+d*q)*sqrt(x)+c*u)),i,0,j),j,
5629
0,n))
5630
/(f*u+d*q)^(2*(n+1))));
5550
5631
0;
5551
5632
5552
/* Problem 702: NEW 10 */
5633
/* Problem 702: NEW 5 */
5553
5634
expand(niceindices(integrate(z^n*(%e^(b*z^2))^q*(%e^(g+f*z+c*z^2))^u,z))-
5554
-(c*u+b*q)^(-n-1)*%e^(g*u-f^2*u^2/(4*(c*u+b*q)))
5635
(-(c*u+b*q)^(-n-1)*%e^(g*u-f^2*u^2/(4*(c*u+b*q)))
5555
5636
*('sum(f^(n-i)*gamma_incomplete((i+1)/2,
5556
5637
-(2*(c*u+b*q)*z+f*u)^2
5557
5638
/(4*(c*u+b*q)))*(-1)^(i-n)
5558
5639
*2^(i-n)*binomial(n,i)*u^(n-i)
5559
*(-c*u-b*q)^((i+1)/2)*(2*(c*u+b*q)*z+f*u)^(i+1)
5560
*abs(2*(c*u+b*q)*z+f*u)^(-i-1),i,0,n))
5561
/2);
5640
*(2*(c*u+b*q)*z+f*u)^(i+1)
5641
/(-(2*(c*u+b*q)*z+f*u)^2/(c*u+b*q))^((i+1)/2),i,0,n))
5642
/2));
5562
5643
0;
5563
5644
5564
/* Problem 703: NEW 11 */
5645
/* Problem 703: NEW 6 */
5565
5646
expand(niceindices(integrate(x^n*(%e^(b*sqrt(x)))^q*(%e^(g+f*x+c*sqrt(x)))^u,x))-
5566
2^(-n-1)*%e^(g*u-(c*u+b*q)^2/(4*f*u))
5647
(2^(-n-1)*%e^(g*u-(c*u+b*q)^2/(4*f*u))
5567
5648
*('sum(4^j*binomial(n,j)
5568
*'sum((-1)^(i-j)*binomial(j,i)*(-f*u)^((j+i+1)/2)
5569
*(c*u+b*q)^(2*n-j-i)
5570
*abs(2*f*u*sqrt(x)+c*u+b*q)^(-j-i-1)
5571
*(2*f*u*sqrt(x)+c*u+b*q)^(j+i)
5572
*(gamma_incomplete((j+i+1)/2,
5573
-(2*f*u*sqrt(x)+c*u+b*q)
5574
^2
5575
/(4*f*u))
5576
*(c*u+b*q)*(2*f*u*sqrt(x)+c*u+b*q)
5577
+2*f
5578
*gamma_incomplete((j+i+2)/2,
5579
-(2*f*u*sqrt(x)
5580
+c*u+b*q)
5581
^2
5582
/(4*f*u))*u
5583
*abs(2*f*u*sqrt(x)+c*u+b*q)
5584
/sqrt(-f*u)),i,0,j),j,0,n))
5585
/(f^(2*(n+1))*u^(2*(n+1))));
5649
*'sum((-1)^(-(j+i+1)/2+j-i)*binomial(j,i)*(f*u)^((j+i+1)/2)
5650
*(c*u+b*q)^(2*n-j-i)
5651
*abs(2*f*u*sqrt(x)+c*u+b*q)
5652
^(-j-i-1)
5653
*(2*f*u*sqrt(x)+c*u+b*q)^(j+i)
5654
*(gamma_incomplete(
5655
(j+i+1)/2,
5656
-(2*f*u*sqrt(x)+c*u+b*q)^2
5657
/(4*f*u))
5658
*(c*u+b*q)
5659
*(2*f*u*sqrt(x)+c*u+b*q)
5660
+2*%i*f
5661
*gamma_incomplete(
5662
(j+i+2)/2,
5663
-(2*f*u*sqrt(x)+c*u+b*q)^2
5664
/(4*f*u))*u
5665
*abs(2*f*u*sqrt(x)+c*u+b*q)
5666
/sqrt(f*u)),i,0,j),j,0,n))
5667
/(f^(2*(n+1))*u^(2*(n+1)))));
5586
5668
0;
5587
5669
5588
/* Problem 704: NEW 10 */
5670
/* Problem 704: NEW 5 */
5589
5671
expand(niceindices(integrate(z^n*(%e^(e+b*z^2))^q*(%e^(g+f*z+c*z^2))^u,z))-
5590
-(c*u+b*q)^(-n-1)*%e^(-f^2*u^2/(4*(c*u+b*q))+g*u+e*q)
5672
(-(c*u+b*q)^(-n-1)*%e^(-f^2*u^2/(4*(c*u+b*q))+g*u+e*q)
5591
5673
*('sum(f^(n-i)*gamma_incomplete((i+1)/2,
5592
5674
-(2*(c*u+b*q)*z+f*u)^2
5593
5675
/(4*(c*u+b*q)))*(-1)^(i-n)
5594
5676
*2^(i-n)*binomial(n,i)*u^(n-i)
5595
*(-c*u-b*q)^((i+1)/2)*(2*(c*u+b*q)*z+f*u)^(i+1)
5596
*abs(2*(c*u+b*q)*z+f*u)^(-i-1),i,0,n))
5597
/2);
5677
*(2*(c*u+b*q)*z+f*u)^(i+1)
5678
/(-(2*(c*u+b*q)*z+f*u)^2/(c*u+b*q))^((i+1)/2),i,0,n))
5679
/2));
5598
5680
0;
5599
5681
5600
/* Problem 705: NEW 11 */
5682
/* Problem 705: NEW 6 */
5601
5683
expand(niceindices(integrate(x^n*(%e^(e+b*sqrt(x)))^q*(%e^(g+f*x+c*sqrt(x)))^u,x))-
5602
2^(-n-1)*%e^(-(c*u+b*q)^2/(4*f*u)+g*u+e*q)
5684
(2^(-n-1)*%e^(-(c*u+b*q)^2/(4*f*u)+g*u+e*q)
5603
5685
*('sum(4^j*binomial(n,j)
5604
*'sum((-1)^(i-j)*binomial(j,i)*(-f*u)^((j+i+1)/2)
5605
*(c*u+b*q)^(2*n-j-i)
5606
*abs(2*f*u*sqrt(x)+c*u+b*q)^(-j-i-1)
5607
*(2*f*u*sqrt(x)+c*u+b*q)^(j+i)
5608
*(gamma_incomplete((j+i+1)/2,
5609
-(2*f*u*sqrt(x)+c*u+b*q)
5610
^2
5611
/(4*f*u))
5612
*(c*u+b*q)*(2*f*u*sqrt(x)+c*u+b*q)
5613
+2*f
5614
*gamma_incomplete((j+i+2)/2,
5615
-(2*f*u*sqrt(x)
5616
+c*u+b*q)
5617
^2
5618
/(4*f*u))*u
5619
*abs(2*f*u*sqrt(x)+c*u+b*q)
5620
/sqrt(-f*u)),i,0,j),j,0,n))
5621
/(f^(2*(n+1))*u^(2*(n+1))));
5686
*'sum((-1)^(-(j+i+1)/2+j-i)*binomial(j,i)*(f*u)^((j+i+1)/2)
5687
*(c*u+b*q)^(2*n-j-i)
5688
*abs(2*f*u*sqrt(x)+c*u+b*q)
5689
^(-j-i-1)
5690
*(2*f*u*sqrt(x)+c*u+b*q)^(j+i)
5691
*(gamma_incomplete(
5692
(j+i+1)/2,
5693
-(2*f*u*sqrt(x)+c*u+b*q)^2
5694
/(4*f*u))
5695
*(c*u+b*q)
5696
*(2*f*u*sqrt(x)+c*u+b*q)
5697
+2*%i*f
5698
*gamma_incomplete(
5699
(j+i+2)/2,
5700
-(2*f*u*sqrt(x)+c*u+b*q)^2
5701
/(4*f*u))*u
5702
*abs(2*f*u*sqrt(x)+c*u+b*q)
5703
/sqrt(f*u)),i,0,j),j,0,n))
5704
/(f^(2*(n+1))*u^(2*(n+1)))));
5622
5705
0;
5623
5706
5624
/* Problem 706: NEW 10 */
5707
/* Problem 706: NEW 5 */
5625
5708
expand(niceindices(integrate(z^n*(%e^(d*z+b*z^2))^q*(%e^(g+f*z+c*z^2))^u,z))-
5626
-(c*u+b*q)^(-n-1)*%e^(g*u-(f*u+d*q)^2/(4*(c*u+b*q)))
5709
(-(c*u+b*q)^(-n-1)*%e^(g*u-(f*u+d*q)^2/(4*(c*u+b*q)))
5627
5710
*('sum(gamma_incomplete((i+1)/2,
5628
5711
-(2*(c*u+b*q)*z+f*u+d*q)^2
5629
5712
/(4*(c*u+b*q)))
5630
*2^(i-n)*binomial(n,i)*(-c*u-b*q)^((i+1)/2)
5631
*(-f*u-d*q)^(n-i)*(2*(c*u+b*q)*z+f*u+d*q)^(i+1)
5632
*abs(2*(c*u+b*q)*z+f*u+d*q)^(-i-1),i,0,n))
5633
/2);
5713
*2^(i-n)*binomial(n,i)*(-f*u-d*q)^(n-i)
5714
*(2*(c*u+b*q)*z+f*u+d*q)^(i+1)
5715
/(-(2*(c*u+b*q)*z+f*u+d*q)^2/(c*u+b*q))^((i+1)/2),i,
5716
0,n))
5717
/2));
5634
5718
0;
5635
5719
5636
/* Problem 707: NEW 11 */
5720
/* Problem 707: NEW 6 */
5637
5721
expand(
5638
5722
niceindices(integrate(x^n*(%e^(d*x+b*sqrt(x)))^q*(%e^(g+f*x+c*sqrt(x)))^u,x))-
5639
2^(-n-1)*%e^(g*u-(c*u+b*q)^2/(4*(f*u+d*q)))
5723
(2^(-n-1)*%e^(g*u-(c*u+b*q)^2/(4*(f*u+d*q)))
5640
5724
*('sum(4^j*binomial(n,j)
5641
*'sum((-1)^(i-j)*binomial(j,i)*(c*u+b*q)^(2*n-j-i)
5642
*(-f*u-d*q)^((j+i+1)/2)
5643
*abs(2*(f*u+d*q)*sqrt(x)+c*u+b*q)^(-j-i-1)
5644
*(2*(f*u+d*q)*sqrt(x)+c*u+b*q)^(j+i)
5645
*(gamma_incomplete((j+i+1)/2,
5646
-(2*(f*u+d*q)*sqrt(x)
5647
+c*u+b*q)
5648
^2
5649
/(4*(f*u+d*q)))
5650
*(c*u+b*q)*(2*(f*u+d*q)*sqrt(x)+c*u+b*q)
5651
+2*gamma_incomplete((j+i+2)/2,
5652
-(2*(f*u+d*q)*sqrt(x)
5653
+c*u+b*q)
5654
^2
5655
/(4*(f*u+d*q)))
5656
*(f*u+d*q)
5657
*abs(2*(f*u+d*q)*sqrt(x)+c*u+b*q)
5658
/sqrt(-f*u-d*q)),i,0,j),j,0,n))
5659
/(f*u+d*q)^(2*(n+1)));
5725
*'sum((-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
5726
*(c*u+b*q)^(2*n-j-i)
5727
*(f*u+d*q)^((j+i+1)/2)
5728
*abs(2*(f*u+d*q)*sqrt(x)+c*u+b*q)
5729
^(-j-i-1)
5730
*(2*(f*u+d*q)*sqrt(x)+c*u+b*q)
5731
^(j+i)
5732
*(gamma_incomplete(
5733
(j+i+1)/2,
5734
-(2*(f*u+d*q)*sqrt(x)+c*u+b*q)^2
5735
/(4*(f*u+d*q)))
5736
*(c*u+b*q)
5737
*(2*(f*u+d*q)*sqrt(x)+c*u+b*q)
5738
+2*%i
5739
*gamma_incomplete(
5740
(j+i+2)/2,
5741
-(2*(f*u+d*q)*sqrt(x)
5742
+c*u+b*q)
5743
^2
5744
/(4*(f*u+d*q)))
5745
*sqrt(f*u+d*q)
5746
*abs(
5747
2*(f*u+d*q)*sqrt(x)
5748
+c*u+b*q)),i,0,j),j,0,n))
5749
/(f*u+d*q)^(2*(n+1))));
5660
5750
0;
5661
5751
5662
/* Problem 708: NEW 10 */
5752
/* Problem 708: NEW 5 */
5663
5753
expand(niceindices(integrate(z^n*(%e^(e+d*z+b*z^2))^q*(%e^(g+f*z+c*z^2))^u,z))-
5664
-(c*u+b*q)^(-n-1)*%e^(-(f*u+d*q)^2/(4*(c*u+b*q))+g*u+e*q)
5754
(-(c*u+b*q)^(-n-1)*%e^(-(f*u+d*q)^2/(4*(c*u+b*q))+g*u+e*q)
5665
5755
*('sum(gamma_incomplete((i+1)/2,
5666
5756
-(2*(c*u+b*q)*z+f*u+d*q)^2
5667
5757
/(4*(c*u+b*q)))
5668
*2^(i-n)*binomial(n,i)*(-c*u-b*q)^((i+1)/2)
5669
*(-f*u-d*q)^(n-i)*(2*(c*u+b*q)*z+f*u+d*q)^(i+1)
5670
*abs(2*(c*u+b*q)*z+f*u+d*q)^(-i-1),i,0,n))
5671
/2);
5758
*2^(i-n)*binomial(n,i)*(-f*u-d*q)^(n-i)
5759
*(2*(c*u+b*q)*z+f*u+d*q)^(i+1)
5760
/(-(2*(c*u+b*q)*z+f*u+d*q)^2/(c*u+b*q))^((i+1)/2),i,
5761
0,n))
5762
/2));
5672
5763
0;
5673
5764
5674
/* Problem 709: NEW 11 */
5765
/* Problem 709: NEW 6 */
5675
5766
expand(
5676
5767
niceindices(integrate(x^n*(%e^(e+d*x+b*sqrt(x)))^q*(%e^(g+f*x+c*sqrt(x)))^u,x))-
5677
2^(-n-1)*%e^(-(c*u+b*q)^2/(4*(f*u+d*q))+g*u+e*q)
5768
(2^(-n-1)*%e^(-(c*u+b*q)^2/(4*(f*u+d*q))+g*u+e*q)
5678
5769
*('sum(4^j*binomial(n,j)
5679
*'sum((-1)^(i-j)*binomial(j,i)*(c*u+b*q)^(2*n-j-i)
5680
*(-f*u-d*q)^((j+i+1)/2)
5681
*abs(2*(f*u+d*q)*sqrt(x)+c*u+b*q)^(-j-i-1)
5682
*(2*(f*u+d*q)*sqrt(x)+c*u+b*q)^(j+i)
5683
*(gamma_incomplete((j+i+1)/2,
5684
-(2*(f*u+d*q)*sqrt(x)
5685
+c*u+b*q)
5686
^2
5687
/(4*(f*u+d*q)))
5688
*(c*u+b*q)*(2*(f*u+d*q)*sqrt(x)+c*u+b*q)
5689
+2*gamma_incomplete((j+i+2)/2,
5690
-(2*(f*u+d*q)*sqrt(x)
5691
+c*u+b*q)
5692
^2
5693
/(4*(f*u+d*q)))
5694
*(f*u+d*q)
5695
*abs(2*(f*u+d*q)*sqrt(x)+c*u+b*q)
5696
/sqrt(-f*u-d*q)),i,0,j),j,0,n))
5697
/(f*u+d*q)^(2*(n+1)));
5770
*'sum((-1)^(-(j+i+1)/2+j-i)*binomial(j,i)
5771
*(c*u+b*q)^(2*n-j-i)
5772
*(f*u+d*q)^((j+i+1)/2)
5773
*abs(2*(f*u+d*q)*sqrt(x)+c*u+b*q)
5774
^(-j-i-1)
5775
*(2*(f*u+d*q)*sqrt(x)+c*u+b*q)
5776
^(j+i)
5777
*(gamma_incomplete(
5778
(j+i+1)/2,
5779
-(2*(f*u+d*q)*sqrt(x)+c*u+b*q)^2
5780
/(4*(f*u+d*q)))
5781
*(c*u+b*q)
5782
*(2*(f*u+d*q)*sqrt(x)+c*u+b*q)
5783
+2*%i
5784
*gamma_incomplete(
5785
(j+i+2)/2,
5786
-(2*(f*u+d*q)*sqrt(x)
5787
+c*u+b*q)
5788
^2
5789
/(4*(f*u+d*q)))
5790
*sqrt(f*u+d*q)
5791
*abs(
5792
2*(f*u+d*q)*sqrt(x)
5793
+c*u+b*q)),i,0,j),j,0,n))
5794
/(f*u+d*q)^(2*(n+1))));
5698
5795
0;
5699
5796
5700
5797
/* Problem 710: OK */
5902
5999
trigsign:true;
5903
6000
true;
5904
6001
5905
kill(all);
5906
done;
Loggerhead is a web-based interface for Breezy
Version: 2.0.1