Integrand size = 35, antiderivative size = 612 \[ \int \left (7+3 x-6 x^2\right )^q \left (1+5 x-2 x^2\right )^3 \left (3+2 x+4 x^2\right ) \, dx=-\frac {\left (2412977088+5623437568 q+5434627508 q^2+2842212284 q^3+871543171 q^4+156578109 q^5+15186393 q^6+607959 q^7\right ) \left (7+3 x-6 x^2\right )^{1+q}}{3456 (1+q) (2+q) (3+q) (4+q) (3+2 q) (5+2 q) (7+2 q) (9+2 q)}-\frac {\left (238010208+569038544 q+528541446 q^2+246189277 q^3+61144323 q^4+7732179 q^5+391383 q^6\right ) x \left (7+3 x-6 x^2\right )^{1+q}}{864 (2+q) (3+q) (4+q) (3+2 q) (5+2 q) (7+2 q) (9+2 q)}-\frac {\left (6468096+9304496 q+5312930 q^2+1517005 q^3+217914 q^4+12639 q^5\right ) x^2 \left (7+3 x-6 x^2\right )^{1+q}}{48 (2+q) (3+q) (4+q) (5+2 q) (7+2 q) (9+2 q)}+\frac {\left (3553404+2916686 q+774803 q^2+64434 q^3-567 q^4\right ) x^3 \left (7+3 x-6 x^2\right )^{1+q}}{108 (3+q) (4+q) (5+2 q) (7+2 q) (9+2 q)}-\frac {\left (67668+41534 q+7637 q^2+381 q^3\right ) x^4 \left (7+3 x-6 x^2\right )^{1+q}}{18 (3+q) (4+q) (7+2 q) (9+2 q)}+\frac {2 \left (10618+5089 q+603 q^2\right ) x^5 \left (7+3 x-6 x^2\right )^{1+q}}{9 (4+q) (7+2 q) (9+2 q)}-\frac {4 (118+27 q) x^6 \left (7+3 x-6 x^2\right )^{1+q}}{3 (4+q) (9+2 q)}+\frac {16 x^7 \left (7+3 x-6 x^2\right )^{1+q}}{3 (9+2 q)}-\frac {2^{-3 (3+q)} 59^q \left (589098472+754659810 q+286430679 q^2+42093666 q^3+1989765 q^4\right ) (1-4 x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-q,\frac {3}{2},\frac {3}{59} (1-4 x)^2\right )}{27 (3+2 q) (5+2 q) (7+2 q) (9+2 q)} \] Output:
-1/3456*(607959*q^7+15186393*q^6+156578109*q^5+871543171*q^4+2842212284*q^ 3+5434627508*q^2+5623437568*q+2412977088)*(-6*x^2+3*x+7)^(1+q)/(1+q)/(2+q) /(3+q)/(4+q)/(3+2*q)/(5+2*q)/(7+2*q)/(9+2*q)-1/864*(391383*q^6+7732179*q^5 +61144323*q^4+246189277*q^3+528541446*q^2+569038544*q+238010208)*x*(-6*x^2 +3*x+7)^(1+q)/(2+q)/(3+q)/(4+q)/(3+2*q)/(5+2*q)/(7+2*q)/(9+2*q)-1/48*(1263 9*q^5+217914*q^4+1517005*q^3+5312930*q^2+9304496*q+6468096)*x^2*(-6*x^2+3* x+7)^(1+q)/(2+q)/(3+q)/(4+q)/(5+2*q)/(7+2*q)/(9+2*q)+1/108*(-567*q^4+64434 *q^3+774803*q^2+2916686*q+3553404)*x^3*(-6*x^2+3*x+7)^(1+q)/(3+q)/(4+q)/(5 +2*q)/(7+2*q)/(9+2*q)-1/18*(381*q^3+7637*q^2+41534*q+67668)*x^4*(-6*x^2+3* x+7)^(1+q)/(3+q)/(4+q)/(7+2*q)/(9+2*q)+2/9*(603*q^2+5089*q+10618)*x^5*(-6* x^2+3*x+7)^(1+q)/(4+q)/(7+2*q)/(9+2*q)-4/3*(118+27*q)*x^6*(-6*x^2+3*x+7)^( 1+q)/(4+q)/(9+2*q)+16*x^7*(-6*x^2+3*x+7)^(1+q)/(27+6*q)-1/27*59^q*(1989765 *q^4+42093666*q^3+286430679*q^2+754659810*q+589098472)*(1-4*x)*hypergeom([ 1/2, -q],[3/2],3/59*(1-4*x)^2)/(2^(9+3*q))/(3+2*q)/(5+2*q)/(7+2*q)/(9+2*q)
Time = 1.91 (sec) , antiderivative size = 466, normalized size of antiderivative = 0.76 \[ \int \left (7+3 x-6 x^2\right )^q \left (1+5 x-2 x^2\right )^3 \left (3+2 x+4 x^2\right ) \, dx=\frac {8^{-3-q} \left (2^{2+3 q} \left (7+3 x-6 x^2\right )^{1+q} \left (9 q^7 \left (-67551-173948 x-202224 x^2-4032 x^3-32512 x^4+205824 x^5-110592 x^6+16384 x^7\right )+192 \left (-12567589-4958546 x-7276608 x^2+3553404 x^3-2030040 x^4+3822480 x^5-1784160 x^6+241920 x^7\right )+3 q^6 \left (-5062131-10831416 x-11976552 x^2+1320160 x^3-2637824 x^4+11385856 x^5-5928960 x^6+860160 x^7\right )+64 q \left (-87866212-50440547 x-67785714 x^2+31847184 x^3-19369368 x^4+38751696 x^5-18276192 x^6+2493504 x^7\right )+q^5 \left (-156578109-275506008 x-299627784 x^2+67908512 x^3-78148416 x^4+260733696 x^5-132129792 x^6+18837504 x^7\right )+4 q^3 \left (-710553071-774730723 x-895043826 x^2+350528200 x^3-261268656 x^4+629564736 x^5-305710848 x^6+42448896 x^7\right )+4 q^2 \left (-1358656877-1097579990 x-1357154316 x^2+596370976 x^3-395119488 x^4+856650048 x^5-409334400 x^6+56286720 x^7\right )+q^4 \left (-871543171-1229334400 x-1358253144 x^2+436506848 x^3-384996096 x^4+1066279680 x^5-527961600 x^6+74188800 x^7\right )\right )+59^q \left (14138363328+47566759040 q+65225773316 q^2+47635860004 q^3+20313207997 q^4+5191733160 q^5+777009114 q^6+61991316 q^7+1989765 q^8\right ) (-1+4 x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-q,\frac {3}{2},\frac {3}{59} (1-4 x)^2\right )\right )}{27 (1+q) (2+q) (3+q) (4+q) (3+2 q) (5+2 q) (7+2 q) (9+2 q)} \] Input:
Integrate[(7 + 3*x - 6*x^2)^q*(1 + 5*x - 2*x^2)^3*(3 + 2*x + 4*x^2),x]
Output:
(8^(-3 - q)*(2^(2 + 3*q)*(7 + 3*x - 6*x^2)^(1 + q)*(9*q^7*(-67551 - 173948 *x - 202224*x^2 - 4032*x^3 - 32512*x^4 + 205824*x^5 - 110592*x^6 + 16384*x ^7) + 192*(-12567589 - 4958546*x - 7276608*x^2 + 3553404*x^3 - 2030040*x^4 + 3822480*x^5 - 1784160*x^6 + 241920*x^7) + 3*q^6*(-5062131 - 10831416*x - 11976552*x^2 + 1320160*x^3 - 2637824*x^4 + 11385856*x^5 - 5928960*x^6 + 860160*x^7) + 64*q*(-87866212 - 50440547*x - 67785714*x^2 + 31847184*x^3 - 19369368*x^4 + 38751696*x^5 - 18276192*x^6 + 2493504*x^7) + q^5*(-1565781 09 - 275506008*x - 299627784*x^2 + 67908512*x^3 - 78148416*x^4 + 260733696 *x^5 - 132129792*x^6 + 18837504*x^7) + 4*q^3*(-710553071 - 774730723*x - 8 95043826*x^2 + 350528200*x^3 - 261268656*x^4 + 629564736*x^5 - 305710848*x ^6 + 42448896*x^7) + 4*q^2*(-1358656877 - 1097579990*x - 1357154316*x^2 + 596370976*x^3 - 395119488*x^4 + 856650048*x^5 - 409334400*x^6 + 56286720*x ^7) + q^4*(-871543171 - 1229334400*x - 1358253144*x^2 + 436506848*x^3 - 38 4996096*x^4 + 1066279680*x^5 - 527961600*x^6 + 74188800*x^7)) + 59^q*(1413 8363328 + 47566759040*q + 65225773316*q^2 + 47635860004*q^3 + 20313207997* q^4 + 5191733160*q^5 + 777009114*q^6 + 61991316*q^7 + 1989765*q^8)*(-1 + 4 *x)*Hypergeometric2F1[1/2, -q, 3/2, (3*(1 - 4*x)^2)/59]))/(27*(1 + q)*(2 + q)*(3 + q)*(4 + q)*(3 + 2*q)*(5 + 2*q)*(7 + 2*q)*(9 + 2*q))
Time = 3.14 (sec) , antiderivative size = 494, normalized size of antiderivative = 0.81, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 27, 2192, 25, 1160, 1090, 237}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (-2 x^2+5 x+1\right )^3 \left (4 x^2+2 x+3\right ) \left (-6 x^2+3 x+7\right )^q \, dx\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {16 x^7 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+9)}-\frac {\int -2 \left (-6 x^2+3 x+7\right )^q \left (48 (27 q+118) x^7-8 (342 q+1637) x^6+492 (2 q+9) x^5-24 (2 q+9) x^4+1179 (2 q+9) x^3+723 (2 q+9) x^2+141 (2 q+9) x+9 (2 q+9)\right )dx}{6 (2 q+9)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \left (-6 x^2+3 x+7\right )^q \left (48 (27 q+118) x^7-8 (342 q+1637) x^6+492 (2 q+9) x^5-24 (2 q+9) x^4+1179 (2 q+9) x^3+723 (2 q+9) x^2+141 (2 q+9) x+9 (2 q+9)\right )dx}{3 (2 q+9)}+\frac {16 x^7 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+9)}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {-\frac {\int -12 \left (-6 x^2+3 x+7\right )^q \left (-4 \left (603 q^2+5089 q+10618\right ) x^6+12 \left (82 q^2+1075 q+3128\right ) x^5-24 (q+4) (2 q+9) x^4+1179 (q+4) (2 q+9) x^3+723 (q+4) (2 q+9) x^2+141 (q+4) (2 q+9) x+9 (q+4) (2 q+9)\right )dx}{12 (q+4)}-\frac {4 (27 q+118) x^6 \left (-6 x^2+3 x+7\right )^{q+1}}{q+4}}{3 (2 q+9)}+\frac {16 x^7 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+9)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \left (-6 x^2+3 x+7\right )^q \left (-4 \left (603 q^2+5089 q+10618\right ) x^6+12 \left (82 q^2+1075 q+3128\right ) x^5-24 (q+4) (2 q+9) x^4+1179 (q+4) (2 q+9) x^3+723 (q+4) (2 q+9) x^2+141 (q+4) (2 q+9) x+9 (q+4) (2 q+9)\right )dx}{q+4}-\frac {4 (27 q+118) x^6 \left (-6 x^2+3 x+7\right )^{q+1}}{q+4}}{3 (2 q+9)}+\frac {16 x^7 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+9)}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {\frac {\frac {2 \left (603 q^2+5089 q+10618\right ) x^5 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+7)}-\frac {\int -2 \left (-6 x^2+3 x+7\right )^q \left (6 \left (381 q^3+7637 q^2+41534 q+67668\right ) x^5-2 \left (144 q^3+22833 q^2+184991 q+380702\right ) x^4+3537 (q+4) (2 q+7) (2 q+9) x^3+2169 (q+4) (2 q+7) (2 q+9) x^2+423 (q+4) (2 q+7) (2 q+9) x+27 (q+4) (2 q+7) (2 q+9)\right )dx}{6 (2 q+7)}}{q+4}-\frac {4 (27 q+118) x^6 \left (-6 x^2+3 x+7\right )^{q+1}}{q+4}}{3 (2 q+9)}+\frac {16 x^7 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+9)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \left (-6 x^2+3 x+7\right )^q \left (6 \left (381 q^3+7637 q^2+41534 q+67668\right ) x^5-2 \left (144 q^3+22833 q^2+184991 q+380702\right ) x^4+3537 (q+4) (2 q+7) (2 q+9) x^3+2169 (q+4) (2 q+7) (2 q+9) x^2+423 (q+4) (2 q+7) (2 q+9) x+27 (q+4) (2 q+7) (2 q+9)\right )dx}{3 (2 q+7)}+\frac {2 \left (603 q^2+5089 q+10618\right ) x^5 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+7)}}{q+4}-\frac {4 (27 q+118) x^6 \left (-6 x^2+3 x+7\right )^{q+1}}{q+4}}{3 (2 q+9)}+\frac {16 x^7 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+9)}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {\frac {\frac {-\frac {\int -6 \left (-6 x^2+3 x+7\right )^q \left (-\left (\left (-567 q^4+64434 q^3+774803 q^2+2916686 q+3553404\right ) x^4\right )+2 \left (14148 q^4+217554 q^3+1291813 q^2+3499501 q+3621324\right ) x^3+4338 (q+3) (q+4) (2 q+7) (2 q+9) x^2+846 (q+3) (q+4) (2 q+7) (2 q+9) x+54 (q+3) (q+4) (2 q+7) (2 q+9)\right )dx}{12 (q+3)}-\frac {\left (381 q^3+7637 q^2+41534 q+67668\right ) x^4 \left (-6 x^2+3 x+7\right )^{q+1}}{2 (q+3)}}{3 (2 q+7)}+\frac {2 \left (603 q^2+5089 q+10618\right ) x^5 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+7)}}{q+4}-\frac {4 (27 q+118) x^6 \left (-6 x^2+3 x+7\right )^{q+1}}{q+4}}{3 (2 q+9)}+\frac {16 x^7 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+9)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \left (-6 x^2+3 x+7\right )^q \left (-\left (\left (-567 q^4+64434 q^3+774803 q^2+2916686 q+3553404\right ) x^4\right )+2 \left (14148 q^4+217554 q^3+1291813 q^2+3499501 q+3621324\right ) x^3+4338 (q+3) (q+4) (2 q+7) (2 q+9) x^2+846 (q+3) (q+4) (2 q+7) (2 q+9) x+54 (q+3) (q+4) (2 q+7) (2 q+9)\right )dx}{2 (q+3)}-\frac {\left (381 q^3+7637 q^2+41534 q+67668\right ) x^4 \left (-6 x^2+3 x+7\right )^{q+1}}{2 (q+3)}}{3 (2 q+7)}+\frac {2 \left (603 q^2+5089 q+10618\right ) x^5 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+7)}}{q+4}-\frac {4 (27 q+118) x^6 \left (-6 x^2+3 x+7\right )^{q+1}}{q+4}}{3 (2 q+9)}+\frac {16 x^7 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+9)}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\left (-567 q^4+64434 q^3+774803 q^2+2916686 q+3553404\right ) x^3 \left (-6 x^2+3 x+7\right )^{q+1}}{6 (2 q+5)}-\frac {\int -3 \left (-6 x^2+3 x+7\right )^q \left (9 \left (12639 q^5+217914 q^4+1517005 q^3+5312930 q^2+9304496 q+6468096\right ) x^3+\left (69408 q^5+1218609 q^4+7964682 q^3+23424079 q^2+28489810 q+7921452\right ) x^2+1692 (q+3) (q+4) (2 q+5) (2 q+7) (2 q+9) x+108 (q+3) (q+4) (2 q+5) (2 q+7) (2 q+9)\right )dx}{6 (2 q+5)}}{2 (q+3)}-\frac {\left (381 q^3+7637 q^2+41534 q+67668\right ) x^4 \left (-6 x^2+3 x+7\right )^{q+1}}{2 (q+3)}}{3 (2 q+7)}+\frac {2 \left (603 q^2+5089 q+10618\right ) x^5 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+7)}}{q+4}-\frac {4 (27 q+118) x^6 \left (-6 x^2+3 x+7\right )^{q+1}}{q+4}}{3 (2 q+9)}+\frac {16 x^7 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+9)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \left (-6 x^2+3 x+7\right )^q \left (9 \left (12639 q^5+217914 q^4+1517005 q^3+5312930 q^2+9304496 q+6468096\right ) x^3+\left (69408 q^5+1218609 q^4+7964682 q^3+23424079 q^2+28489810 q+7921452\right ) x^2+1692 (q+3) (q+4) (2 q+5) (2 q+7) (2 q+9) x+108 (q+3) (q+4) (2 q+5) (2 q+7) (2 q+9)\right )dx}{2 (2 q+5)}+\frac {\left (-567 q^4+64434 q^3+774803 q^2+2916686 q+3553404\right ) x^3 \left (-6 x^2+3 x+7\right )^{q+1}}{6 (2 q+5)}}{2 (q+3)}-\frac {\left (381 q^3+7637 q^2+41534 q+67668\right ) x^4 \left (-6 x^2+3 x+7\right )^{q+1}}{2 (q+3)}}{3 (2 q+7)}+\frac {2 \left (603 q^2+5089 q+10618\right ) x^5 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+7)}}{q+4}-\frac {4 (27 q+118) x^6 \left (-6 x^2+3 x+7\right )^{q+1}}{q+4}}{3 (2 q+9)}+\frac {16 x^7 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+9)}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {-\frac {\int -3 \left (-6 x^2+3 x+7\right )^q \left (\left (391383 q^6+7732179 q^5+61144323 q^4+246189277 q^3+528541446 q^2+569038544 q+238010208\right ) x^2+6 \left (9024 q^6+264441 q^5+2935398 q^4+16557955 q^3+51050246 q^2+82112384 q+53804352\right ) x+432 (q+2) (q+3) (q+4) (2 q+5) (2 q+7) (2 q+9)\right )dx}{12 (q+2)}-\frac {3 \left (12639 q^5+217914 q^4+1517005 q^3+5312930 q^2+9304496 q+6468096\right ) x^2 \left (-6 x^2+3 x+7\right )^{q+1}}{4 (q+2)}}{2 (2 q+5)}+\frac {\left (-567 q^4+64434 q^3+774803 q^2+2916686 q+3553404\right ) x^3 \left (-6 x^2+3 x+7\right )^{q+1}}{6 (2 q+5)}}{2 (q+3)}-\frac {\left (381 q^3+7637 q^2+41534 q+67668\right ) x^4 \left (-6 x^2+3 x+7\right )^{q+1}}{2 (q+3)}}{3 (2 q+7)}+\frac {2 \left (603 q^2+5089 q+10618\right ) x^5 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+7)}}{q+4}-\frac {4 (27 q+118) x^6 \left (-6 x^2+3 x+7\right )^{q+1}}{q+4}}{3 (2 q+9)}+\frac {16 x^7 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+9)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\int \left (-6 x^2+3 x+7\right )^q \left (\left (391383 q^6+7732179 q^5+61144323 q^4+246189277 q^3+528541446 q^2+569038544 q+238010208\right ) x^2+6 \left (9024 q^6+264441 q^5+2935398 q^4+16557955 q^3+51050246 q^2+82112384 q+53804352\right ) x+432 (q+2) (q+3) (q+4) (2 q+5) (2 q+7) (2 q+9)\right )dx}{4 (q+2)}-\frac {3 \left (12639 q^5+217914 q^4+1517005 q^3+5312930 q^2+9304496 q+6468096\right ) x^2 \left (-6 x^2+3 x+7\right )^{q+1}}{4 (q+2)}}{2 (2 q+5)}+\frac {\left (-567 q^4+64434 q^3+774803 q^2+2916686 q+3553404\right ) x^3 \left (-6 x^2+3 x+7\right )^{q+1}}{6 (2 q+5)}}{2 (q+3)}-\frac {\left (381 q^3+7637 q^2+41534 q+67668\right ) x^4 \left (-6 x^2+3 x+7\right )^{q+1}}{2 (q+3)}}{3 (2 q+7)}+\frac {2 \left (603 q^2+5089 q+10618\right ) x^5 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+7)}}{q+4}-\frac {4 (27 q+118) x^6 \left (-6 x^2+3 x+7\right )^{q+1}}{q+4}}{3 (2 q+9)}+\frac {16 x^7 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+9)}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {-\frac {\int -\left (\left (41472 q^7+3610593 q^6+61818309 q^5+465024021 q^4+1827961387 q^3+3873373770 q^2+4139520752 q+3 \left (607959 q^7+15186393 q^6+156578109 q^5+871543171 q^4+2842212284 q^3+5434627508 q^2+5623437568 q+2412977088\right ) x+1724858016\right ) \left (-6 x^2+3 x+7\right )^q\right )dx}{6 (2 q+3)}-\frac {\left (391383 q^6+7732179 q^5+61144323 q^4+246189277 q^3+528541446 q^2+569038544 q+238010208\right ) x \left (-6 x^2+3 x+7\right )^{q+1}}{6 (2 q+3)}}{4 (q+2)}-\frac {3 \left (12639 q^5+217914 q^4+1517005 q^3+5312930 q^2+9304496 q+6468096\right ) x^2 \left (-6 x^2+3 x+7\right )^{q+1}}{4 (q+2)}}{2 (2 q+5)}+\frac {\left (-567 q^4+64434 q^3+774803 q^2+2916686 q+3553404\right ) x^3 \left (-6 x^2+3 x+7\right )^{q+1}}{6 (2 q+5)}}{2 (q+3)}-\frac {\left (381 q^3+7637 q^2+41534 q+67668\right ) x^4 \left (-6 x^2+3 x+7\right )^{q+1}}{2 (q+3)}}{3 (2 q+7)}+\frac {2 \left (603 q^2+5089 q+10618\right ) x^5 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+7)}}{q+4}-\frac {4 (27 q+118) x^6 \left (-6 x^2+3 x+7\right )^{q+1}}{q+4}}{3 (2 q+9)}+\frac {16 x^7 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+9)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {\int \left (41472 q^7+3610593 q^6+61818309 q^5+465024021 q^4+1827961387 q^3+3873373770 q^2+4139520752 q+3 \left (607959 q^7+15186393 q^6+156578109 q^5+871543171 q^4+2842212284 q^3+5434627508 q^2+5623437568 q+2412977088\right ) x+1724858016\right ) \left (-6 x^2+3 x+7\right )^qdx}{6 (2 q+3)}-\frac {\left (391383 q^6+7732179 q^5+61144323 q^4+246189277 q^3+528541446 q^2+569038544 q+238010208\right ) x \left (-6 x^2+3 x+7\right )^{q+1}}{6 (2 q+3)}}{4 (q+2)}-\frac {3 \left (12639 q^5+217914 q^4+1517005 q^3+5312930 q^2+9304496 q+6468096\right ) x^2 \left (-6 x^2+3 x+7\right )^{q+1}}{4 (q+2)}}{2 (2 q+5)}+\frac {\left (-567 q^4+64434 q^3+774803 q^2+2916686 q+3553404\right ) x^3 \left (-6 x^2+3 x+7\right )^{q+1}}{6 (2 q+5)}}{2 (q+3)}-\frac {\left (381 q^3+7637 q^2+41534 q+67668\right ) x^4 \left (-6 x^2+3 x+7\right )^{q+1}}{2 (q+3)}}{3 (2 q+7)}+\frac {2 \left (603 q^2+5089 q+10618\right ) x^5 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+7)}}{q+4}-\frac {4 (27 q+118) x^6 \left (-6 x^2+3 x+7\right )^{q+1}}{q+4}}{3 (2 q+9)}+\frac {16 x^7 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+9)}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {\frac {1}{4} (q+2) (q+3) (q+4) \left (1989765 q^4+42093666 q^3+286430679 q^2+754659810 q+589098472\right ) \int \left (-6 x^2+3 x+7\right )^qdx-\frac {\left (607959 q^7+15186393 q^6+156578109 q^5+871543171 q^4+2842212284 q^3+5434627508 q^2+5623437568 q+2412977088\right ) \left (-6 x^2+3 x+7\right )^{q+1}}{4 (q+1)}}{6 (2 q+3)}-\frac {\left (391383 q^6+7732179 q^5+61144323 q^4+246189277 q^3+528541446 q^2+569038544 q+238010208\right ) x \left (-6 x^2+3 x+7\right )^{q+1}}{6 (2 q+3)}}{4 (q+2)}-\frac {3 \left (12639 q^5+217914 q^4+1517005 q^3+5312930 q^2+9304496 q+6468096\right ) x^2 \left (-6 x^2+3 x+7\right )^{q+1}}{4 (q+2)}}{2 (2 q+5)}+\frac {\left (-567 q^4+64434 q^3+774803 q^2+2916686 q+3553404\right ) x^3 \left (-6 x^2+3 x+7\right )^{q+1}}{6 (2 q+5)}}{2 (q+3)}-\frac {\left (381 q^3+7637 q^2+41534 q+67668\right ) x^4 \left (-6 x^2+3 x+7\right )^{q+1}}{2 (q+3)}}{3 (2 q+7)}+\frac {2 \left (603 q^2+5089 q+10618\right ) x^5 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+7)}}{q+4}-\frac {4 (27 q+118) x^6 \left (-6 x^2+3 x+7\right )^{q+1}}{q+4}}{3 (2 q+9)}+\frac {16 x^7 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+9)}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {-\frac {1}{3} 2^{-3 q-4} 59^q (q+2) (q+3) (q+4) \left (1989765 q^4+42093666 q^3+286430679 q^2+754659810 q+589098472\right ) \int \left (1-\frac {1}{177} (3-12 x)^2\right )^qd(3-12 x)-\frac {\left (607959 q^7+15186393 q^6+156578109 q^5+871543171 q^4+2842212284 q^3+5434627508 q^2+5623437568 q+2412977088\right ) \left (-6 x^2+3 x+7\right )^{q+1}}{4 (q+1)}}{6 (2 q+3)}-\frac {\left (391383 q^6+7732179 q^5+61144323 q^4+246189277 q^3+528541446 q^2+569038544 q+238010208\right ) x \left (-6 x^2+3 x+7\right )^{q+1}}{6 (2 q+3)}}{4 (q+2)}-\frac {3 \left (12639 q^5+217914 q^4+1517005 q^3+5312930 q^2+9304496 q+6468096\right ) x^2 \left (-6 x^2+3 x+7\right )^{q+1}}{4 (q+2)}}{2 (2 q+5)}+\frac {\left (-567 q^4+64434 q^3+774803 q^2+2916686 q+3553404\right ) x^3 \left (-6 x^2+3 x+7\right )^{q+1}}{6 (2 q+5)}}{2 (q+3)}-\frac {\left (381 q^3+7637 q^2+41534 q+67668\right ) x^4 \left (-6 x^2+3 x+7\right )^{q+1}}{2 (q+3)}}{3 (2 q+7)}+\frac {2 \left (603 q^2+5089 q+10618\right ) x^5 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+7)}}{q+4}-\frac {4 (27 q+118) x^6 \left (-6 x^2+3 x+7\right )^{q+1}}{q+4}}{3 (2 q+9)}+\frac {16 x^7 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+9)}\) |
| \(\Big \downarrow \) 237 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {-\frac {1}{3} 2^{-3 q-4} 59^q (q+2) (q+3) (q+4) \left (1989765 q^4+42093666 q^3+286430679 q^2+754659810 q+589098472\right ) (3-12 x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-q,\frac {3}{2},\frac {1}{177} (3-12 x)^2\right )-\frac {\left (607959 q^7+15186393 q^6+156578109 q^5+871543171 q^4+2842212284 q^3+5434627508 q^2+5623437568 q+2412977088\right ) \left (-6 x^2+3 x+7\right )^{q+1}}{4 (q+1)}}{6 (2 q+3)}-\frac {\left (391383 q^6+7732179 q^5+61144323 q^4+246189277 q^3+528541446 q^2+569038544 q+238010208\right ) x \left (-6 x^2+3 x+7\right )^{q+1}}{6 (2 q+3)}}{4 (q+2)}-\frac {3 \left (12639 q^5+217914 q^4+1517005 q^3+5312930 q^2+9304496 q+6468096\right ) x^2 \left (-6 x^2+3 x+7\right )^{q+1}}{4 (q+2)}}{2 (2 q+5)}+\frac {\left (-567 q^4+64434 q^3+774803 q^2+2916686 q+3553404\right ) x^3 \left (-6 x^2+3 x+7\right )^{q+1}}{6 (2 q+5)}}{2 (q+3)}-\frac {\left (381 q^3+7637 q^2+41534 q+67668\right ) x^4 \left (-6 x^2+3 x+7\right )^{q+1}}{2 (q+3)}}{3 (2 q+7)}+\frac {2 \left (603 q^2+5089 q+10618\right ) x^5 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+7)}}{q+4}-\frac {4 (27 q+118) x^6 \left (-6 x^2+3 x+7\right )^{q+1}}{q+4}}{3 (2 q+9)}+\frac {16 x^7 \left (-6 x^2+3 x+7\right )^{q+1}}{3 (2 q+9)}\) |
Input:
Int[(7 + 3*x - 6*x^2)^q*(1 + 5*x - 2*x^2)^3*(3 + 2*x + 4*x^2),x]
Output:
(16*x^7*(7 + 3*x - 6*x^2)^(1 + q))/(3*(9 + 2*q)) + ((-4*(118 + 27*q)*x^6*( 7 + 3*x - 6*x^2)^(1 + q))/(4 + q) + ((2*(10618 + 5089*q + 603*q^2)*x^5*(7 + 3*x - 6*x^2)^(1 + q))/(3*(7 + 2*q)) + (-1/2*((67668 + 41534*q + 7637*q^2 + 381*q^3)*x^4*(7 + 3*x - 6*x^2)^(1 + q))/(3 + q) + (((3553404 + 2916686* q + 774803*q^2 + 64434*q^3 - 567*q^4)*x^3*(7 + 3*x - 6*x^2)^(1 + q))/(6*(5 + 2*q)) + ((-3*(6468096 + 9304496*q + 5312930*q^2 + 1517005*q^3 + 217914* q^4 + 12639*q^5)*x^2*(7 + 3*x - 6*x^2)^(1 + q))/(4*(2 + q)) + (-1/6*((2380 10208 + 569038544*q + 528541446*q^2 + 246189277*q^3 + 61144323*q^4 + 77321 79*q^5 + 391383*q^6)*x*(7 + 3*x - 6*x^2)^(1 + q))/(3 + 2*q) + (-1/4*((2412 977088 + 5623437568*q + 5434627508*q^2 + 2842212284*q^3 + 871543171*q^4 + 156578109*q^5 + 15186393*q^6 + 607959*q^7)*(7 + 3*x - 6*x^2)^(1 + q))/(1 + q) - (2^(-4 - 3*q)*59^q*(2 + q)*(3 + q)*(4 + q)*(589098472 + 754659810*q + 286430679*q^2 + 42093666*q^3 + 1989765*q^4)*(3 - 12*x)*Hypergeometric2F1 [1/2, -q, 3/2, (3 - 12*x)^2/177])/3)/(6*(3 + 2*q)))/(4*(2 + q)))/(2*(5 + 2 *q)))/(2*(3 + q)))/(3*(7 + 2*q)))/(4 + q))/(3*(9 + 2*q))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[- p, 1/2, 1/2 + 1, (-b)*(x^2/a)], x] /; FreeQ[{a, b, p}, x] && !IntegerQ[2*p ] && GtQ[a, 0]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1)) Int[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b *e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c , p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && !LeQ[p, -1]
\[\int \left (-6 x^{2}+3 x +7\right )^{q} \left (-2 x^{2}+5 x +1\right )^{3} \left (4 x^{2}+2 x +3\right )d x\]
Input:
int((-6*x^2+3*x+7)^q*(-2*x^2+5*x+1)^3*(4*x^2+2*x+3),x)
Output:
int((-6*x^2+3*x+7)^q*(-2*x^2+5*x+1)^3*(4*x^2+2*x+3),x)
\[ \int \left (7+3 x-6 x^2\right )^q \left (1+5 x-2 x^2\right )^3 \left (3+2 x+4 x^2\right ) \, dx=\int { -{\left (4 \, x^{2} + 2 \, x + 3\right )} {\left (2 \, x^{2} - 5 \, x - 1\right )}^{3} {\left (-6 \, x^{2} + 3 \, x + 7\right )}^{q} \,d x } \] Input:
integrate((-6*x^2+3*x+7)^q*(-2*x^2+5*x+1)^3*(4*x^2+2*x+3),x, algorithm="fr icas")
Output:
integral(-(32*x^8 - 224*x^7 + 456*x^6 - 164*x^5 + 8*x^4 - 393*x^3 - 241*x^ 2 - 47*x - 3)*(-6*x^2 + 3*x + 7)^q, x)
\[ \int \left (7+3 x-6 x^2\right )^q \left (1+5 x-2 x^2\right )^3 \left (3+2 x+4 x^2\right ) \, dx=- \int \left (- 47 x \left (- 6 x^{2} + 3 x + 7\right )^{q}\right )\, dx - \int \left (- 241 x^{2} \left (- 6 x^{2} + 3 x + 7\right )^{q}\right )\, dx - \int \left (- 393 x^{3} \left (- 6 x^{2} + 3 x + 7\right )^{q}\right )\, dx - \int 8 x^{4} \left (- 6 x^{2} + 3 x + 7\right )^{q}\, dx - \int \left (- 164 x^{5} \left (- 6 x^{2} + 3 x + 7\right )^{q}\right )\, dx - \int 456 x^{6} \left (- 6 x^{2} + 3 x + 7\right )^{q}\, dx - \int \left (- 224 x^{7} \left (- 6 x^{2} + 3 x + 7\right )^{q}\right )\, dx - \int 32 x^{8} \left (- 6 x^{2} + 3 x + 7\right )^{q}\, dx - \int \left (- 3 \left (- 6 x^{2} + 3 x + 7\right )^{q}\right )\, dx \] Input:
integrate((-6*x**2+3*x+7)**q*(-2*x**2+5*x+1)**3*(4*x**2+2*x+3),x)
Output:
-Integral(-47*x*(-6*x**2 + 3*x + 7)**q, x) - Integral(-241*x**2*(-6*x**2 + 3*x + 7)**q, x) - Integral(-393*x**3*(-6*x**2 + 3*x + 7)**q, x) - Integra l(8*x**4*(-6*x**2 + 3*x + 7)**q, x) - Integral(-164*x**5*(-6*x**2 + 3*x + 7)**q, x) - Integral(456*x**6*(-6*x**2 + 3*x + 7)**q, x) - Integral(-224*x **7*(-6*x**2 + 3*x + 7)**q, x) - Integral(32*x**8*(-6*x**2 + 3*x + 7)**q, x) - Integral(-3*(-6*x**2 + 3*x + 7)**q, x)
\[ \int \left (7+3 x-6 x^2\right )^q \left (1+5 x-2 x^2\right )^3 \left (3+2 x+4 x^2\right ) \, dx=\int { -{\left (4 \, x^{2} + 2 \, x + 3\right )} {\left (2 \, x^{2} - 5 \, x - 1\right )}^{3} {\left (-6 \, x^{2} + 3 \, x + 7\right )}^{q} \,d x } \] Input:
integrate((-6*x^2+3*x+7)^q*(-2*x^2+5*x+1)^3*(4*x^2+2*x+3),x, algorithm="ma xima")
Output:
-integrate((4*x^2 + 2*x + 3)*(2*x^2 - 5*x - 1)^3*(-6*x^2 + 3*x + 7)^q, x)
\[ \int \left (7+3 x-6 x^2\right )^q \left (1+5 x-2 x^2\right )^3 \left (3+2 x+4 x^2\right ) \, dx=\int { -{\left (4 \, x^{2} + 2 \, x + 3\right )} {\left (2 \, x^{2} - 5 \, x - 1\right )}^{3} {\left (-6 \, x^{2} + 3 \, x + 7\right )}^{q} \,d x } \] Input:
integrate((-6*x^2+3*x+7)^q*(-2*x^2+5*x+1)^3*(4*x^2+2*x+3),x, algorithm="gi ac")
Output:
integrate(-(4*x^2 + 2*x + 3)*(2*x^2 - 5*x - 1)^3*(-6*x^2 + 3*x + 7)^q, x)
Timed out. \[ \int \left (7+3 x-6 x^2\right )^q \left (1+5 x-2 x^2\right )^3 \left (3+2 x+4 x^2\right ) \, dx=\int {\left (-2\,x^2+5\,x+1\right )}^3\,\left (4\,x^2+2\,x+3\right )\,{\left (-6\,x^2+3\,x+7\right )}^q \,d x \] Input:
int((5*x - 2*x^2 + 1)^3*(2*x + 4*x^2 + 3)*(3*x - 6*x^2 + 7)^q,x)
Output:
int((5*x - 2*x^2 + 1)^3*(2*x + 4*x^2 + 3)*(3*x - 6*x^2 + 7)^q, x)
\[ \int \left (7+3 x-6 x^2\right )^q \left (1+5 x-2 x^2\right )^3 \left (3+2 x+4 x^2\right ) \, dx=\text {too large to display} \] Input:
int((-6*x^2+3*x+7)^q*(-2*x^2+5*x+1)^3*(4*x^2+2*x+3),x)
Output:
( - 5308416*( - 6*x**2 + 3*x + 7)**q*q**8*x**9 + 38486016*( - 6*x**2 + 3*x + 7)**q*q**8*x**8 - 78409728*( - 6*x**2 + 3*x + 7)**q*q**8*x**7 + 2073600 *( - 6*x**2 + 3*x + 7)**q*q**8*x**6 + 73840896*( - 6*x**2 + 3*x + 7)**q*q* *8*x**5 + 52577856*( - 6*x**2 + 3*x + 7)**q*q**8*x**4 + 22074768*( - 6*x** 2 + 3*x + 7)**q*q**8*x**3 - 82733724*( - 6*x**2 + 3*x + 7)**q*q**8*x**2 - 70726311*( - 6*x**2 + 3*x + 7)**q*q**8*x + 2322432*( - 6*x**2 + 3*x + 7)** q*q**8 - 95551488*( - 6*x**2 + 3*x + 7)**q*q**7*x**9 + 706019328*( - 6*x** 2 + 3*x + 7)**q*q**7*x**8 - 1480660992*( - 6*x**2 + 3*x + 7)**q*q**7*x**7 + 153709056*( - 6*x**2 + 3*x + 7)**q*q**7*x**6 + 1186518528*( - 6*x**2 + 3 *x + 7)**q*q**7*x**5 + 1058679360*( - 6*x**2 + 3*x + 7)**q*q**7*x**4 + 700 436664*( - 6*x**2 + 3*x + 7)**q*q**7*x**3 - 1588598730*( - 6*x**2 + 3*x + 7)**q*q**7*x**2 - 1490487345*( - 6*x**2 + 3*x + 7)**q*q**7*x + 217282779*( - 6*x**2 + 3*x + 7)**q*q**7 - 724598784*( - 6*x**2 + 3*x + 7)**q*q**6*x** 9 + 5439135744*( - 6*x**2 + 3*x + 7)**q*q**6*x**8 - 11694302208*( - 6*x**2 + 3*x + 7)**q*q**6*x**7 + 2033434368*( - 6*x**2 + 3*x + 7)**q*q**6*x**6 + 7674236352*( - 6*x**2 + 3*x + 7)**q*q**6*x**5 + 9242915184*( - 6*x**2 + 3 *x + 7)**q*q**6*x**4 + 7721773320*( - 6*x**2 + 3*x + 7)**q*q**6*x**3 - 126 80279082*( - 6*x**2 + 3*x + 7)**q*q**6*x**2 - 12877687701*( - 6*x**2 + 3*x + 7)**q*q**6*x + 3982932765*( - 6*x**2 + 3*x + 7)**q*q**6 - 3009871872*( - 6*x**2 + 3*x + 7)**q*q**5*x**9 + 22889889792*( - 6*x**2 + 3*x + 7)**q...