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3.15.16 \(\int (b+2 c x) (d+e x)^{3/2} (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=427 \[ \frac {2 (d+e x)^{11/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{11 e^8}+\frac {2 c^2 (d+e x)^{15/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^8}-\frac {10 c (d+e x)^{13/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{13 e^8}-\frac {2 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^8}+\frac {2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^8}-\frac {2 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^8}-\frac {14 c^3 (d+e x)^{17/2} (2 c d-b e)}{17 e^8}+\frac {4 c^4 (d+e x)^{19/2}}{19 e^8} \]

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Rubi [A]  time = 0.24, antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {771} \begin {gather*} \frac {2 (d+e x)^{11/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{11 e^8}+\frac {2 c^2 (d+e x)^{15/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^8}-\frac {10 c (d+e x)^{13/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{13 e^8}-\frac {2 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^8}+\frac {2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^8}-\frac {2 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^8}-\frac {14 c^3 (d+e x)^{17/2} (2 c d-b e)}{17 e^8}+\frac {4 c^4 (d+e x)^{19/2}}{19 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(d + e*x)^(3/2)*(a + b*x + c*x^2)^3,x]

[Out]

(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(5/2))/(5*e^8) + (2*(c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^
2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(7/2))/(7*e^8) - (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^
2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(9/2))/(3*e^8) + (2*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*
d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^(11/2))/(
11*e^8) - (10*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(13/2))/(13*e^8) + (2*c^2*
(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(15/2))/(5*e^8) - (14*c^3*(2*c*d - b*e)*(d + e*x)^(17
/2))/(17*e^8) + (4*c^4*(d + e*x)^(19/2))/(19*e^8)

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int (b+2 c x) (d+e x)^{3/2} \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}{e^7}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{5/2}}{e^7}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right ) (d+e x)^{7/2}}{e^7}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{9/2}}{e^7}+\frac {5 c (2 c d-b e) \left (-7 c^2 d^2-b^2 e^2+c e (7 b d-3 a e)\right ) (d+e x)^{11/2}}{e^7}+\frac {3 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{13/2}}{e^7}-\frac {7 c^3 (2 c d-b e) (d+e x)^{15/2}}{e^7}+\frac {2 c^4 (d+e x)^{17/2}}{e^7}\right ) \, dx\\ &=-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{5/2}}{5 e^8}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{7/2}}{7 e^8}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{9/2}}{3 e^8}+\frac {2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{11/2}}{11 e^8}-\frac {10 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{13/2}}{13 e^8}+\frac {2 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{15/2}}{5 e^8}-\frac {14 c^3 (2 c d-b e) (d+e x)^{17/2}}{17 e^8}+\frac {4 c^4 (d+e x)^{19/2}}{19 e^8}\\ \end {align*}

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Mathematica [A]  time = 0.62, size = 600, normalized size = 1.41 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (-969 c^2 e^2 \left (26 a^2 e^2 \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )-5 a b e \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+b^2 \left (256 d^5-640 d^4 e x+1120 d^3 e^2 x^2-1680 d^2 e^3 x^3+2310 d e^4 x^4-3003 e^5 x^5\right )\right )+323 c e^3 \left (858 a^3 e^3 (5 e x-2 d)+429 a^2 b e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+156 a b^2 e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+5 b^3 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )+4199 b e^4 \left (231 a^3 e^3+99 a^2 b e^2 (5 e x-2 d)+11 a b^2 e \left (8 d^2-20 d e x+35 e^2 x^2\right )+b^3 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )\right )+19 c^3 e \left (34 a e \left (-256 d^5+640 d^4 e x-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3-2310 d e^4 x^4+3003 e^5 x^5\right )+7 b \left (1024 d^6-2560 d^5 e x+4480 d^4 e^2 x^2-6720 d^3 e^3 x^3+9240 d^2 e^4 x^4-12012 d e^5 x^5+15015 e^6 x^6\right )\right )-14 c^4 \left (2048 d^7-5120 d^6 e x+8960 d^5 e^2 x^2-13440 d^4 e^3 x^3+18480 d^3 e^4 x^4-24024 d^2 e^5 x^5+30030 d e^6 x^6-36465 e^7 x^7\right )\right )}{4849845 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)*(d + e*x)^(3/2)*(a + b*x + c*x^2)^3,x]

[Out]

(2*(d + e*x)^(5/2)*(-14*c^4*(2048*d^7 - 5120*d^6*e*x + 8960*d^5*e^2*x^2 - 13440*d^4*e^3*x^3 + 18480*d^3*e^4*x^
4 - 24024*d^2*e^5*x^5 + 30030*d*e^6*x^6 - 36465*e^7*x^7) + 4199*b*e^4*(231*a^3*e^3 + 99*a^2*b*e^2*(-2*d + 5*e*
x) + 11*a*b^2*e*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + b^3*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3)) + 3
23*c*e^3*(858*a^3*e^3*(-2*d + 5*e*x) + 429*a^2*b*e^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 156*a*b^2*e*(-16*d^3 +
40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 5*b^3*(128*d^4 - 320*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 11
55*e^4*x^4)) - 969*c^2*e^2*(26*a^2*e^2*(16*d^3 - 40*d^2*e*x + 70*d*e^2*x^2 - 105*e^3*x^3) - 5*a*b*e*(128*d^4 -
 320*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4) + b^2*(256*d^5 - 640*d^4*e*x + 1120*d^3*e^2*x^2
 - 1680*d^2*e^3*x^3 + 2310*d*e^4*x^4 - 3003*e^5*x^5)) + 19*c^3*e*(34*a*e*(-256*d^5 + 640*d^4*e*x - 1120*d^3*e^
2*x^2 + 1680*d^2*e^3*x^3 - 2310*d*e^4*x^4 + 3003*e^5*x^5) + 7*b*(1024*d^6 - 2560*d^5*e*x + 4480*d^4*e^2*x^2 -
6720*d^3*e^3*x^3 + 9240*d^2*e^4*x^4 - 12012*d*e^5*x^5 + 15015*e^6*x^6))))/(4849845*e^8)

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IntegrateAlgebraic [B]  time = 0.36, size = 951, normalized size = 2.23 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (-1939938 c^4 d^7+6789783 b c^3 e d^6+9699690 c^4 (d+e x) d^6-5819814 a c^3 e^2 d^5-8729721 b^2 c^2 e^2 d^5-22632610 c^4 (d+e x)^2 d^5-29099070 b c^3 e (d+e x) d^5+14549535 a b c^2 e^3 d^4+4849845 b^3 c e^3 d^4+30862650 c^4 (d+e x)^3 d^4+56581525 b c^3 e (d+e x)^2 d^4+20785050 a c^3 e^2 (d+e x) d^4+31177575 b^2 c^2 e^2 (d+e x) d^4-969969 b^4 e^4 d^3-5819814 a^2 c^2 e^4 d^3-11639628 a b^2 c e^4 d^3-26114550 c^4 (d+e x)^4 d^3-61725300 b c^3 e (d+e x)^3 d^3-32332300 a c^3 e^2 (d+e x)^2 d^3-48498450 b^2 c^2 e^2 (d+e x)^2 d^3-41570100 a b c^2 e^3 (d+e x) d^3-13856700 b^3 c e^3 (d+e x) d^3+2909907 a b^3 e^5 d^2+8729721 a^2 b c e^5 d^2+13579566 c^4 (d+e x)^5 d^2+39171825 b c^3 e (d+e x)^4 d^2+26453700 a c^3 e^2 (d+e x)^3 d^2+39680550 b^2 c^2 e^2 (d+e x)^3 d^2+48498450 a b c^2 e^3 (d+e x)^2 d^2+16166150 b^3 c e^3 (d+e x)^2 d^2+2078505 b^4 e^4 (d+e x) d^2+12471030 a^2 c^2 e^4 (d+e x) d^2+24942060 a b^2 c e^4 (d+e x) d^2-2909907 a^2 b^2 e^6 d-1939938 a^3 c e^6 d-3993990 c^4 (d+e x)^6 d-13579566 b c^3 e (d+e x)^5 d-11191950 a c^3 e^2 (d+e x)^4 d-16787925 b^2 c^2 e^2 (d+e x)^4 d-26453700 a b c^2 e^3 (d+e x)^3 d-8817900 b^3 c e^3 (d+e x)^3 d-1616615 b^4 e^4 (d+e x)^2 d-9699690 a^2 c^2 e^4 (d+e x)^2 d-19399380 a b^2 c e^4 (d+e x)^2 d-4157010 a b^3 e^5 (d+e x) d-12471030 a^2 b c e^5 (d+e x) d+969969 a^3 b e^7+510510 c^4 (d+e x)^7+1996995 b c^3 e (d+e x)^6+1939938 a c^3 e^2 (d+e x)^5+2909907 b^2 c^2 e^2 (d+e x)^5+5595975 a b c^2 e^3 (d+e x)^4+1865325 b^3 c e^3 (d+e x)^4+440895 b^4 e^4 (d+e x)^3+2645370 a^2 c^2 e^4 (d+e x)^3+5290740 a b^2 c e^4 (d+e x)^3+1616615 a b^3 e^5 (d+e x)^2+4849845 a^2 b c e^5 (d+e x)^2+2078505 a^2 b^2 e^6 (d+e x)+1385670 a^3 c e^6 (d+e x)\right )}{4849845 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b + 2*c*x)*(d + e*x)^(3/2)*(a + b*x + c*x^2)^3,x]

[Out]

(2*(d + e*x)^(5/2)*(-1939938*c^4*d^7 + 6789783*b*c^3*d^6*e - 8729721*b^2*c^2*d^5*e^2 - 5819814*a*c^3*d^5*e^2 +
 4849845*b^3*c*d^4*e^3 + 14549535*a*b*c^2*d^4*e^3 - 969969*b^4*d^3*e^4 - 11639628*a*b^2*c*d^3*e^4 - 5819814*a^
2*c^2*d^3*e^4 + 2909907*a*b^3*d^2*e^5 + 8729721*a^2*b*c*d^2*e^5 - 2909907*a^2*b^2*d*e^6 - 1939938*a^3*c*d*e^6
+ 969969*a^3*b*e^7 + 9699690*c^4*d^6*(d + e*x) - 29099070*b*c^3*d^5*e*(d + e*x) + 31177575*b^2*c^2*d^4*e^2*(d
+ e*x) + 20785050*a*c^3*d^4*e^2*(d + e*x) - 13856700*b^3*c*d^3*e^3*(d + e*x) - 41570100*a*b*c^2*d^3*e^3*(d + e
*x) + 2078505*b^4*d^2*e^4*(d + e*x) + 24942060*a*b^2*c*d^2*e^4*(d + e*x) + 12471030*a^2*c^2*d^2*e^4*(d + e*x)
- 4157010*a*b^3*d*e^5*(d + e*x) - 12471030*a^2*b*c*d*e^5*(d + e*x) + 2078505*a^2*b^2*e^6*(d + e*x) + 1385670*a
^3*c*e^6*(d + e*x) - 22632610*c^4*d^5*(d + e*x)^2 + 56581525*b*c^3*d^4*e*(d + e*x)^2 - 48498450*b^2*c^2*d^3*e^
2*(d + e*x)^2 - 32332300*a*c^3*d^3*e^2*(d + e*x)^2 + 16166150*b^3*c*d^2*e^3*(d + e*x)^2 + 48498450*a*b*c^2*d^2
*e^3*(d + e*x)^2 - 1616615*b^4*d*e^4*(d + e*x)^2 - 19399380*a*b^2*c*d*e^4*(d + e*x)^2 - 9699690*a^2*c^2*d*e^4*
(d + e*x)^2 + 1616615*a*b^3*e^5*(d + e*x)^2 + 4849845*a^2*b*c*e^5*(d + e*x)^2 + 30862650*c^4*d^4*(d + e*x)^3 -
 61725300*b*c^3*d^3*e*(d + e*x)^3 + 39680550*b^2*c^2*d^2*e^2*(d + e*x)^3 + 26453700*a*c^3*d^2*e^2*(d + e*x)^3
- 8817900*b^3*c*d*e^3*(d + e*x)^3 - 26453700*a*b*c^2*d*e^3*(d + e*x)^3 + 440895*b^4*e^4*(d + e*x)^3 + 5290740*
a*b^2*c*e^4*(d + e*x)^3 + 2645370*a^2*c^2*e^4*(d + e*x)^3 - 26114550*c^4*d^3*(d + e*x)^4 + 39171825*b*c^3*d^2*
e*(d + e*x)^4 - 16787925*b^2*c^2*d*e^2*(d + e*x)^4 - 11191950*a*c^3*d*e^2*(d + e*x)^4 + 1865325*b^3*c*e^3*(d +
 e*x)^4 + 5595975*a*b*c^2*e^3*(d + e*x)^4 + 13579566*c^4*d^2*(d + e*x)^5 - 13579566*b*c^3*d*e*(d + e*x)^5 + 29
09907*b^2*c^2*e^2*(d + e*x)^5 + 1939938*a*c^3*e^2*(d + e*x)^5 - 3993990*c^4*d*(d + e*x)^6 + 1996995*b*c^3*e*(d
 + e*x)^6 + 510510*c^4*(d + e*x)^7))/(4849845*e^8)

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fricas [B]  time = 0.42, size = 958, normalized size = 2.24 \begin {gather*} \frac {2 \, {\left (510510 \, c^{4} e^{9} x^{9} - 28672 \, c^{4} d^{9} + 136192 \, b c^{3} d^{8} e + 969969 \, a^{3} b d^{2} e^{7} - 82688 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{7} e^{2} + 206720 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{6} e^{3} - 67184 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{5} e^{4} + 369512 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{4} e^{5} - 277134 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{3} e^{6} + 15015 \, {\left (40 \, c^{4} d e^{8} + 133 \, b c^{3} e^{9}\right )} x^{8} + 3003 \, {\left (2 \, c^{4} d^{2} e^{7} + 798 \, b c^{3} d e^{8} + 323 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{9}\right )} x^{7} - 231 \, {\left (28 \, c^{4} d^{3} e^{6} - 133 \, b c^{3} d^{2} e^{7} - 5168 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{8} - 8075 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{9}\right )} x^{6} + 21 \, {\left (336 \, c^{4} d^{4} e^{5} - 1596 \, b c^{3} d^{3} e^{6} + 969 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{7} + 113050 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{8} + 20995 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{9}\right )} x^{5} - 35 \, {\left (224 \, c^{4} d^{5} e^{4} - 1064 \, b c^{3} d^{4} e^{5} + 646 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{6} - 1615 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{7} - 16796 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{8} - 46189 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{9}\right )} x^{4} + 5 \, {\left (1792 \, c^{4} d^{6} e^{3} - 8512 \, b c^{3} d^{5} e^{4} + 5168 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{5} - 12920 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{6} + 4199 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{7} + 461890 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{8} + 138567 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{9}\right )} x^{3} - 3 \, {\left (3584 \, c^{4} d^{7} e^{2} - 17024 \, b c^{3} d^{6} e^{3} - 323323 \, a^{3} b e^{9} + 10336 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{4} - 25840 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{5} + 8398 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{6} - 46189 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{7} - 369512 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{8}\right )} x^{2} + {\left (14336 \, c^{4} d^{8} e - 68096 \, b c^{3} d^{7} e^{2} + 1939938 \, a^{3} b d e^{8} + 41344 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{3} - 103360 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{4} + 33592 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{5} - 184756 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{6} + 138567 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{7}\right )} x\right )} \sqrt {e x + d}}{4849845 \, e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(3/2)*(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

2/4849845*(510510*c^4*e^9*x^9 - 28672*c^4*d^9 + 136192*b*c^3*d^8*e + 969969*a^3*b*d^2*e^7 - 82688*(3*b^2*c^2 +
 2*a*c^3)*d^7*e^2 + 206720*(b^3*c + 3*a*b*c^2)*d^6*e^3 - 67184*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^5*e^4 + 369512
*(a*b^3 + 3*a^2*b*c)*d^4*e^5 - 277134*(3*a^2*b^2 + 2*a^3*c)*d^3*e^6 + 15015*(40*c^4*d*e^8 + 133*b*c^3*e^9)*x^8
 + 3003*(2*c^4*d^2*e^7 + 798*b*c^3*d*e^8 + 323*(3*b^2*c^2 + 2*a*c^3)*e^9)*x^7 - 231*(28*c^4*d^3*e^6 - 133*b*c^
3*d^2*e^7 - 5168*(3*b^2*c^2 + 2*a*c^3)*d*e^8 - 8075*(b^3*c + 3*a*b*c^2)*e^9)*x^6 + 21*(336*c^4*d^4*e^5 - 1596*
b*c^3*d^3*e^6 + 969*(3*b^2*c^2 + 2*a*c^3)*d^2*e^7 + 113050*(b^3*c + 3*a*b*c^2)*d*e^8 + 20995*(b^4 + 12*a*b^2*c
 + 6*a^2*c^2)*e^9)*x^5 - 35*(224*c^4*d^5*e^4 - 1064*b*c^3*d^4*e^5 + 646*(3*b^2*c^2 + 2*a*c^3)*d^3*e^6 - 1615*(
b^3*c + 3*a*b*c^2)*d^2*e^7 - 16796*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^8 - 46189*(a*b^3 + 3*a^2*b*c)*e^9)*x^4 +
 5*(1792*c^4*d^6*e^3 - 8512*b*c^3*d^5*e^4 + 5168*(3*b^2*c^2 + 2*a*c^3)*d^4*e^5 - 12920*(b^3*c + 3*a*b*c^2)*d^3
*e^6 + 4199*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^7 + 461890*(a*b^3 + 3*a^2*b*c)*d*e^8 + 138567*(3*a^2*b^2 + 2*
a^3*c)*e^9)*x^3 - 3*(3584*c^4*d^7*e^2 - 17024*b*c^3*d^6*e^3 - 323323*a^3*b*e^9 + 10336*(3*b^2*c^2 + 2*a*c^3)*d
^5*e^4 - 25840*(b^3*c + 3*a*b*c^2)*d^4*e^5 + 8398*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^6 - 46189*(a*b^3 + 3*a^
2*b*c)*d^2*e^7 - 369512*(3*a^2*b^2 + 2*a^3*c)*d*e^8)*x^2 + (14336*c^4*d^8*e - 68096*b*c^3*d^7*e^2 + 1939938*a^
3*b*d*e^8 + 41344*(3*b^2*c^2 + 2*a*c^3)*d^6*e^3 - 103360*(b^3*c + 3*a*b*c^2)*d^5*e^4 + 33592*(b^4 + 12*a*b^2*c
 + 6*a^2*c^2)*d^4*e^5 - 184756*(a*b^3 + 3*a^2*b*c)*d^3*e^6 + 138567*(3*a^2*b^2 + 2*a^3*c)*d^2*e^7)*x)*sqrt(e*x
 + d)/e^8

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giac [B]  time = 0.37, size = 3123, normalized size = 7.31

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(3/2)*(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

2/14549535*(14549535*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*b^2*d^2*e^(-1) + 9699690*((x*e + d)^(3/2) - 3*s
qrt(x*e + d)*d)*a^3*c*d^2*e^(-1) + 2909907*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a
*b^3*d^2*e^(-2) + 8729721*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*b*c*d^2*e^(-2)
 + 415701*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*b^4*d^2*e
^(-3) + 4988412*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a*b
^2*c*d^2*e^(-3) + 2494206*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d
)*d^3)*a^2*c^2*d^2*e^(-3) + 230945*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420
*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*b^3*c*d^2*e^(-4) + 692835*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7
/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a*b*c^2*d^2*e^(-4) + 188955
*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e
 + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*b^2*c^2*d^2*e^(-5) + 125970*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2
)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a
*c^3*d^2*e^(-5) + 33915*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e
 + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*b*c^3*d^2*e^(-
6) + 4522*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)
*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*
d^7)*c^4*d^2*e^(-7) + 5819814*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*b^2*d*e^(-
1) + 3879876*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3*c*d*e^(-1) + 2494206*(5*(x*
e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a*b^3*d*e^(-2) + 7482618*
(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^2*b*c*d*e^(-2) +
92378*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sq
rt(x*e + d)*d^4)*b^4*d*e^(-3) + 1108536*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2
- 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a*b^2*c*d*e^(-3) + 554268*(35*(x*e + d)^(9/2) - 180*(x*e +
d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a^2*c^2*d*e^(-3) + 209
950*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(
x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*b^3*c*d*e^(-4) + 629850*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)
*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a*
b*c^2*d*e^(-4) + 87210*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e
+ d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*b^2*c^2*d*e^(-5
) + 58140*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^
3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*a*c^3*d*e^(-5) + 31654*(429*
(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x
*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*b*c^3*d*e^
(-6) + 532*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^
(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*
e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*c^4*d*e^(-7) + 14549535*sqrt(x*e + d)*a^3*b*d^2 + 9699690*((x*e +
 d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3*b*d + 1247103*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2
)*d^2 - 35*sqrt(x*e + d)*d^3)*a^2*b^2*e^(-1) + 831402*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)
^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^3*c*e^(-1) + 138567*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*
e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a*b^3*e^(-2) + 415701*(35*(x*e + d)^(9/2)
- 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a^2*b*c*e
^(-2) + 20995*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^
3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*b^4*e^(-3) + 251940*(63*(x*e + d)^(11/2) - 385*(x*e + d)
^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d
^5)*a*b^2*c*e^(-3) + 125970*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e
 + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a^2*c^2*e^(-3) + 24225*(231*(x*e + d)^(13/
2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4
- 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*b^3*c*e^(-4) + 72675*(231*(x*e + d)^(13/2) - 1638*(x*e +
d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^
(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*a*b*c^2*e^(-4) + 20349*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 1
2285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5
+ 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*b^2*c^2*e^(-5) + 13566*(429*(x*e + d)^(15/2) - 3465*(x*e
 + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x
*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*a*c^3*e^(-5) + 931*(6435*(x*e + d)^(17
/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^
(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt
(x*e + d)*d^8)*b*c^3*e^(-6) + 126*(12155*(x*e + d)^(19/2) - 122265*(x*e + d)^(17/2)*d + 554268*(x*e + d)^(15/2
)*d^2 - 1492260*(x*e + d)^(13/2)*d^3 + 2645370*(x*e + d)^(11/2)*d^4 - 3233230*(x*e + d)^(9/2)*d^5 + 2771340*(x
*e + d)^(7/2)*d^6 - 1662804*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8 - 230945*sqrt(x*e + d)*d^9)*c^4*e
^(-7) + 969969*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3*b)*e^(-1)

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maple [B]  time = 0.05, size = 795, normalized size = 1.86 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (510510 c^{4} e^{7} x^{7}+1996995 b \,c^{3} e^{7} x^{6}-420420 c^{4} d \,e^{6} x^{6}+1939938 a \,c^{3} e^{7} x^{5}+2909907 b^{2} c^{2} e^{7} x^{5}-1597596 b \,c^{3} d \,e^{6} x^{5}+336336 c^{4} d^{2} e^{5} x^{5}+5595975 a b \,c^{2} e^{7} x^{4}-1492260 a \,c^{3} d \,e^{6} x^{4}+1865325 b^{3} c \,e^{7} x^{4}-2238390 b^{2} c^{2} d \,e^{6} x^{4}+1228920 b \,c^{3} d^{2} e^{5} x^{4}-258720 c^{4} d^{3} e^{4} x^{4}+2645370 a^{2} c^{2} e^{7} x^{3}+5290740 a \,b^{2} c \,e^{7} x^{3}-4069800 a b \,c^{2} d \,e^{6} x^{3}+1085280 a \,c^{3} d^{2} e^{5} x^{3}+440895 b^{4} e^{7} x^{3}-1356600 b^{3} c d \,e^{6} x^{3}+1627920 b^{2} c^{2} d^{2} e^{5} x^{3}-893760 b \,c^{3} d^{3} e^{4} x^{3}+188160 c^{4} d^{4} e^{3} x^{3}+4849845 a^{2} b c \,e^{7} x^{2}-1763580 a^{2} c^{2} d \,e^{6} x^{2}+1616615 a \,b^{3} e^{7} x^{2}-3527160 a \,b^{2} c d \,e^{6} x^{2}+2713200 a b \,c^{2} d^{2} e^{5} x^{2}-723520 a \,c^{3} d^{3} e^{4} x^{2}-293930 b^{4} d \,e^{6} x^{2}+904400 b^{3} c \,d^{2} e^{5} x^{2}-1085280 b^{2} c^{2} d^{3} e^{4} x^{2}+595840 b \,c^{3} d^{4} e^{3} x^{2}-125440 c^{4} d^{5} e^{2} x^{2}+1385670 a^{3} c \,e^{7} x +2078505 a^{2} b^{2} e^{7} x -2771340 a^{2} b c d \,e^{6} x +1007760 a^{2} c^{2} d^{2} e^{5} x -923780 a \,b^{3} d \,e^{6} x +2015520 a \,b^{2} c \,d^{2} e^{5} x -1550400 a b \,c^{2} d^{3} e^{4} x +413440 a \,c^{3} d^{4} e^{3} x +167960 b^{4} d^{2} e^{5} x -516800 b^{3} c \,d^{3} e^{4} x +620160 b^{2} c^{2} d^{4} e^{3} x -340480 b \,c^{3} d^{5} e^{2} x +71680 c^{4} d^{6} e x +969969 b \,a^{3} e^{7}-554268 a^{3} c d \,e^{6}-831402 a^{2} b^{2} d \,e^{6}+1108536 a^{2} b c \,d^{2} e^{5}-403104 a^{2} c^{2} d^{3} e^{4}+369512 a \,b^{3} d^{2} e^{5}-806208 a \,b^{2} c \,d^{3} e^{4}+620160 a b \,c^{2} d^{4} e^{3}-165376 a \,c^{3} d^{5} e^{2}-67184 b^{4} d^{3} e^{4}+206720 b^{3} c \,d^{4} e^{3}-248064 b^{2} c^{2} d^{5} e^{2}+136192 b \,c^{3} d^{6} e -28672 c^{4} d^{7}\right )}{4849845 e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(e*x+d)^(3/2)*(c*x^2+b*x+a)^3,x)

[Out]

2/4849845*(e*x+d)^(5/2)*(510510*c^4*e^7*x^7+1996995*b*c^3*e^7*x^6-420420*c^4*d*e^6*x^6+1939938*a*c^3*e^7*x^5+2
909907*b^2*c^2*e^7*x^5-1597596*b*c^3*d*e^6*x^5+336336*c^4*d^2*e^5*x^5+5595975*a*b*c^2*e^7*x^4-1492260*a*c^3*d*
e^6*x^4+1865325*b^3*c*e^7*x^4-2238390*b^2*c^2*d*e^6*x^4+1228920*b*c^3*d^2*e^5*x^4-258720*c^4*d^3*e^4*x^4+26453
70*a^2*c^2*e^7*x^3+5290740*a*b^2*c*e^7*x^3-4069800*a*b*c^2*d*e^6*x^3+1085280*a*c^3*d^2*e^5*x^3+440895*b^4*e^7*
x^3-1356600*b^3*c*d*e^6*x^3+1627920*b^2*c^2*d^2*e^5*x^3-893760*b*c^3*d^3*e^4*x^3+188160*c^4*d^4*e^3*x^3+484984
5*a^2*b*c*e^7*x^2-1763580*a^2*c^2*d*e^6*x^2+1616615*a*b^3*e^7*x^2-3527160*a*b^2*c*d*e^6*x^2+2713200*a*b*c^2*d^
2*e^5*x^2-723520*a*c^3*d^3*e^4*x^2-293930*b^4*d*e^6*x^2+904400*b^3*c*d^2*e^5*x^2-1085280*b^2*c^2*d^3*e^4*x^2+5
95840*b*c^3*d^4*e^3*x^2-125440*c^4*d^5*e^2*x^2+1385670*a^3*c*e^7*x+2078505*a^2*b^2*e^7*x-2771340*a^2*b*c*d*e^6
*x+1007760*a^2*c^2*d^2*e^5*x-923780*a*b^3*d*e^6*x+2015520*a*b^2*c*d^2*e^5*x-1550400*a*b*c^2*d^3*e^4*x+413440*a
*c^3*d^4*e^3*x+167960*b^4*d^2*e^5*x-516800*b^3*c*d^3*e^4*x+620160*b^2*c^2*d^4*e^3*x-340480*b*c^3*d^5*e^2*x+716
80*c^4*d^6*e*x+969969*a^3*b*e^7-554268*a^3*c*d*e^6-831402*a^2*b^2*d*e^6+1108536*a^2*b*c*d^2*e^5-403104*a^2*c^2
*d^3*e^4+369512*a*b^3*d^2*e^5-806208*a*b^2*c*d^3*e^4+620160*a*b*c^2*d^4*e^3-165376*a*c^3*d^5*e^2-67184*b^4*d^3
*e^4+206720*b^3*c*d^4*e^3-248064*b^2*c^2*d^5*e^2+136192*b*c^3*d^6*e-28672*c^4*d^7)/e^8

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maxima [A]  time = 0.54, size = 645, normalized size = 1.51 \begin {gather*} \frac {2 \, {\left (510510 \, {\left (e x + d\right )}^{\frac {19}{2}} c^{4} - 1996995 \, {\left (2 \, c^{4} d - b c^{3} e\right )} {\left (e x + d\right )}^{\frac {17}{2}} + 969969 \, {\left (14 \, c^{4} d^{2} - 14 \, b c^{3} d e + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {15}{2}} - 1865325 \, {\left (14 \, c^{4} d^{3} - 21 \, b c^{3} d^{2} e + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2} - {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 440895 \, {\left (70 \, c^{4} d^{4} - 140 \, b c^{3} d^{3} e + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{2} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 1616615 \, {\left (14 \, c^{4} d^{5} - 35 \, b c^{3} d^{4} e + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{2} - 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{4} - {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{5}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 692835 \, {\left (14 \, c^{4} d^{6} - 42 \, b c^{3} d^{5} e + 15 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{2} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{3} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{4} - 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{6}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 969969 \, {\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e - a^{3} b e^{7} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{4849845 \, e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(3/2)*(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

2/4849845*(510510*(e*x + d)^(19/2)*c^4 - 1996995*(2*c^4*d - b*c^3*e)*(e*x + d)^(17/2) + 969969*(14*c^4*d^2 - 1
4*b*c^3*d*e + (3*b^2*c^2 + 2*a*c^3)*e^2)*(e*x + d)^(15/2) - 1865325*(14*c^4*d^3 - 21*b*c^3*d^2*e + 3*(3*b^2*c^
2 + 2*a*c^3)*d*e^2 - (b^3*c + 3*a*b*c^2)*e^3)*(e*x + d)^(13/2) + 440895*(70*c^4*d^4 - 140*b*c^3*d^3*e + 30*(3*
b^2*c^2 + 2*a*c^3)*d^2*e^2 - 20*(b^3*c + 3*a*b*c^2)*d*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^4)*(e*x + d)^(11/
2) - 1616615*(14*c^4*d^5 - 35*b*c^3*d^4*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^3*e^2 - 10*(b^3*c + 3*a*b*c^2)*d^2*e^3
+ (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^4 - (a*b^3 + 3*a^2*b*c)*e^5)*(e*x + d)^(9/2) + 692835*(14*c^4*d^6 - 42*b*
c^3*d^5*e + 15*(3*b^2*c^2 + 2*a*c^3)*d^4*e^2 - 20*(b^3*c + 3*a*b*c^2)*d^3*e^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^
2)*d^2*e^4 - 6*(a*b^3 + 3*a^2*b*c)*d*e^5 + (3*a^2*b^2 + 2*a^3*c)*e^6)*(e*x + d)^(7/2) - 969969*(2*c^4*d^7 - 7*
b*c^3*d^6*e - a^3*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c + 3*a*b*c^2)*d^4*e^3 + (b^4 + 12*a*b^2*c
+ 6*a^2*c^2)*d^3*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a^3*c)*d*e^6)*(e*x + d)^(5/2))/e^8

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mupad [B]  time = 0.12, size = 444, normalized size = 1.04 \begin {gather*} \frac {{\left (d+e\,x\right )}^{15/2}\,\left (18\,b^2\,c^2\,e^2-84\,b\,c^3\,d\,e+84\,c^4\,d^2+12\,a\,c^3\,e^2\right )}{15\,e^8}+\frac {4\,c^4\,{\left (d+e\,x\right )}^{19/2}}{19\,e^8}-\frac {\left (28\,c^4\,d-14\,b\,c^3\,e\right )\,{\left (d+e\,x\right )}^{17/2}}{17\,e^8}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (12\,a^2\,c^2\,e^4+24\,a\,b^2\,c\,e^4-120\,a\,b\,c^2\,d\,e^3+120\,a\,c^3\,d^2\,e^2+2\,b^4\,e^4-40\,b^3\,c\,d\,e^3+180\,b^2\,c^2\,d^2\,e^2-280\,b\,c^3\,d^3\,e+140\,c^4\,d^4\right )}{11\,e^8}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^3}{5\,e^8}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (3\,a^2\,c\,e^4+a\,b^2\,e^4-10\,a\,b\,c\,d\,e^3+10\,a\,c^2\,d^2\,e^2-b^3\,d\,e^3+8\,b^2\,c\,d^2\,e^2-14\,b\,c^2\,d^3\,e+7\,c^3\,d^4\right )}{3\,e^8}+\frac {2\,{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2\,\left (3\,b^2\,e^2-14\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )}{7\,e^8}+\frac {10\,c\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{13/2}\,\left (b^2\,e^2-7\,b\,c\,d\,e+7\,c^2\,d^2+3\,a\,c\,e^2\right )}{13\,e^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x)*(d + e*x)^(3/2)*(a + b*x + c*x^2)^3,x)

[Out]

((d + e*x)^(15/2)*(84*c^4*d^2 + 12*a*c^3*e^2 + 18*b^2*c^2*e^2 - 84*b*c^3*d*e))/(15*e^8) + (4*c^4*(d + e*x)^(19
/2))/(19*e^8) - ((28*c^4*d - 14*b*c^3*e)*(d + e*x)^(17/2))/(17*e^8) + ((d + e*x)^(11/2)*(2*b^4*e^4 + 140*c^4*d
^4 + 12*a^2*c^2*e^4 + 120*a*c^3*d^2*e^2 + 180*b^2*c^2*d^2*e^2 + 24*a*b^2*c*e^4 - 280*b*c^3*d^3*e - 40*b^3*c*d*
e^3 - 120*a*b*c^2*d*e^3))/(11*e^8) + (2*(b*e - 2*c*d)*(d + e*x)^(5/2)*(a*e^2 + c*d^2 - b*d*e)^3)/(5*e^8) + (2*
(b*e - 2*c*d)*(d + e*x)^(9/2)*(7*c^3*d^4 + a*b^2*e^4 + 3*a^2*c*e^4 - b^3*d*e^3 + 10*a*c^2*d^2*e^2 + 8*b^2*c*d^
2*e^2 - 14*b*c^2*d^3*e - 10*a*b*c*d*e^3))/(3*e^8) + (2*(d + e*x)^(7/2)*(a*e^2 + c*d^2 - b*d*e)^2*(3*b^2*e^2 +
14*c^2*d^2 + 2*a*c*e^2 - 14*b*c*d*e))/(7*e^8) + (10*c*(b*e - 2*c*d)*(d + e*x)^(13/2)*(b^2*e^2 + 7*c^2*d^2 + 3*
a*c*e^2 - 7*b*c*d*e))/(13*e^8)

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sympy [A]  time = 77.35, size = 2122, normalized size = 4.97

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)**(3/2)*(c*x**2+b*x+a)**3,x)

[Out]

a**3*b*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a**3*b*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 4*a**3*c*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 4*a**3*c*(d**2*(d + e
*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 6*a**2*b**2*d*(-d*(d + e*x)**(3/2)/3 + (d +
 e*x)**(5/2)/5)/e**2 + 6*a**2*b**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**
2 + 18*a**2*b*c*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 18*a**2*b*c*(
-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 12*
a**2*c**2*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/
9)/e**4 + 12*a**2*c**2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*
(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 6*a*b**3*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5
 + (d + e*x)**(7/2)/7)/e**3 + 6*a*b**3*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**
(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 24*a*b**2*c*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d
*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 24*a*b**2*c*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2
)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 30*a*b*c**2*d*(d**4*(d
 + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)*
*(11/2)/11)/e**5 + 30*a*b*c**2*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7
+ 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5 + 12*a*c**3*d*(-d**5*(d +
 e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*
x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**6 + 12*a*c**3*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 +
15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)
/13 + (d + e*x)**(15/2)/15)/e**6 + 2*b**4*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e
*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 2*b**4*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2
*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 10*b**3*c*d*(d**4*(d + e*x)**(3/2)
/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e*
*5 + 10*b**3*c*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e
*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5 + 18*b**2*c**2*d*(-d**5*(d + e*x)**(3/2)/
3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11
 + (d + e*x)**(13/2)/13)/e**6 + 18*b**2*c**2*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d
 + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d +
 e*x)**(15/2)/15)/e**6 + 14*b*c**3*d*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)*
*(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(
15/2)/15)/e**7 + 14*b*c**3*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 3
5*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2
)/15 + (d + e*x)**(17/2)/17)/e**7 + 4*c**4*d*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d
 + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*
d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**8 + 4*c**4*(d**8*(d + e*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/2
)/5 + 4*d**6*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)*
*(13/2)/13 + 28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)**(19/2)/19)/e**8

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