3.12.83 \(\int (d+e x)^8 (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=173 \[ -\frac {3 b^5 (d+e x)^{14} (b d-a e)}{7 e^7}+\frac {15 b^4 (d+e x)^{13} (b d-a e)^2}{13 e^7}-\frac {5 b^3 (d+e x)^{12} (b d-a e)^3}{3 e^7}+\frac {15 b^2 (d+e x)^{11} (b d-a e)^4}{11 e^7}-\frac {3 b (d+e x)^{10} (b d-a e)^5}{5 e^7}+\frac {(d+e x)^9 (b d-a e)^6}{9 e^7}+\frac {b^6 (d+e x)^{15}}{15 e^7} \]

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Rubi [A]  time = 0.52, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 43} \begin {gather*} -\frac {3 b^5 (d+e x)^{14} (b d-a e)}{7 e^7}+\frac {15 b^4 (d+e x)^{13} (b d-a e)^2}{13 e^7}-\frac {5 b^3 (d+e x)^{12} (b d-a e)^3}{3 e^7}+\frac {15 b^2 (d+e x)^{11} (b d-a e)^4}{11 e^7}-\frac {3 b (d+e x)^{10} (b d-a e)^5}{5 e^7}+\frac {(d+e x)^9 (b d-a e)^6}{9 e^7}+\frac {b^6 (d+e x)^{15}}{15 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^8*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((b*d - a*e)^6*(d + e*x)^9)/(9*e^7) - (3*b*(b*d - a*e)^5*(d + e*x)^10)/(5*e^7) + (15*b^2*(b*d - a*e)^4*(d + e*
x)^11)/(11*e^7) - (5*b^3*(b*d - a*e)^3*(d + e*x)^12)/(3*e^7) + (15*b^4*(b*d - a*e)^2*(d + e*x)^13)/(13*e^7) -
(3*b^5*(b*d - a*e)*(d + e*x)^14)/(7*e^7) + (b^6*(d + e*x)^15)/(15*e^7)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^6 (d+e x)^8 \, dx\\ &=\int \left (\frac {(-b d+a e)^6 (d+e x)^8}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^9}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{10}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{11}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{12}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{13}}{e^6}+\frac {b^6 (d+e x)^{14}}{e^6}\right ) \, dx\\ &=\frac {(b d-a e)^6 (d+e x)^9}{9 e^7}-\frac {3 b (b d-a e)^5 (d+e x)^{10}}{5 e^7}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{11}}{11 e^7}-\frac {5 b^3 (b d-a e)^3 (d+e x)^{12}}{3 e^7}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{13}}{13 e^7}-\frac {3 b^5 (b d-a e) (d+e x)^{14}}{7 e^7}+\frac {b^6 (d+e x)^{15}}{15 e^7}\\ \end {align*}

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Mathematica [B]  time = 0.11, size = 771, normalized size = 4.46 \begin {gather*} a^6 d^8 x+a^5 d^7 x^2 (4 a e+3 b d)+\frac {1}{13} b^4 e^6 x^{13} \left (15 a^2 e^2+48 a b d e+28 b^2 d^2\right )+\frac {1}{3} a^4 d^6 x^3 \left (28 a^2 e^2+48 a b d e+15 b^2 d^2\right )+\frac {1}{3} b^3 e^5 x^{12} \left (5 a^3 e^3+30 a^2 b d e^2+42 a b^2 d^2 e+14 b^3 d^3\right )+a^3 d^5 x^4 \left (14 a^3 e^3+42 a^2 b d e^2+30 a b^2 d^2 e+5 b^3 d^3\right )+\frac {1}{11} b^2 e^4 x^{11} \left (15 a^4 e^4+160 a^3 b d e^3+420 a^2 b^2 d^2 e^2+336 a b^3 d^3 e+70 b^4 d^4\right )+\frac {1}{5} a^2 d^4 x^5 \left (70 a^4 e^4+336 a^3 b d e^3+420 a^2 b^2 d^2 e^2+160 a b^3 d^3 e+15 b^4 d^4\right )+\frac {1}{5} b e^3 x^{10} \left (3 a^5 e^5+60 a^4 b d e^4+280 a^3 b^2 d^2 e^3+420 a^2 b^3 d^3 e^2+210 a b^4 d^4 e+28 b^5 d^5\right )+\frac {1}{3} a d^3 x^6 \left (28 a^5 e^5+210 a^4 b d e^4+420 a^3 b^2 d^2 e^3+280 a^2 b^3 d^3 e^2+60 a b^4 d^4 e+3 b^5 d^5\right )+\frac {1}{9} e^2 x^9 \left (a^6 e^6+48 a^5 b d e^5+420 a^4 b^2 d^2 e^4+1120 a^3 b^3 d^3 e^3+1050 a^2 b^4 d^4 e^2+336 a b^5 d^5 e+28 b^6 d^6\right )+d e x^8 \left (a^6 e^6+21 a^5 b d e^5+105 a^4 b^2 d^2 e^4+175 a^3 b^3 d^3 e^3+105 a^2 b^4 d^4 e^2+21 a b^5 d^5 e+b^6 d^6\right )+\frac {1}{7} d^2 x^7 \left (28 a^6 e^6+336 a^5 b d e^5+1050 a^4 b^2 d^2 e^4+1120 a^3 b^3 d^3 e^3+420 a^2 b^4 d^4 e^2+48 a b^5 d^5 e+b^6 d^6\right )+\frac {1}{7} b^5 e^7 x^{14} (3 a e+4 b d)+\frac {1}{15} b^6 e^8 x^{15} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^8*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

a^6*d^8*x + a^5*d^7*(3*b*d + 4*a*e)*x^2 + (a^4*d^6*(15*b^2*d^2 + 48*a*b*d*e + 28*a^2*e^2)*x^3)/3 + a^3*d^5*(5*
b^3*d^3 + 30*a*b^2*d^2*e + 42*a^2*b*d*e^2 + 14*a^3*e^3)*x^4 + (a^2*d^4*(15*b^4*d^4 + 160*a*b^3*d^3*e + 420*a^2
*b^2*d^2*e^2 + 336*a^3*b*d*e^3 + 70*a^4*e^4)*x^5)/5 + (a*d^3*(3*b^5*d^5 + 60*a*b^4*d^4*e + 280*a^2*b^3*d^3*e^2
+ 420*a^3*b^2*d^2*e^3 + 210*a^4*b*d*e^4 + 28*a^5*e^5)*x^6)/3 + (d^2*(b^6*d^6 + 48*a*b^5*d^5*e + 420*a^2*b^4*d
^4*e^2 + 1120*a^3*b^3*d^3*e^3 + 1050*a^4*b^2*d^2*e^4 + 336*a^5*b*d*e^5 + 28*a^6*e^6)*x^7)/7 + d*e*(b^6*d^6 + 2
1*a*b^5*d^5*e + 105*a^2*b^4*d^4*e^2 + 175*a^3*b^3*d^3*e^3 + 105*a^4*b^2*d^2*e^4 + 21*a^5*b*d*e^5 + a^6*e^6)*x^
8 + (e^2*(28*b^6*d^6 + 336*a*b^5*d^5*e + 1050*a^2*b^4*d^4*e^2 + 1120*a^3*b^3*d^3*e^3 + 420*a^4*b^2*d^2*e^4 + 4
8*a^5*b*d*e^5 + a^6*e^6)*x^9)/9 + (b*e^3*(28*b^5*d^5 + 210*a*b^4*d^4*e + 420*a^2*b^3*d^3*e^2 + 280*a^3*b^2*d^2
*e^3 + 60*a^4*b*d*e^4 + 3*a^5*e^5)*x^10)/5 + (b^2*e^4*(70*b^4*d^4 + 336*a*b^3*d^3*e + 420*a^2*b^2*d^2*e^2 + 16
0*a^3*b*d*e^3 + 15*a^4*e^4)*x^11)/11 + (b^3*e^5*(14*b^3*d^3 + 42*a*b^2*d^2*e + 30*a^2*b*d*e^2 + 5*a^3*e^3)*x^1
2)/3 + (b^4*e^6*(28*b^2*d^2 + 48*a*b*d*e + 15*a^2*e^2)*x^13)/13 + (b^5*e^7*(4*b*d + 3*a*e)*x^14)/7 + (b^6*e^8*
x^15)/15

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^8 \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^8*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

IntegrateAlgebraic[(d + e*x)^8*(a^2 + 2*a*b*x + b^2*x^2)^3, x]

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fricas [B]  time = 0.36, size = 906, normalized size = 5.24 \begin {gather*} \frac {1}{15} x^{15} e^{8} b^{6} + \frac {4}{7} x^{14} e^{7} d b^{6} + \frac {3}{7} x^{14} e^{8} b^{5} a + \frac {28}{13} x^{13} e^{6} d^{2} b^{6} + \frac {48}{13} x^{13} e^{7} d b^{5} a + \frac {15}{13} x^{13} e^{8} b^{4} a^{2} + \frac {14}{3} x^{12} e^{5} d^{3} b^{6} + 14 x^{12} e^{6} d^{2} b^{5} a + 10 x^{12} e^{7} d b^{4} a^{2} + \frac {5}{3} x^{12} e^{8} b^{3} a^{3} + \frac {70}{11} x^{11} e^{4} d^{4} b^{6} + \frac {336}{11} x^{11} e^{5} d^{3} b^{5} a + \frac {420}{11} x^{11} e^{6} d^{2} b^{4} a^{2} + \frac {160}{11} x^{11} e^{7} d b^{3} a^{3} + \frac {15}{11} x^{11} e^{8} b^{2} a^{4} + \frac {28}{5} x^{10} e^{3} d^{5} b^{6} + 42 x^{10} e^{4} d^{4} b^{5} a + 84 x^{10} e^{5} d^{3} b^{4} a^{2} + 56 x^{10} e^{6} d^{2} b^{3} a^{3} + 12 x^{10} e^{7} d b^{2} a^{4} + \frac {3}{5} x^{10} e^{8} b a^{5} + \frac {28}{9} x^{9} e^{2} d^{6} b^{6} + \frac {112}{3} x^{9} e^{3} d^{5} b^{5} a + \frac {350}{3} x^{9} e^{4} d^{4} b^{4} a^{2} + \frac {1120}{9} x^{9} e^{5} d^{3} b^{3} a^{3} + \frac {140}{3} x^{9} e^{6} d^{2} b^{2} a^{4} + \frac {16}{3} x^{9} e^{7} d b a^{5} + \frac {1}{9} x^{9} e^{8} a^{6} + x^{8} e d^{7} b^{6} + 21 x^{8} e^{2} d^{6} b^{5} a + 105 x^{8} e^{3} d^{5} b^{4} a^{2} + 175 x^{8} e^{4} d^{4} b^{3} a^{3} + 105 x^{8} e^{5} d^{3} b^{2} a^{4} + 21 x^{8} e^{6} d^{2} b a^{5} + x^{8} e^{7} d a^{6} + \frac {1}{7} x^{7} d^{8} b^{6} + \frac {48}{7} x^{7} e d^{7} b^{5} a + 60 x^{7} e^{2} d^{6} b^{4} a^{2} + 160 x^{7} e^{3} d^{5} b^{3} a^{3} + 150 x^{7} e^{4} d^{4} b^{2} a^{4} + 48 x^{7} e^{5} d^{3} b a^{5} + 4 x^{7} e^{6} d^{2} a^{6} + x^{6} d^{8} b^{5} a + 20 x^{6} e d^{7} b^{4} a^{2} + \frac {280}{3} x^{6} e^{2} d^{6} b^{3} a^{3} + 140 x^{6} e^{3} d^{5} b^{2} a^{4} + 70 x^{6} e^{4} d^{4} b a^{5} + \frac {28}{3} x^{6} e^{5} d^{3} a^{6} + 3 x^{5} d^{8} b^{4} a^{2} + 32 x^{5} e d^{7} b^{3} a^{3} + 84 x^{5} e^{2} d^{6} b^{2} a^{4} + \frac {336}{5} x^{5} e^{3} d^{5} b a^{5} + 14 x^{5} e^{4} d^{4} a^{6} + 5 x^{4} d^{8} b^{3} a^{3} + 30 x^{4} e d^{7} b^{2} a^{4} + 42 x^{4} e^{2} d^{6} b a^{5} + 14 x^{4} e^{3} d^{5} a^{6} + 5 x^{3} d^{8} b^{2} a^{4} + 16 x^{3} e d^{7} b a^{5} + \frac {28}{3} x^{3} e^{2} d^{6} a^{6} + 3 x^{2} d^{8} b a^{5} + 4 x^{2} e d^{7} a^{6} + x d^{8} a^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

1/15*x^15*e^8*b^6 + 4/7*x^14*e^7*d*b^6 + 3/7*x^14*e^8*b^5*a + 28/13*x^13*e^6*d^2*b^6 + 48/13*x^13*e^7*d*b^5*a
+ 15/13*x^13*e^8*b^4*a^2 + 14/3*x^12*e^5*d^3*b^6 + 14*x^12*e^6*d^2*b^5*a + 10*x^12*e^7*d*b^4*a^2 + 5/3*x^12*e^
8*b^3*a^3 + 70/11*x^11*e^4*d^4*b^6 + 336/11*x^11*e^5*d^3*b^5*a + 420/11*x^11*e^6*d^2*b^4*a^2 + 160/11*x^11*e^7
*d*b^3*a^3 + 15/11*x^11*e^8*b^2*a^4 + 28/5*x^10*e^3*d^5*b^6 + 42*x^10*e^4*d^4*b^5*a + 84*x^10*e^5*d^3*b^4*a^2
+ 56*x^10*e^6*d^2*b^3*a^3 + 12*x^10*e^7*d*b^2*a^4 + 3/5*x^10*e^8*b*a^5 + 28/9*x^9*e^2*d^6*b^6 + 112/3*x^9*e^3*
d^5*b^5*a + 350/3*x^9*e^4*d^4*b^4*a^2 + 1120/9*x^9*e^5*d^3*b^3*a^3 + 140/3*x^9*e^6*d^2*b^2*a^4 + 16/3*x^9*e^7*
d*b*a^5 + 1/9*x^9*e^8*a^6 + x^8*e*d^7*b^6 + 21*x^8*e^2*d^6*b^5*a + 105*x^8*e^3*d^5*b^4*a^2 + 175*x^8*e^4*d^4*b
^3*a^3 + 105*x^8*e^5*d^3*b^2*a^4 + 21*x^8*e^6*d^2*b*a^5 + x^8*e^7*d*a^6 + 1/7*x^7*d^8*b^6 + 48/7*x^7*e*d^7*b^5
*a + 60*x^7*e^2*d^6*b^4*a^2 + 160*x^7*e^3*d^5*b^3*a^3 + 150*x^7*e^4*d^4*b^2*a^4 + 48*x^7*e^5*d^3*b*a^5 + 4*x^7
*e^6*d^2*a^6 + x^6*d^8*b^5*a + 20*x^6*e*d^7*b^4*a^2 + 280/3*x^6*e^2*d^6*b^3*a^3 + 140*x^6*e^3*d^5*b^2*a^4 + 70
*x^6*e^4*d^4*b*a^5 + 28/3*x^6*e^5*d^3*a^6 + 3*x^5*d^8*b^4*a^2 + 32*x^5*e*d^7*b^3*a^3 + 84*x^5*e^2*d^6*b^2*a^4
+ 336/5*x^5*e^3*d^5*b*a^5 + 14*x^5*e^4*d^4*a^6 + 5*x^4*d^8*b^3*a^3 + 30*x^4*e*d^7*b^2*a^4 + 42*x^4*e^2*d^6*b*a
^5 + 14*x^4*e^3*d^5*a^6 + 5*x^3*d^8*b^2*a^4 + 16*x^3*e*d^7*b*a^5 + 28/3*x^3*e^2*d^6*a^6 + 3*x^2*d^8*b*a^5 + 4*
x^2*e*d^7*a^6 + x*d^8*a^6

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giac [B]  time = 0.20, size = 864, normalized size = 4.99 \begin {gather*} \frac {1}{15} \, b^{6} x^{15} e^{8} + \frac {4}{7} \, b^{6} d x^{14} e^{7} + \frac {28}{13} \, b^{6} d^{2} x^{13} e^{6} + \frac {14}{3} \, b^{6} d^{3} x^{12} e^{5} + \frac {70}{11} \, b^{6} d^{4} x^{11} e^{4} + \frac {28}{5} \, b^{6} d^{5} x^{10} e^{3} + \frac {28}{9} \, b^{6} d^{6} x^{9} e^{2} + b^{6} d^{7} x^{8} e + \frac {1}{7} \, b^{6} d^{8} x^{7} + \frac {3}{7} \, a b^{5} x^{14} e^{8} + \frac {48}{13} \, a b^{5} d x^{13} e^{7} + 14 \, a b^{5} d^{2} x^{12} e^{6} + \frac {336}{11} \, a b^{5} d^{3} x^{11} e^{5} + 42 \, a b^{5} d^{4} x^{10} e^{4} + \frac {112}{3} \, a b^{5} d^{5} x^{9} e^{3} + 21 \, a b^{5} d^{6} x^{8} e^{2} + \frac {48}{7} \, a b^{5} d^{7} x^{7} e + a b^{5} d^{8} x^{6} + \frac {15}{13} \, a^{2} b^{4} x^{13} e^{8} + 10 \, a^{2} b^{4} d x^{12} e^{7} + \frac {420}{11} \, a^{2} b^{4} d^{2} x^{11} e^{6} + 84 \, a^{2} b^{4} d^{3} x^{10} e^{5} + \frac {350}{3} \, a^{2} b^{4} d^{4} x^{9} e^{4} + 105 \, a^{2} b^{4} d^{5} x^{8} e^{3} + 60 \, a^{2} b^{4} d^{6} x^{7} e^{2} + 20 \, a^{2} b^{4} d^{7} x^{6} e + 3 \, a^{2} b^{4} d^{8} x^{5} + \frac {5}{3} \, a^{3} b^{3} x^{12} e^{8} + \frac {160}{11} \, a^{3} b^{3} d x^{11} e^{7} + 56 \, a^{3} b^{3} d^{2} x^{10} e^{6} + \frac {1120}{9} \, a^{3} b^{3} d^{3} x^{9} e^{5} + 175 \, a^{3} b^{3} d^{4} x^{8} e^{4} + 160 \, a^{3} b^{3} d^{5} x^{7} e^{3} + \frac {280}{3} \, a^{3} b^{3} d^{6} x^{6} e^{2} + 32 \, a^{3} b^{3} d^{7} x^{5} e + 5 \, a^{3} b^{3} d^{8} x^{4} + \frac {15}{11} \, a^{4} b^{2} x^{11} e^{8} + 12 \, a^{4} b^{2} d x^{10} e^{7} + \frac {140}{3} \, a^{4} b^{2} d^{2} x^{9} e^{6} + 105 \, a^{4} b^{2} d^{3} x^{8} e^{5} + 150 \, a^{4} b^{2} d^{4} x^{7} e^{4} + 140 \, a^{4} b^{2} d^{5} x^{6} e^{3} + 84 \, a^{4} b^{2} d^{6} x^{5} e^{2} + 30 \, a^{4} b^{2} d^{7} x^{4} e + 5 \, a^{4} b^{2} d^{8} x^{3} + \frac {3}{5} \, a^{5} b x^{10} e^{8} + \frac {16}{3} \, a^{5} b d x^{9} e^{7} + 21 \, a^{5} b d^{2} x^{8} e^{6} + 48 \, a^{5} b d^{3} x^{7} e^{5} + 70 \, a^{5} b d^{4} x^{6} e^{4} + \frac {336}{5} \, a^{5} b d^{5} x^{5} e^{3} + 42 \, a^{5} b d^{6} x^{4} e^{2} + 16 \, a^{5} b d^{7} x^{3} e + 3 \, a^{5} b d^{8} x^{2} + \frac {1}{9} \, a^{6} x^{9} e^{8} + a^{6} d x^{8} e^{7} + 4 \, a^{6} d^{2} x^{7} e^{6} + \frac {28}{3} \, a^{6} d^{3} x^{6} e^{5} + 14 \, a^{6} d^{4} x^{5} e^{4} + 14 \, a^{6} d^{5} x^{4} e^{3} + \frac {28}{3} \, a^{6} d^{6} x^{3} e^{2} + 4 \, a^{6} d^{7} x^{2} e + a^{6} d^{8} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

1/15*b^6*x^15*e^8 + 4/7*b^6*d*x^14*e^7 + 28/13*b^6*d^2*x^13*e^6 + 14/3*b^6*d^3*x^12*e^5 + 70/11*b^6*d^4*x^11*e
^4 + 28/5*b^6*d^5*x^10*e^3 + 28/9*b^6*d^6*x^9*e^2 + b^6*d^7*x^8*e + 1/7*b^6*d^8*x^7 + 3/7*a*b^5*x^14*e^8 + 48/
13*a*b^5*d*x^13*e^7 + 14*a*b^5*d^2*x^12*e^6 + 336/11*a*b^5*d^3*x^11*e^5 + 42*a*b^5*d^4*x^10*e^4 + 112/3*a*b^5*
d^5*x^9*e^3 + 21*a*b^5*d^6*x^8*e^2 + 48/7*a*b^5*d^7*x^7*e + a*b^5*d^8*x^6 + 15/13*a^2*b^4*x^13*e^8 + 10*a^2*b^
4*d*x^12*e^7 + 420/11*a^2*b^4*d^2*x^11*e^6 + 84*a^2*b^4*d^3*x^10*e^5 + 350/3*a^2*b^4*d^4*x^9*e^4 + 105*a^2*b^4
*d^5*x^8*e^3 + 60*a^2*b^4*d^6*x^7*e^2 + 20*a^2*b^4*d^7*x^6*e + 3*a^2*b^4*d^8*x^5 + 5/3*a^3*b^3*x^12*e^8 + 160/
11*a^3*b^3*d*x^11*e^7 + 56*a^3*b^3*d^2*x^10*e^6 + 1120/9*a^3*b^3*d^3*x^9*e^5 + 175*a^3*b^3*d^4*x^8*e^4 + 160*a
^3*b^3*d^5*x^7*e^3 + 280/3*a^3*b^3*d^6*x^6*e^2 + 32*a^3*b^3*d^7*x^5*e + 5*a^3*b^3*d^8*x^4 + 15/11*a^4*b^2*x^11
*e^8 + 12*a^4*b^2*d*x^10*e^7 + 140/3*a^4*b^2*d^2*x^9*e^6 + 105*a^4*b^2*d^3*x^8*e^5 + 150*a^4*b^2*d^4*x^7*e^4 +
140*a^4*b^2*d^5*x^6*e^3 + 84*a^4*b^2*d^6*x^5*e^2 + 30*a^4*b^2*d^7*x^4*e + 5*a^4*b^2*d^8*x^3 + 3/5*a^5*b*x^10*
e^8 + 16/3*a^5*b*d*x^9*e^7 + 21*a^5*b*d^2*x^8*e^6 + 48*a^5*b*d^3*x^7*e^5 + 70*a^5*b*d^4*x^6*e^4 + 336/5*a^5*b*
d^5*x^5*e^3 + 42*a^5*b*d^6*x^4*e^2 + 16*a^5*b*d^7*x^3*e + 3*a^5*b*d^8*x^2 + 1/9*a^6*x^9*e^8 + a^6*d*x^8*e^7 +
4*a^6*d^2*x^7*e^6 + 28/3*a^6*d^3*x^6*e^5 + 14*a^6*d^4*x^5*e^4 + 14*a^6*d^5*x^4*e^3 + 28/3*a^6*d^6*x^3*e^2 + 4*
a^6*d^7*x^2*e + a^6*d^8*x

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maple [B]  time = 0.04, size = 803, normalized size = 4.64 \begin {gather*} \frac {b^{6} e^{8} x^{15}}{15}+a^{6} d^{8} x +\frac {\left (6 e^{8} a \,b^{5}+8 d \,e^{7} b^{6}\right ) x^{14}}{14}+\frac {\left (15 e^{8} a^{2} b^{4}+48 d \,e^{7} a \,b^{5}+28 d^{2} e^{6} b^{6}\right ) x^{13}}{13}+\frac {\left (20 e^{8} a^{3} b^{3}+120 d \,e^{7} a^{2} b^{4}+168 d^{2} e^{6} a \,b^{5}+56 d^{3} e^{5} b^{6}\right ) x^{12}}{12}+\frac {\left (15 e^{8} a^{4} b^{2}+160 d \,e^{7} a^{3} b^{3}+420 d^{2} e^{6} a^{2} b^{4}+336 d^{3} e^{5} a \,b^{5}+70 d^{4} e^{4} b^{6}\right ) x^{11}}{11}+\frac {\left (6 e^{8} a^{5} b +120 d \,e^{7} a^{4} b^{2}+560 d^{2} e^{6} a^{3} b^{3}+840 d^{3} e^{5} a^{2} b^{4}+420 d^{4} e^{4} a \,b^{5}+56 d^{5} e^{3} b^{6}\right ) x^{10}}{10}+\frac {\left (e^{8} a^{6}+48 d \,e^{7} a^{5} b +420 d^{2} e^{6} a^{4} b^{2}+1120 d^{3} e^{5} a^{3} b^{3}+1050 d^{4} e^{4} a^{2} b^{4}+336 d^{5} e^{3} a \,b^{5}+28 d^{6} e^{2} b^{6}\right ) x^{9}}{9}+\frac {\left (8 d \,e^{7} a^{6}+168 d^{2} e^{6} a^{5} b +840 d^{3} e^{5} a^{4} b^{2}+1400 d^{4} e^{4} a^{3} b^{3}+840 d^{5} e^{3} a^{2} b^{4}+168 d^{6} e^{2} a \,b^{5}+8 d^{7} e \,b^{6}\right ) x^{8}}{8}+\frac {\left (28 d^{2} e^{6} a^{6}+336 d^{3} e^{5} a^{5} b +1050 d^{4} e^{4} a^{4} b^{2}+1120 d^{5} e^{3} a^{3} b^{3}+420 d^{6} e^{2} a^{2} b^{4}+48 d^{7} e a \,b^{5}+d^{8} b^{6}\right ) x^{7}}{7}+\frac {\left (56 d^{3} e^{5} a^{6}+420 d^{4} e^{4} a^{5} b +840 d^{5} e^{3} a^{4} b^{2}+560 d^{6} e^{2} a^{3} b^{3}+120 d^{7} e \,a^{2} b^{4}+6 d^{8} a \,b^{5}\right ) x^{6}}{6}+\frac {\left (70 d^{4} e^{4} a^{6}+336 d^{5} e^{3} a^{5} b +420 d^{6} e^{2} a^{4} b^{2}+160 d^{7} e \,a^{3} b^{3}+15 d^{8} a^{2} b^{4}\right ) x^{5}}{5}+\frac {\left (56 d^{5} e^{3} a^{6}+168 d^{6} e^{2} a^{5} b +120 d^{7} e \,a^{4} b^{2}+20 d^{8} a^{3} b^{3}\right ) x^{4}}{4}+\frac {\left (28 d^{6} e^{2} a^{6}+48 d^{7} e \,a^{5} b +15 d^{8} a^{4} b^{2}\right ) x^{3}}{3}+\frac {\left (8 d^{7} e \,a^{6}+6 d^{8} a^{5} b \right ) x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^8*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1/15*e^8*b^6*x^15+1/14*(6*a*b^5*e^8+8*b^6*d*e^7)*x^14+1/13*(15*a^2*b^4*e^8+48*a*b^5*d*e^7+28*b^6*d^2*e^6)*x^13
+1/12*(20*a^3*b^3*e^8+120*a^2*b^4*d*e^7+168*a*b^5*d^2*e^6+56*b^6*d^3*e^5)*x^12+1/11*(15*a^4*b^2*e^8+160*a^3*b^
3*d*e^7+420*a^2*b^4*d^2*e^6+336*a*b^5*d^3*e^5+70*b^6*d^4*e^4)*x^11+1/10*(6*a^5*b*e^8+120*a^4*b^2*d*e^7+560*a^3
*b^3*d^2*e^6+840*a^2*b^4*d^3*e^5+420*a*b^5*d^4*e^4+56*b^6*d^5*e^3)*x^10+1/9*(a^6*e^8+48*a^5*b*d*e^7+420*a^4*b^
2*d^2*e^6+1120*a^3*b^3*d^3*e^5+1050*a^2*b^4*d^4*e^4+336*a*b^5*d^5*e^3+28*b^6*d^6*e^2)*x^9+1/8*(8*a^6*d*e^7+168
*a^5*b*d^2*e^6+840*a^4*b^2*d^3*e^5+1400*a^3*b^3*d^4*e^4+840*a^2*b^4*d^5*e^3+168*a*b^5*d^6*e^2+8*b^6*d^7*e)*x^8
+1/7*(28*a^6*d^2*e^6+336*a^5*b*d^3*e^5+1050*a^4*b^2*d^4*e^4+1120*a^3*b^3*d^5*e^3+420*a^2*b^4*d^6*e^2+48*a*b^5*
d^7*e+b^6*d^8)*x^7+1/6*(56*a^6*d^3*e^5+420*a^5*b*d^4*e^4+840*a^4*b^2*d^5*e^3+560*a^3*b^3*d^6*e^2+120*a^2*b^4*d
^7*e+6*a*b^5*d^8)*x^6+1/5*(70*a^6*d^4*e^4+336*a^5*b*d^5*e^3+420*a^4*b^2*d^6*e^2+160*a^3*b^3*d^7*e+15*a^2*b^4*d
^8)*x^5+1/4*(56*a^6*d^5*e^3+168*a^5*b*d^6*e^2+120*a^4*b^2*d^7*e+20*a^3*b^3*d^8)*x^4+1/3*(28*a^6*d^6*e^2+48*a^5
*b*d^7*e+15*a^4*b^2*d^8)*x^3+1/2*(8*a^6*d^7*e+6*a^5*b*d^8)*x^2+d^8*a^6*x

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maxima [B]  time = 1.66, size = 797, normalized size = 4.61 \begin {gather*} \frac {1}{15} \, b^{6} e^{8} x^{15} + a^{6} d^{8} x + \frac {1}{7} \, {\left (4 \, b^{6} d e^{7} + 3 \, a b^{5} e^{8}\right )} x^{14} + \frac {1}{13} \, {\left (28 \, b^{6} d^{2} e^{6} + 48 \, a b^{5} d e^{7} + 15 \, a^{2} b^{4} e^{8}\right )} x^{13} + \frac {1}{3} \, {\left (14 \, b^{6} d^{3} e^{5} + 42 \, a b^{5} d^{2} e^{6} + 30 \, a^{2} b^{4} d e^{7} + 5 \, a^{3} b^{3} e^{8}\right )} x^{12} + \frac {1}{11} \, {\left (70 \, b^{6} d^{4} e^{4} + 336 \, a b^{5} d^{3} e^{5} + 420 \, a^{2} b^{4} d^{2} e^{6} + 160 \, a^{3} b^{3} d e^{7} + 15 \, a^{4} b^{2} e^{8}\right )} x^{11} + \frac {1}{5} \, {\left (28 \, b^{6} d^{5} e^{3} + 210 \, a b^{5} d^{4} e^{4} + 420 \, a^{2} b^{4} d^{3} e^{5} + 280 \, a^{3} b^{3} d^{2} e^{6} + 60 \, a^{4} b^{2} d e^{7} + 3 \, a^{5} b e^{8}\right )} x^{10} + \frac {1}{9} \, {\left (28 \, b^{6} d^{6} e^{2} + 336 \, a b^{5} d^{5} e^{3} + 1050 \, a^{2} b^{4} d^{4} e^{4} + 1120 \, a^{3} b^{3} d^{3} e^{5} + 420 \, a^{4} b^{2} d^{2} e^{6} + 48 \, a^{5} b d e^{7} + a^{6} e^{8}\right )} x^{9} + {\left (b^{6} d^{7} e + 21 \, a b^{5} d^{6} e^{2} + 105 \, a^{2} b^{4} d^{5} e^{3} + 175 \, a^{3} b^{3} d^{4} e^{4} + 105 \, a^{4} b^{2} d^{3} e^{5} + 21 \, a^{5} b d^{2} e^{6} + a^{6} d e^{7}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{8} + 48 \, a b^{5} d^{7} e + 420 \, a^{2} b^{4} d^{6} e^{2} + 1120 \, a^{3} b^{3} d^{5} e^{3} + 1050 \, a^{4} b^{2} d^{4} e^{4} + 336 \, a^{5} b d^{3} e^{5} + 28 \, a^{6} d^{2} e^{6}\right )} x^{7} + \frac {1}{3} \, {\left (3 \, a b^{5} d^{8} + 60 \, a^{2} b^{4} d^{7} e + 280 \, a^{3} b^{3} d^{6} e^{2} + 420 \, a^{4} b^{2} d^{5} e^{3} + 210 \, a^{5} b d^{4} e^{4} + 28 \, a^{6} d^{3} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (15 \, a^{2} b^{4} d^{8} + 160 \, a^{3} b^{3} d^{7} e + 420 \, a^{4} b^{2} d^{6} e^{2} + 336 \, a^{5} b d^{5} e^{3} + 70 \, a^{6} d^{4} e^{4}\right )} x^{5} + {\left (5 \, a^{3} b^{3} d^{8} + 30 \, a^{4} b^{2} d^{7} e + 42 \, a^{5} b d^{6} e^{2} + 14 \, a^{6} d^{5} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (15 \, a^{4} b^{2} d^{8} + 48 \, a^{5} b d^{7} e + 28 \, a^{6} d^{6} e^{2}\right )} x^{3} + {\left (3 \, a^{5} b d^{8} + 4 \, a^{6} d^{7} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

1/15*b^6*e^8*x^15 + a^6*d^8*x + 1/7*(4*b^6*d*e^7 + 3*a*b^5*e^8)*x^14 + 1/13*(28*b^6*d^2*e^6 + 48*a*b^5*d*e^7 +
15*a^2*b^4*e^8)*x^13 + 1/3*(14*b^6*d^3*e^5 + 42*a*b^5*d^2*e^6 + 30*a^2*b^4*d*e^7 + 5*a^3*b^3*e^8)*x^12 + 1/11
*(70*b^6*d^4*e^4 + 336*a*b^5*d^3*e^5 + 420*a^2*b^4*d^2*e^6 + 160*a^3*b^3*d*e^7 + 15*a^4*b^2*e^8)*x^11 + 1/5*(2
8*b^6*d^5*e^3 + 210*a*b^5*d^4*e^4 + 420*a^2*b^4*d^3*e^5 + 280*a^3*b^3*d^2*e^6 + 60*a^4*b^2*d*e^7 + 3*a^5*b*e^8
)*x^10 + 1/9*(28*b^6*d^6*e^2 + 336*a*b^5*d^5*e^3 + 1050*a^2*b^4*d^4*e^4 + 1120*a^3*b^3*d^3*e^5 + 420*a^4*b^2*d
^2*e^6 + 48*a^5*b*d*e^7 + a^6*e^8)*x^9 + (b^6*d^7*e + 21*a*b^5*d^6*e^2 + 105*a^2*b^4*d^5*e^3 + 175*a^3*b^3*d^4
*e^4 + 105*a^4*b^2*d^3*e^5 + 21*a^5*b*d^2*e^6 + a^6*d*e^7)*x^8 + 1/7*(b^6*d^8 + 48*a*b^5*d^7*e + 420*a^2*b^4*d
^6*e^2 + 1120*a^3*b^3*d^5*e^3 + 1050*a^4*b^2*d^4*e^4 + 336*a^5*b*d^3*e^5 + 28*a^6*d^2*e^6)*x^7 + 1/3*(3*a*b^5*
d^8 + 60*a^2*b^4*d^7*e + 280*a^3*b^3*d^6*e^2 + 420*a^4*b^2*d^5*e^3 + 210*a^5*b*d^4*e^4 + 28*a^6*d^3*e^5)*x^6 +
1/5*(15*a^2*b^4*d^8 + 160*a^3*b^3*d^7*e + 420*a^4*b^2*d^6*e^2 + 336*a^5*b*d^5*e^3 + 70*a^6*d^4*e^4)*x^5 + (5*
a^3*b^3*d^8 + 30*a^4*b^2*d^7*e + 42*a^5*b*d^6*e^2 + 14*a^6*d^5*e^3)*x^4 + 1/3*(15*a^4*b^2*d^8 + 48*a^5*b*d^7*e
+ 28*a^6*d^6*e^2)*x^3 + (3*a^5*b*d^8 + 4*a^6*d^7*e)*x^2

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mupad [B]  time = 0.77, size = 768, normalized size = 4.44 \begin {gather*} x^7\,\left (4\,a^6\,d^2\,e^6+48\,a^5\,b\,d^3\,e^5+150\,a^4\,b^2\,d^4\,e^4+160\,a^3\,b^3\,d^5\,e^3+60\,a^2\,b^4\,d^6\,e^2+\frac {48\,a\,b^5\,d^7\,e}{7}+\frac {b^6\,d^8}{7}\right )+x^9\,\left (\frac {a^6\,e^8}{9}+\frac {16\,a^5\,b\,d\,e^7}{3}+\frac {140\,a^4\,b^2\,d^2\,e^6}{3}+\frac {1120\,a^3\,b^3\,d^3\,e^5}{9}+\frac {350\,a^2\,b^4\,d^4\,e^4}{3}+\frac {112\,a\,b^5\,d^5\,e^3}{3}+\frac {28\,b^6\,d^6\,e^2}{9}\right )+x^5\,\left (14\,a^6\,d^4\,e^4+\frac {336\,a^5\,b\,d^5\,e^3}{5}+84\,a^4\,b^2\,d^6\,e^2+32\,a^3\,b^3\,d^7\,e+3\,a^2\,b^4\,d^8\right )+x^{11}\,\left (\frac {15\,a^4\,b^2\,e^8}{11}+\frac {160\,a^3\,b^3\,d\,e^7}{11}+\frac {420\,a^2\,b^4\,d^2\,e^6}{11}+\frac {336\,a\,b^5\,d^3\,e^5}{11}+\frac {70\,b^6\,d^4\,e^4}{11}\right )+x^6\,\left (\frac {28\,a^6\,d^3\,e^5}{3}+70\,a^5\,b\,d^4\,e^4+140\,a^4\,b^2\,d^5\,e^3+\frac {280\,a^3\,b^3\,d^6\,e^2}{3}+20\,a^2\,b^4\,d^7\,e+a\,b^5\,d^8\right )+x^{10}\,\left (\frac {3\,a^5\,b\,e^8}{5}+12\,a^4\,b^2\,d\,e^7+56\,a^3\,b^3\,d^2\,e^6+84\,a^2\,b^4\,d^3\,e^5+42\,a\,b^5\,d^4\,e^4+\frac {28\,b^6\,d^5\,e^3}{5}\right )+x^8\,\left (a^6\,d\,e^7+21\,a^5\,b\,d^2\,e^6+105\,a^4\,b^2\,d^3\,e^5+175\,a^3\,b^3\,d^4\,e^4+105\,a^2\,b^4\,d^5\,e^3+21\,a\,b^5\,d^6\,e^2+b^6\,d^7\,e\right )+a^6\,d^8\,x+\frac {b^6\,e^8\,x^{15}}{15}+a^3\,d^5\,x^4\,\left (14\,a^3\,e^3+42\,a^2\,b\,d\,e^2+30\,a\,b^2\,d^2\,e+5\,b^3\,d^3\right )+\frac {b^3\,e^5\,x^{12}\,\left (5\,a^3\,e^3+30\,a^2\,b\,d\,e^2+42\,a\,b^2\,d^2\,e+14\,b^3\,d^3\right )}{3}+a^5\,d^7\,x^2\,\left (4\,a\,e+3\,b\,d\right )+\frac {b^5\,e^7\,x^{14}\,\left (3\,a\,e+4\,b\,d\right )}{7}+\frac {a^4\,d^6\,x^3\,\left (28\,a^2\,e^2+48\,a\,b\,d\,e+15\,b^2\,d^2\right )}{3}+\frac {b^4\,e^6\,x^{13}\,\left (15\,a^2\,e^2+48\,a\,b\,d\,e+28\,b^2\,d^2\right )}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^8*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

x^7*((b^6*d^8)/7 + 4*a^6*d^2*e^6 + 48*a^5*b*d^3*e^5 + 60*a^2*b^4*d^6*e^2 + 160*a^3*b^3*d^5*e^3 + 150*a^4*b^2*d
^4*e^4 + (48*a*b^5*d^7*e)/7) + x^9*((a^6*e^8)/9 + (28*b^6*d^6*e^2)/9 + (112*a*b^5*d^5*e^3)/3 + (350*a^2*b^4*d^
4*e^4)/3 + (1120*a^3*b^3*d^3*e^5)/9 + (140*a^4*b^2*d^2*e^6)/3 + (16*a^5*b*d*e^7)/3) + x^5*(3*a^2*b^4*d^8 + 14*
a^6*d^4*e^4 + 32*a^3*b^3*d^7*e + (336*a^5*b*d^5*e^3)/5 + 84*a^4*b^2*d^6*e^2) + x^11*((15*a^4*b^2*e^8)/11 + (70
*b^6*d^4*e^4)/11 + (336*a*b^5*d^3*e^5)/11 + (160*a^3*b^3*d*e^7)/11 + (420*a^2*b^4*d^2*e^6)/11) + x^6*(a*b^5*d^
8 + (28*a^6*d^3*e^5)/3 + 20*a^2*b^4*d^7*e + 70*a^5*b*d^4*e^4 + (280*a^3*b^3*d^6*e^2)/3 + 140*a^4*b^2*d^5*e^3)
+ x^10*((3*a^5*b*e^8)/5 + (28*b^6*d^5*e^3)/5 + 42*a*b^5*d^4*e^4 + 12*a^4*b^2*d*e^7 + 84*a^2*b^4*d^3*e^5 + 56*a
^3*b^3*d^2*e^6) + x^8*(a^6*d*e^7 + b^6*d^7*e + 21*a*b^5*d^6*e^2 + 21*a^5*b*d^2*e^6 + 105*a^2*b^4*d^5*e^3 + 175
*a^3*b^3*d^4*e^4 + 105*a^4*b^2*d^3*e^5) + a^6*d^8*x + (b^6*e^8*x^15)/15 + a^3*d^5*x^4*(14*a^3*e^3 + 5*b^3*d^3
+ 30*a*b^2*d^2*e + 42*a^2*b*d*e^2) + (b^3*e^5*x^12*(5*a^3*e^3 + 14*b^3*d^3 + 42*a*b^2*d^2*e + 30*a^2*b*d*e^2))
/3 + a^5*d^7*x^2*(4*a*e + 3*b*d) + (b^5*e^7*x^14*(3*a*e + 4*b*d))/7 + (a^4*d^6*x^3*(28*a^2*e^2 + 15*b^2*d^2 +
48*a*b*d*e))/3 + (b^4*e^6*x^13*(15*a^2*e^2 + 28*b^2*d^2 + 48*a*b*d*e))/13

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sympy [B]  time = 0.20, size = 884, normalized size = 5.11 \begin {gather*} a^{6} d^{8} x + \frac {b^{6} e^{8} x^{15}}{15} + x^{14} \left (\frac {3 a b^{5} e^{8}}{7} + \frac {4 b^{6} d e^{7}}{7}\right ) + x^{13} \left (\frac {15 a^{2} b^{4} e^{8}}{13} + \frac {48 a b^{5} d e^{7}}{13} + \frac {28 b^{6} d^{2} e^{6}}{13}\right ) + x^{12} \left (\frac {5 a^{3} b^{3} e^{8}}{3} + 10 a^{2} b^{4} d e^{7} + 14 a b^{5} d^{2} e^{6} + \frac {14 b^{6} d^{3} e^{5}}{3}\right ) + x^{11} \left (\frac {15 a^{4} b^{2} e^{8}}{11} + \frac {160 a^{3} b^{3} d e^{7}}{11} + \frac {420 a^{2} b^{4} d^{2} e^{6}}{11} + \frac {336 a b^{5} d^{3} e^{5}}{11} + \frac {70 b^{6} d^{4} e^{4}}{11}\right ) + x^{10} \left (\frac {3 a^{5} b e^{8}}{5} + 12 a^{4} b^{2} d e^{7} + 56 a^{3} b^{3} d^{2} e^{6} + 84 a^{2} b^{4} d^{3} e^{5} + 42 a b^{5} d^{4} e^{4} + \frac {28 b^{6} d^{5} e^{3}}{5}\right ) + x^{9} \left (\frac {a^{6} e^{8}}{9} + \frac {16 a^{5} b d e^{7}}{3} + \frac {140 a^{4} b^{2} d^{2} e^{6}}{3} + \frac {1120 a^{3} b^{3} d^{3} e^{5}}{9} + \frac {350 a^{2} b^{4} d^{4} e^{4}}{3} + \frac {112 a b^{5} d^{5} e^{3}}{3} + \frac {28 b^{6} d^{6} e^{2}}{9}\right ) + x^{8} \left (a^{6} d e^{7} + 21 a^{5} b d^{2} e^{6} + 105 a^{4} b^{2} d^{3} e^{5} + 175 a^{3} b^{3} d^{4} e^{4} + 105 a^{2} b^{4} d^{5} e^{3} + 21 a b^{5} d^{6} e^{2} + b^{6} d^{7} e\right ) + x^{7} \left (4 a^{6} d^{2} e^{6} + 48 a^{5} b d^{3} e^{5} + 150 a^{4} b^{2} d^{4} e^{4} + 160 a^{3} b^{3} d^{5} e^{3} + 60 a^{2} b^{4} d^{6} e^{2} + \frac {48 a b^{5} d^{7} e}{7} + \frac {b^{6} d^{8}}{7}\right ) + x^{6} \left (\frac {28 a^{6} d^{3} e^{5}}{3} + 70 a^{5} b d^{4} e^{4} + 140 a^{4} b^{2} d^{5} e^{3} + \frac {280 a^{3} b^{3} d^{6} e^{2}}{3} + 20 a^{2} b^{4} d^{7} e + a b^{5} d^{8}\right ) + x^{5} \left (14 a^{6} d^{4} e^{4} + \frac {336 a^{5} b d^{5} e^{3}}{5} + 84 a^{4} b^{2} d^{6} e^{2} + 32 a^{3} b^{3} d^{7} e + 3 a^{2} b^{4} d^{8}\right ) + x^{4} \left (14 a^{6} d^{5} e^{3} + 42 a^{5} b d^{6} e^{2} + 30 a^{4} b^{2} d^{7} e + 5 a^{3} b^{3} d^{8}\right ) + x^{3} \left (\frac {28 a^{6} d^{6} e^{2}}{3} + 16 a^{5} b d^{7} e + 5 a^{4} b^{2} d^{8}\right ) + x^{2} \left (4 a^{6} d^{7} e + 3 a^{5} b d^{8}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**8*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

a**6*d**8*x + b**6*e**8*x**15/15 + x**14*(3*a*b**5*e**8/7 + 4*b**6*d*e**7/7) + x**13*(15*a**2*b**4*e**8/13 + 4
8*a*b**5*d*e**7/13 + 28*b**6*d**2*e**6/13) + x**12*(5*a**3*b**3*e**8/3 + 10*a**2*b**4*d*e**7 + 14*a*b**5*d**2*
e**6 + 14*b**6*d**3*e**5/3) + x**11*(15*a**4*b**2*e**8/11 + 160*a**3*b**3*d*e**7/11 + 420*a**2*b**4*d**2*e**6/
11 + 336*a*b**5*d**3*e**5/11 + 70*b**6*d**4*e**4/11) + x**10*(3*a**5*b*e**8/5 + 12*a**4*b**2*d*e**7 + 56*a**3*
b**3*d**2*e**6 + 84*a**2*b**4*d**3*e**5 + 42*a*b**5*d**4*e**4 + 28*b**6*d**5*e**3/5) + x**9*(a**6*e**8/9 + 16*
a**5*b*d*e**7/3 + 140*a**4*b**2*d**2*e**6/3 + 1120*a**3*b**3*d**3*e**5/9 + 350*a**2*b**4*d**4*e**4/3 + 112*a*b
**5*d**5*e**3/3 + 28*b**6*d**6*e**2/9) + x**8*(a**6*d*e**7 + 21*a**5*b*d**2*e**6 + 105*a**4*b**2*d**3*e**5 + 1
75*a**3*b**3*d**4*e**4 + 105*a**2*b**4*d**5*e**3 + 21*a*b**5*d**6*e**2 + b**6*d**7*e) + x**7*(4*a**6*d**2*e**6
+ 48*a**5*b*d**3*e**5 + 150*a**4*b**2*d**4*e**4 + 160*a**3*b**3*d**5*e**3 + 60*a**2*b**4*d**6*e**2 + 48*a*b**
5*d**7*e/7 + b**6*d**8/7) + x**6*(28*a**6*d**3*e**5/3 + 70*a**5*b*d**4*e**4 + 140*a**4*b**2*d**5*e**3 + 280*a*
*3*b**3*d**6*e**2/3 + 20*a**2*b**4*d**7*e + a*b**5*d**8) + x**5*(14*a**6*d**4*e**4 + 336*a**5*b*d**5*e**3/5 +
84*a**4*b**2*d**6*e**2 + 32*a**3*b**3*d**7*e + 3*a**2*b**4*d**8) + x**4*(14*a**6*d**5*e**3 + 42*a**5*b*d**6*e*
*2 + 30*a**4*b**2*d**7*e + 5*a**3*b**3*d**8) + x**3*(28*a**6*d**6*e**2/3 + 16*a**5*b*d**7*e + 5*a**4*b**2*d**8
) + x**2*(4*a**6*d**7*e + 3*a**5*b*d**8)

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