3.19.27 \(\int \frac {(a+b x+c x^2)^4}{(d+e x)^8} \, dx\)

Optimal. Leaf size=424 \[ -\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{3 e^9 (d+e x)^3}-\frac {2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (d+e x)}+\frac {2 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^2}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^4}-\frac {2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9 (d+e x)^5}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^9 (d+e x)^6}-\frac {\left (a e^2-b d e+c d^2\right )^4}{7 e^9 (d+e x)^7}-\frac {4 c^3 (2 c d-b e) \log (d+e x)}{e^9}+\frac {c^4 x}{e^8} \]

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Rubi [A]  time = 0.48, antiderivative size = 424, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {698} \begin {gather*} -\frac {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}{3 e^9 (d+e x)^3}-\frac {2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (d+e x)}+\frac {2 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^2}+\frac {(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 )}{e^9 (d+e x)^4}-\frac {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 )}{5 e^9 (d+e x)^5}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^9 (d+e x)^6}-\frac {\left (a e^2-b d e+c d^2\right )^4}{7 e^9 (d+e x)^7}-\frac {4 c^3 (2 c d-b e) \log (d+e x)}{e^9}+\frac {c^4 x}{e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^4/(d + e*x)^8,x]

[Out]

(c^4*x)/e^8 - (c*d^2 - b*d*e + a*e^2)^4/(7*e^9*(d + e*x)^7) + (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(3*e
^9*(d + e*x)^6) - (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)))/(5*e^9*(d + e*x
)^5) + ((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)))/(e^9*(d + e*x)^4) -
(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))/(3*e^9*(d + e*x)^3) + (2*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e)))/(
e^9*(d + e*x)^2) - (2*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e)))/(e^9*(d + e*x)) - (4*c^3*(2*c*d - b*
e)*Log[d + e*x])/e^9

Rule 698

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

Rubi steps

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

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Mathematica [A]  time = 0.51, size = 748, normalized size = 1.76 \begin {gather*} -\frac {6 c^2 e^2 \left (a^2 e^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 a b e \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )+15 b^2 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )\right )+c e^3 \left (4 a^3 e^3 \left (d^2+7 d e x+21 e^2 x^2\right )+9 a^2 b e^2 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+12 a b^2 e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+10 b^3 \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )+e^4 \left (15 a^4 e^4+10 a^3 b e^3 (d+7 e x)+6 a^2 b^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 a b^3 e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+b^4 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+c^3 e \left (60 a e \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )-b d \left (1089 d^6+7203 d^5 e x+20139 d^4 e^2 x^2+30625 d^3 e^3 x^3+26950 d^2 e^4 x^4+13230 d e^5 x^5+2940 e^6 x^6\right )\right )+420 c^3 (d+e x)^7 (2 c d-b e) \log (d+e x)+c^4 \left (1443 d^8+9261 d^7 e x+24843 d^6 e^2 x^2+35525 d^5 e^3 x^3+28175 d^4 e^4 x^4+11025 d^3 e^5 x^5+735 d^2 e^6 x^6-735 d e^7 x^7-105 e^8 x^8\right )}{105 e^9 (d+e x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^4/(d + e*x)^8,x]

[Out]

-1/105*(c^4*(1443*d^8 + 9261*d^7*e*x + 24843*d^6*e^2*x^2 + 35525*d^5*e^3*x^3 + 28175*d^4*e^4*x^4 + 11025*d^3*e
^5*x^5 + 735*d^2*e^6*x^6 - 735*d*e^7*x^7 - 105*e^8*x^8) + e^4*(15*a^4*e^4 + 10*a^3*b*e^3*(d + 7*e*x) + 6*a^2*b
^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*a*b^3*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + b^4*(d^4 + 7*d
^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4)) + c*e^3*(4*a^3*e^3*(d^2 + 7*d*e*x + 21*e^2*x^2) + 9*a^2*
b*e^2*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 12*a*b^2*e*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*
x^3 + 35*e^4*x^4) + 10*b^3*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)) +
6*c^2*e^2*(a^2*e^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 5*a*b*e*(d^5 + 7*d^4*e*x +
21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5) + 15*b^2*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*
d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6)) + c^3*e*(60*a*e*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 +
35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6) - b*d*(1089*d^6 + 7203*d^5*e*x + 20139*d^4*e^2*x^2
+ 30625*d^3*e^3*x^3 + 26950*d^2*e^4*x^4 + 13230*d*e^5*x^5 + 2940*e^6*x^6)) + 420*c^3*(2*c*d - b*e)*(d + e*x)^
7*Log[d + e*x])/(e^9*(d + e*x)^7)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x + c*x^2)^4/(d + e*x)^8,x]

[Out]

IntegrateAlgebraic[(a + b*x + c*x^2)^4/(d + e*x)^8, x]

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fricas [B]  time = 0.40, size = 1082, normalized size = 2.55

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^8,x, algorithm="fricas")

[Out]

1/105*(105*c^4*e^8*x^8 + 735*c^4*d*e^7*x^7 - 1443*c^4*d^8 + 1089*b*c^3*d^7*e - 10*a^3*b*d*e^7 - 15*a^4*e^8 - 3
0*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 10*(b^3*c + 3*a*b*c^2)*d^5*e^3 - (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 3*
(a*b^3 + 3*a^2*b*c)*d^3*e^5 - 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 - 105*(7*c^4*d^2*e^6 - 28*b*c^3*d*e^7 + 2*(3*b^2
*c^2 + 2*a*c^3)*e^8)*x^6 - 105*(105*c^4*d^3*e^5 - 126*b*c^3*d^2*e^6 + 6*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 2*(b^3*c
+ 3*a*b*c^2)*e^8)*x^5 - 35*(805*c^4*d^4*e^4 - 770*b*c^3*d^3*e^5 + 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 + 10*(b^3*
c + 3*a*b*c^2)*d*e^7 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 - 35*(1015*c^4*d^5*e^3 - 875*b*c^3*d^4*e^4 + 30
*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 + 10*(b^3*c + 3*a*b*c^2)*d^2*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 + 3*(a*
b^3 + 3*a^2*b*c)*e^8)*x^3 - 21*(1183*c^4*d^6*e^2 - 959*b*c^3*d^5*e^3 + 30*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 + 10*(
b^3*c + 3*a*b*c^2)*d^3*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 + 3*(a*b^3 + 3*a^2*b*c)*d*e^7 + 2*(3*a^2*b
^2 + 2*a^3*c)*e^8)*x^2 - 7*(1323*c^4*d^7*e - 1029*b*c^3*d^6*e^2 + 10*a^3*b*e^8 + 30*(3*b^2*c^2 + 2*a*c^3)*d^5*
e^3 + 10*(b^3*c + 3*a*b*c^2)*d^4*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 + 3*(a*b^3 + 3*a^2*b*c)*d^2*e^6
+ 2*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x - 420*(2*c^4*d^8 - b*c^3*d^7*e + (2*c^4*d*e^7 - b*c^3*e^8)*x^7 + 7*(2*c^4*d
^2*e^6 - b*c^3*d*e^7)*x^6 + 21*(2*c^4*d^3*e^5 - b*c^3*d^2*e^6)*x^5 + 35*(2*c^4*d^4*e^4 - b*c^3*d^3*e^5)*x^4 +
35*(2*c^4*d^5*e^3 - b*c^3*d^4*e^4)*x^3 + 21*(2*c^4*d^6*e^2 - b*c^3*d^5*e^3)*x^2 + 7*(2*c^4*d^7*e - b*c^3*d^6*e
^2)*x)*log(e*x + d))/(e^16*x^7 + 7*d*e^15*x^6 + 21*d^2*e^14*x^5 + 35*d^3*e^13*x^4 + 35*d^4*e^12*x^3 + 21*d^5*e
^11*x^2 + 7*d^6*e^10*x + d^7*e^9)

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giac [B]  time = 0.17, size = 833, normalized size = 1.96 \begin {gather*} c^{4} x e^{\left (-8\right )} - 4 \, {\left (2 \, c^{4} d - b c^{3} e\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (1443 \, c^{4} d^{8} - 1089 \, b c^{3} d^{7} e + 90 \, b^{2} c^{2} d^{6} e^{2} + 60 \, a c^{3} d^{6} e^{2} + 10 \, b^{3} c d^{5} e^{3} + 30 \, a b c^{2} d^{5} e^{3} + b^{4} d^{4} e^{4} + 12 \, a b^{2} c d^{4} e^{4} + 6 \, a^{2} c^{2} d^{4} e^{4} + 3 \, a b^{3} d^{3} e^{5} + 9 \, a^{2} b c d^{3} e^{5} + 210 \, {\left (14 \, c^{4} d^{2} e^{6} - 14 \, b c^{3} d e^{7} + 3 \, b^{2} c^{2} e^{8} + 2 \, a c^{3} e^{8}\right )} x^{6} + 6 \, a^{2} b^{2} d^{2} e^{6} + 4 \, a^{3} c d^{2} e^{6} + 210 \, {\left (70 \, c^{4} d^{3} e^{5} - 63 \, b c^{3} d^{2} e^{6} + 9 \, b^{2} c^{2} d e^{7} + 6 \, a c^{3} d e^{7} + b^{3} c e^{8} + 3 \, a b c^{2} e^{8}\right )} x^{5} + 10 \, a^{3} b d e^{7} + 35 \, {\left (910 \, c^{4} d^{4} e^{4} - 770 \, b c^{3} d^{3} e^{5} + 90 \, b^{2} c^{2} d^{2} e^{6} + 60 \, a c^{3} d^{2} e^{6} + 10 \, b^{3} c d e^{7} + 30 \, a b c^{2} d e^{7} + b^{4} e^{8} + 12 \, a b^{2} c e^{8} + 6 \, a^{2} c^{2} e^{8}\right )} x^{4} + 15 \, a^{4} e^{8} + 35 \, {\left (1078 \, c^{4} d^{5} e^{3} - 875 \, b c^{3} d^{4} e^{4} + 90 \, b^{2} c^{2} d^{3} e^{5} + 60 \, a c^{3} d^{3} e^{5} + 10 \, b^{3} c d^{2} e^{6} + 30 \, a b c^{2} d^{2} e^{6} + b^{4} d e^{7} + 12 \, a b^{2} c d e^{7} + 6 \, a^{2} c^{2} d e^{7} + 3 \, a b^{3} e^{8} + 9 \, a^{2} b c e^{8}\right )} x^{3} + 21 \, {\left (1218 \, c^{4} d^{6} e^{2} - 959 \, b c^{3} d^{5} e^{3} + 90 \, b^{2} c^{2} d^{4} e^{4} + 60 \, a c^{3} d^{4} e^{4} + 10 \, b^{3} c d^{3} e^{5} + 30 \, a b c^{2} d^{3} e^{5} + b^{4} d^{2} e^{6} + 12 \, a b^{2} c d^{2} e^{6} + 6 \, a^{2} c^{2} d^{2} e^{6} + 3 \, a b^{3} d e^{7} + 9 \, a^{2} b c d e^{7} + 6 \, a^{2} b^{2} e^{8} + 4 \, a^{3} c e^{8}\right )} x^{2} + 7 \, {\left (1338 \, c^{4} d^{7} e - 1029 \, b c^{3} d^{6} e^{2} + 90 \, b^{2} c^{2} d^{5} e^{3} + 60 \, a c^{3} d^{5} e^{3} + 10 \, b^{3} c d^{4} e^{4} + 30 \, a b c^{2} d^{4} e^{4} + b^{4} d^{3} e^{5} + 12 \, a b^{2} c d^{3} e^{5} + 6 \, a^{2} c^{2} d^{3} e^{5} + 3 \, a b^{3} d^{2} e^{6} + 9 \, a^{2} b c d^{2} e^{6} + 6 \, a^{2} b^{2} d e^{7} + 4 \, a^{3} c d e^{7} + 10 \, a^{3} b e^{8}\right )} x\right )} e^{\left (-9\right )}}{105 \, {\left (x e + d\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^8,x, algorithm="giac")

[Out]

c^4*x*e^(-8) - 4*(2*c^4*d - b*c^3*e)*e^(-9)*log(abs(x*e + d)) - 1/105*(1443*c^4*d^8 - 1089*b*c^3*d^7*e + 90*b^
2*c^2*d^6*e^2 + 60*a*c^3*d^6*e^2 + 10*b^3*c*d^5*e^3 + 30*a*b*c^2*d^5*e^3 + b^4*d^4*e^4 + 12*a*b^2*c*d^4*e^4 +
6*a^2*c^2*d^4*e^4 + 3*a*b^3*d^3*e^5 + 9*a^2*b*c*d^3*e^5 + 210*(14*c^4*d^2*e^6 - 14*b*c^3*d*e^7 + 3*b^2*c^2*e^8
+ 2*a*c^3*e^8)*x^6 + 6*a^2*b^2*d^2*e^6 + 4*a^3*c*d^2*e^6 + 210*(70*c^4*d^3*e^5 - 63*b*c^3*d^2*e^6 + 9*b^2*c^2
*d*e^7 + 6*a*c^3*d*e^7 + b^3*c*e^8 + 3*a*b*c^2*e^8)*x^5 + 10*a^3*b*d*e^7 + 35*(910*c^4*d^4*e^4 - 770*b*c^3*d^3
*e^5 + 90*b^2*c^2*d^2*e^6 + 60*a*c^3*d^2*e^6 + 10*b^3*c*d*e^7 + 30*a*b*c^2*d*e^7 + b^4*e^8 + 12*a*b^2*c*e^8 +
6*a^2*c^2*e^8)*x^4 + 15*a^4*e^8 + 35*(1078*c^4*d^5*e^3 - 875*b*c^3*d^4*e^4 + 90*b^2*c^2*d^3*e^5 + 60*a*c^3*d^3
*e^5 + 10*b^3*c*d^2*e^6 + 30*a*b*c^2*d^2*e^6 + b^4*d*e^7 + 12*a*b^2*c*d*e^7 + 6*a^2*c^2*d*e^7 + 3*a*b^3*e^8 +
9*a^2*b*c*e^8)*x^3 + 21*(1218*c^4*d^6*e^2 - 959*b*c^3*d^5*e^3 + 90*b^2*c^2*d^4*e^4 + 60*a*c^3*d^4*e^4 + 10*b^3
*c*d^3*e^5 + 30*a*b*c^2*d^3*e^5 + b^4*d^2*e^6 + 12*a*b^2*c*d^2*e^6 + 6*a^2*c^2*d^2*e^6 + 3*a*b^3*d*e^7 + 9*a^2
*b*c*d*e^7 + 6*a^2*b^2*e^8 + 4*a^3*c*e^8)*x^2 + 7*(1338*c^4*d^7*e - 1029*b*c^3*d^6*e^2 + 90*b^2*c^2*d^5*e^3 +
60*a*c^3*d^5*e^3 + 10*b^3*c*d^4*e^4 + 30*a*b*c^2*d^4*e^4 + b^4*d^3*e^5 + 12*a*b^2*c*d^3*e^5 + 6*a^2*c^2*d^3*e^
5 + 3*a*b^3*d^2*e^6 + 9*a^2*b*c*d^2*e^6 + 6*a^2*b^2*d*e^7 + 4*a^3*c*d*e^7 + 10*a^3*b*e^8)*x)*e^(-9)/(x*e + d)^
7

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maple [B]  time = 0.06, size = 1374, normalized size = 3.24

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^4/(e*x+d)^8,x)

[Out]

-6/e^4/(e*x+d)^6*a^2*b*c*d^2+12/7/e^4/(e*x+d)^7*d^3*a^2*b*c-12/7/e^5/(e*x+d)^7*d^4*a*b^2*c+20/e^6/(e*x+d)^3*a*
b*c^2*d+12/e^5/(e*x+d)^4*a*b^2*c*d-30/e^6/(e*x+d)^4*a*b*c^2*d^2+8/e^5/(e*x+d)^6*a*b^2*c*d^3-10/e^6/(e*x+d)^6*a
*b*c^2*d^4+36/5/e^4/(e*x+d)^5*a^2*b*c*d-72/5/e^5/(e*x+d)^5*a*b^2*c*d^2+24/e^6/(e*x+d)^5*a*b*c^2*d^3+12/7/e^6/(
e*x+d)^7*d^5*a*b*c^2-28*c^4/e^9/(e*x+d)*d^2-1/7/e^5/(e*x+d)^7*d^4*b^4-1/7/e^9/(e*x+d)^7*c^4*d^8+4*c^3/e^8*ln(e
*x+d)*b-8*c^4/e^9*ln(e*x+d)*d-1/e^4/(e*x+d)^4*a*b^3+1/e^5/(e*x+d)^4*b^4*d+14/e^9/(e*x+d)^4*c^4*d^5-4/5/e^3/(e*
x+d)^5*a^3*c-6/5/e^3/(e*x+d)^5*a^2*b^2-6/5/e^5/(e*x+d)^5*b^4*d^2-28/5/e^9/(e*x+d)^5*c^4*d^6-2*c/e^6/(e*x+d)^2*
b^3+28*c^4/e^9/(e*x+d)^2*d^3-2/e^5/(e*x+d)^3*c^2*a^2-70/3/e^9/(e*x+d)^3*c^4*d^4+4/3/e^9/(e*x+d)^6*c^4*d^7-2/3/
e^2/(e*x+d)^6*a^3*b+2/3/e^5/(e*x+d)^6*b^4*d^3-4*c^3/e^7/(e*x+d)*a-6*c^2/e^7/(e*x+d)*b^2-6/7/e^5/(e*x+d)^7*a^2*
c^2*d^4+4/7/e^4/(e*x+d)^7*d^3*a*b^3-4/7/e^7/(e*x+d)^7*a*c^3*d^6+4/7/e^6/(e*x+d)^7*d^5*b^3*c-6/7/e^7/(e*x+d)^7*
d^6*b^2*c^2-36/5/e^5/(e*x+d)^5*a^2*c^2*d^2+12/5/e^4/(e*x+d)^5*a*b^3*d-12/e^7/(e*x+d)^5*a*c^3*d^4-1/3/(e*x+d)^3
*b^4/e^5-35/e^8/(e*x+d)^4*b*c^3*d^4+4/7/e^8/(e*x+d)^7*d^7*b*c^3+4/3/e^3/(e*x+d)^6*a^3*c*d+2/e^3/(e*x+d)^6*a^2*
b^2*d+4/e^5/(e*x+d)^6*a^2*c^2*d^3+4/e^7/(e*x+d)^6*a*c^3*d^5+28*c^3/e^8/(e*x+d)*b*d+4/7/e^2/(e*x+d)^7*d*a^3*b-4
/7/e^3/(e*x+d)^7*a^3*c*d^2-6/7/e^3/(e*x+d)^7*d^2*a^2*b^2-1/7/e/(e*x+d)^7*a^4-6*c^2/e^6/(e*x+d)^2*a*b+12*c^3/e^
7/(e*x+d)^2*a*d+18*c^2/e^7/(e*x+d)^2*b^2*d-42*c^3/e^8/(e*x+d)^2*b*d^2-4/e^5/(e*x+d)^3*a*b^2*c-20/e^7/(e*x+d)^3
*a*c^3*d^2-2/e^4/(e*x+d)^6*a*b^3*d^2+20/3/e^6/(e*x+d)^3*b^3*c*d-30/e^7/(e*x+d)^3*b^2*c^2*d^2-10/3/e^6/(e*x+d)^
6*b^3*c*d^4+6/e^7/(e*x+d)^6*b^2*c^2*d^5+140/3/e^8/(e*x+d)^3*b*c^3*d^3-3/e^4/(e*x+d)^4*a^2*b*c+6/e^5/(e*x+d)^4*
a^2*c^2*d+20/e^7/(e*x+d)^4*a*c^3*d^3-10/e^6/(e*x+d)^4*b^3*c*d^2+30/e^7/(e*x+d)^4*b^2*c^2*d^3+8/e^6/(e*x+d)^5*b
^3*c*d^3-18/e^7/(e*x+d)^5*b^2*c^2*d^4+84/5/e^8/(e*x+d)^5*b*c^3*d^5-14/3/e^8/(e*x+d)^6*b*c^3*d^6+c^4*x/e^8

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maxima [B]  time = 1.38, size = 872, normalized size = 2.06 \begin {gather*} -\frac {1443 \, c^{4} d^{8} - 1089 \, b c^{3} d^{7} e + 10 \, a^{3} b d e^{7} + 15 \, a^{4} e^{8} + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 210 \, {\left (14 \, c^{4} d^{2} e^{6} - 14 \, b c^{3} d e^{7} + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{8}\right )} x^{6} + 210 \, {\left (70 \, c^{4} d^{3} e^{5} - 63 \, b c^{3} d^{2} e^{6} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{7} + {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{8}\right )} x^{5} + 35 \, {\left (910 \, c^{4} d^{4} e^{4} - 770 \, b c^{3} d^{3} e^{5} + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{6} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{7} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{8}\right )} x^{4} + 35 \, {\left (1078 \, c^{4} d^{5} e^{3} - 875 \, b c^{3} d^{4} e^{4} + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{6} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{7} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} + 21 \, {\left (1218 \, c^{4} d^{6} e^{2} - 959 \, b c^{3} d^{5} e^{3} + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{5} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{6} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{7} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} + 7 \, {\left (1338 \, c^{4} d^{7} e - 1029 \, b c^{3} d^{6} e^{2} + 10 \, a^{3} b e^{8} + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} + 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{5} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} + 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x}{105 \, {\left (e^{16} x^{7} + 7 \, d e^{15} x^{6} + 21 \, d^{2} e^{14} x^{5} + 35 \, d^{3} e^{13} x^{4} + 35 \, d^{4} e^{12} x^{3} + 21 \, d^{5} e^{11} x^{2} + 7 \, d^{6} e^{10} x + d^{7} e^{9}\right )}} + \frac {c^{4} x}{e^{8}} - \frac {4 \, {\left (2 \, c^{4} d - b c^{3} e\right )} \log \left (e x + d\right )}{e^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^8,x, algorithm="maxima")

[Out]

-1/105*(1443*c^4*d^8 - 1089*b*c^3*d^7*e + 10*a^3*b*d*e^7 + 15*a^4*e^8 + 30*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 10*
(b^3*c + 3*a*b*c^2)*d^5*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 3*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 2*(3*a^
2*b^2 + 2*a^3*c)*d^2*e^6 + 210*(14*c^4*d^2*e^6 - 14*b*c^3*d*e^7 + (3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 210*(70*c^4
*d^3*e^5 - 63*b*c^3*d^2*e^6 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + (b^3*c + 3*a*b*c^2)*e^8)*x^5 + 35*(910*c^4*d^4*e
^4 - 770*b*c^3*d^3*e^5 + 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 + 10*(b^3*c + 3*a*b*c^2)*d*e^7 + (b^4 + 12*a*b^2*c +
6*a^2*c^2)*e^8)*x^4 + 35*(1078*c^4*d^5*e^3 - 875*b*c^3*d^4*e^4 + 30*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 + 10*(b^3*c
+ 3*a*b*c^2)*d^2*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 + 3*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 21*(1218*c^4*d
^6*e^2 - 959*b*c^3*d^5*e^3 + 30*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 + 10*(b^3*c + 3*a*b*c^2)*d^3*e^5 + (b^4 + 12*a*b
^2*c + 6*a^2*c^2)*d^2*e^6 + 3*(a*b^3 + 3*a^2*b*c)*d*e^7 + 2*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 7*(1338*c^4*d^7*e
- 1029*b*c^3*d^6*e^2 + 10*a^3*b*e^8 + 30*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 + 10*(b^3*c + 3*a*b*c^2)*d^4*e^4 + (b^
4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 + 3*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 2*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^16*
x^7 + 7*d*e^15*x^6 + 21*d^2*e^14*x^5 + 35*d^3*e^13*x^4 + 35*d^4*e^12*x^3 + 21*d^5*e^11*x^2 + 7*d^6*e^10*x + d^
7*e^9) + c^4*x/e^8 - 4*(2*c^4*d - b*c^3*e)*log(e*x + d)/e^9

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mupad [B]  time = 0.95, size = 1306, normalized size = 3.08 \begin {gather*} -\frac {\frac {a^4\,e^8}{7}+\frac {481\,c^4\,d^8}{35}+8\,c^4\,d^8\,\ln \left (d+e\,x\right )+\frac {b^4\,d^4\,e^4}{105}+\frac {b^4\,e^8\,x^4}{3}-c^4\,e^8\,x^8+\frac {a\,b^3\,d^3\,e^5}{35}+\frac {4\,a\,c^3\,d^6\,e^2}{7}+\frac {4\,a^3\,c\,d^2\,e^6}{105}+\frac {2\,b^3\,c\,d^5\,e^3}{21}+a\,b^3\,e^8\,x^3+\frac {4\,a^3\,c\,e^8\,x^2}{5}+4\,a\,c^3\,e^8\,x^6+2\,b^3\,c\,e^8\,x^5+\frac {b^4\,d^3\,e^5\,x}{15}+\frac {b^4\,d\,e^7\,x^3}{3}-7\,c^4\,d\,e^7\,x^7+\frac {2\,a^2\,b^2\,d^2\,e^6}{35}+\frac {2\,a^2\,c^2\,d^4\,e^4}{35}+\frac {6\,b^2\,c^2\,d^6\,e^2}{7}+\frac {6\,a^2\,b^2\,e^8\,x^2}{5}+2\,a^2\,c^2\,e^8\,x^4+6\,b^2\,c^2\,e^8\,x^6+\frac {b^4\,d^2\,e^6\,x^2}{5}+\frac {1183\,c^4\,d^6\,e^2\,x^2}{5}+\frac {1015\,c^4\,d^5\,e^3\,x^3}{3}+\frac {805\,c^4\,d^4\,e^4\,x^4}{3}+105\,c^4\,d^3\,e^5\,x^5+7\,c^4\,d^2\,e^6\,x^6+\frac {2\,a^3\,b\,d\,e^7}{21}-\frac {363\,b\,c^3\,d^7\,e}{35}+\frac {2\,a^3\,b\,e^8\,x}{3}+\frac {441\,c^4\,d^7\,e\,x}{5}-4\,b\,c^3\,d^7\,e\,\ln \left (d+e\,x\right )+\frac {4\,a^3\,c\,d\,e^7\,x}{15}+56\,c^4\,d^7\,e\,x\,\ln \left (d+e\,x\right )+\frac {6\,a^2\,c^2\,d^2\,e^6\,x^2}{5}+18\,b^2\,c^2\,d^4\,e^4\,x^2+30\,b^2\,c^2\,d^3\,e^5\,x^3+30\,b^2\,c^2\,d^2\,e^6\,x^4+\frac {2\,a\,b\,c^2\,d^5\,e^3}{7}+\frac {4\,a\,b^2\,c\,d^4\,e^4}{35}+\frac {3\,a^2\,b\,c\,d^3\,e^5}{35}+3\,a^2\,b\,c\,e^8\,x^3+4\,a\,b^2\,c\,e^8\,x^4+6\,a\,b\,c^2\,e^8\,x^5+\frac {a\,b^3\,d^2\,e^6\,x}{5}+\frac {2\,a^2\,b^2\,d\,e^7\,x}{5}+\frac {3\,a\,b^3\,d\,e^7\,x^2}{5}+4\,a\,c^3\,d^5\,e^3\,x+12\,a\,c^3\,d\,e^7\,x^5-\frac {343\,b\,c^3\,d^6\,e^2\,x}{5}+\frac {2\,b^3\,c\,d^4\,e^4\,x}{3}+\frac {10\,b^3\,c\,d\,e^7\,x^4}{3}-28\,b\,c^3\,d\,e^7\,x^6-4\,b\,c^3\,e^8\,x^7\,\ln \left (d+e\,x\right )+8\,c^4\,d\,e^7\,x^7\,\ln \left (d+e\,x\right )+\frac {2\,a^2\,c^2\,d^3\,e^5\,x}{5}+12\,a\,c^3\,d^4\,e^4\,x^2+20\,a\,c^3\,d^3\,e^5\,x^3+2\,a^2\,c^2\,d\,e^7\,x^3+20\,a\,c^3\,d^2\,e^6\,x^4+6\,b^2\,c^2\,d^5\,e^3\,x-\frac {959\,b\,c^3\,d^5\,e^3\,x^2}{5}+2\,b^3\,c\,d^3\,e^5\,x^2-\frac {875\,b\,c^3\,d^4\,e^4\,x^3}{3}+\frac {10\,b^3\,c\,d^2\,e^6\,x^3}{3}-\frac {770\,b\,c^3\,d^3\,e^5\,x^4}{3}-126\,b\,c^3\,d^2\,e^6\,x^5+18\,b^2\,c^2\,d\,e^7\,x^5+168\,c^4\,d^6\,e^2\,x^2\,\ln \left (d+e\,x\right )+280\,c^4\,d^5\,e^3\,x^3\,\ln \left (d+e\,x\right )+280\,c^4\,d^4\,e^4\,x^4\,\ln \left (d+e\,x\right )+168\,c^4\,d^3\,e^5\,x^5\,\ln \left (d+e\,x\right )+56\,c^4\,d^2\,e^6\,x^6\,\ln \left (d+e\,x\right )+6\,a\,b\,c^2\,d^3\,e^5\,x^2+\frac {12\,a\,b^2\,c\,d^2\,e^6\,x^2}{5}+10\,a\,b\,c^2\,d^2\,e^6\,x^3-84\,b\,c^3\,d^5\,e^3\,x^2\,\ln \left (d+e\,x\right )-140\,b\,c^3\,d^4\,e^4\,x^3\,\ln \left (d+e\,x\right )-140\,b\,c^3\,d^3\,e^5\,x^4\,\ln \left (d+e\,x\right )-84\,b\,c^3\,d^2\,e^6\,x^5\,\ln \left (d+e\,x\right )+2\,a\,b\,c^2\,d^4\,e^4\,x+\frac {4\,a\,b^2\,c\,d^3\,e^5\,x}{5}+\frac {3\,a^2\,b\,c\,d^2\,e^6\,x}{5}+\frac {9\,a^2\,b\,c\,d\,e^7\,x^2}{5}+4\,a\,b^2\,c\,d\,e^7\,x^3+10\,a\,b\,c^2\,d\,e^7\,x^4-28\,b\,c^3\,d^6\,e^2\,x\,\ln \left (d+e\,x\right )-28\,b\,c^3\,d\,e^7\,x^6\,\ln \left (d+e\,x\right )}{e^9\,{\left (d+e\,x\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^4/(d + e*x)^8,x)

[Out]

-((a^4*e^8)/7 + (481*c^4*d^8)/35 + 8*c^4*d^8*log(d + e*x) + (b^4*d^4*e^4)/105 + (b^4*e^8*x^4)/3 - c^4*e^8*x^8
+ (a*b^3*d^3*e^5)/35 + (4*a*c^3*d^6*e^2)/7 + (4*a^3*c*d^2*e^6)/105 + (2*b^3*c*d^5*e^3)/21 + a*b^3*e^8*x^3 + (4
*a^3*c*e^8*x^2)/5 + 4*a*c^3*e^8*x^6 + 2*b^3*c*e^8*x^5 + (b^4*d^3*e^5*x)/15 + (b^4*d*e^7*x^3)/3 - 7*c^4*d*e^7*x
^7 + (2*a^2*b^2*d^2*e^6)/35 + (2*a^2*c^2*d^4*e^4)/35 + (6*b^2*c^2*d^6*e^2)/7 + (6*a^2*b^2*e^8*x^2)/5 + 2*a^2*c
^2*e^8*x^4 + 6*b^2*c^2*e^8*x^6 + (b^4*d^2*e^6*x^2)/5 + (1183*c^4*d^6*e^2*x^2)/5 + (1015*c^4*d^5*e^3*x^3)/3 + (
805*c^4*d^4*e^4*x^4)/3 + 105*c^4*d^3*e^5*x^5 + 7*c^4*d^2*e^6*x^6 + (2*a^3*b*d*e^7)/21 - (363*b*c^3*d^7*e)/35 +
(2*a^3*b*e^8*x)/3 + (441*c^4*d^7*e*x)/5 - 4*b*c^3*d^7*e*log(d + e*x) + (4*a^3*c*d*e^7*x)/15 + 56*c^4*d^7*e*x*
log(d + e*x) + (6*a^2*c^2*d^2*e^6*x^2)/5 + 18*b^2*c^2*d^4*e^4*x^2 + 30*b^2*c^2*d^3*e^5*x^3 + 30*b^2*c^2*d^2*e^
6*x^4 + (2*a*b*c^2*d^5*e^3)/7 + (4*a*b^2*c*d^4*e^4)/35 + (3*a^2*b*c*d^3*e^5)/35 + 3*a^2*b*c*e^8*x^3 + 4*a*b^2*
c*e^8*x^4 + 6*a*b*c^2*e^8*x^5 + (a*b^3*d^2*e^6*x)/5 + (2*a^2*b^2*d*e^7*x)/5 + (3*a*b^3*d*e^7*x^2)/5 + 4*a*c^3*
d^5*e^3*x + 12*a*c^3*d*e^7*x^5 - (343*b*c^3*d^6*e^2*x)/5 + (2*b^3*c*d^4*e^4*x)/3 + (10*b^3*c*d*e^7*x^4)/3 - 28
*b*c^3*d*e^7*x^6 - 4*b*c^3*e^8*x^7*log(d + e*x) + 8*c^4*d*e^7*x^7*log(d + e*x) + (2*a^2*c^2*d^3*e^5*x)/5 + 12*
a*c^3*d^4*e^4*x^2 + 20*a*c^3*d^3*e^5*x^3 + 2*a^2*c^2*d*e^7*x^3 + 20*a*c^3*d^2*e^6*x^4 + 6*b^2*c^2*d^5*e^3*x -
(959*b*c^3*d^5*e^3*x^2)/5 + 2*b^3*c*d^3*e^5*x^2 - (875*b*c^3*d^4*e^4*x^3)/3 + (10*b^3*c*d^2*e^6*x^3)/3 - (770*
b*c^3*d^3*e^5*x^4)/3 - 126*b*c^3*d^2*e^6*x^5 + 18*b^2*c^2*d*e^7*x^5 + 168*c^4*d^6*e^2*x^2*log(d + e*x) + 280*c
^4*d^5*e^3*x^3*log(d + e*x) + 280*c^4*d^4*e^4*x^4*log(d + e*x) + 168*c^4*d^3*e^5*x^5*log(d + e*x) + 56*c^4*d^2
*e^6*x^6*log(d + e*x) + 6*a*b*c^2*d^3*e^5*x^2 + (12*a*b^2*c*d^2*e^6*x^2)/5 + 10*a*b*c^2*d^2*e^6*x^3 - 84*b*c^3
*d^5*e^3*x^2*log(d + e*x) - 140*b*c^3*d^4*e^4*x^3*log(d + e*x) - 140*b*c^3*d^3*e^5*x^4*log(d + e*x) - 84*b*c^3
*d^2*e^6*x^5*log(d + e*x) + 2*a*b*c^2*d^4*e^4*x + (4*a*b^2*c*d^3*e^5*x)/5 + (3*a^2*b*c*d^2*e^6*x)/5 + (9*a^2*b
*c*d*e^7*x^2)/5 + 4*a*b^2*c*d*e^7*x^3 + 10*a*b*c^2*d*e^7*x^4 - 28*b*c^3*d^6*e^2*x*log(d + e*x) - 28*b*c^3*d*e^
7*x^6*log(d + e*x))/(e^9*(d + e*x)^7)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**4/(e*x+d)**8,x)

[Out]

Timed out

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