3.1326 \(\int \frac{(A+B x) (a+c x^2)^3}{(d+e x)^8} \, dx\)

Optimal. Leaf size=327 \[ \frac{c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{4 e^8 (d+e x)^4}-\frac{3 c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{2 e^8 (d+e x)^2}+\frac{c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{3 e^8 (d+e x)^3}+\frac{3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{5 e^8 (d+e x)^5}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{6 e^8 (d+e x)^6}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{7 e^8 (d+e x)^7}+\frac{c^3 (7 B d-A e)}{e^8 (d+e x)}+\frac{B c^3 \log (d+e x)}{e^8} \]

[Out]

((B*d - A*e)*(c*d^2 + a*e^2)^3)/(7*e^8*(d + e*x)^7) - ((c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(6
*e^8*(d + e*x)^6) + (3*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(5*e^8*(d + e*x)^5
) + (c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4)))/(4*e^8*(d + e*x)^4) + (c
^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3))/(3*e^8*(d + e*x)^3) - (3*c^2*(7*B*c*d^2 - 2*A*c*d*e
+ a*B*e^2))/(2*e^8*(d + e*x)^2) + (c^3*(7*B*d - A*e))/(e^8*(d + e*x)) + (B*c^3*Log[d + e*x])/e^8

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Rubi [A]  time = 0.297088, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {772} \[ \frac{c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{4 e^8 (d+e x)^4}-\frac{3 c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{2 e^8 (d+e x)^2}+\frac{c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{3 e^8 (d+e x)^3}+\frac{3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{5 e^8 (d+e x)^5}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{6 e^8 (d+e x)^6}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{7 e^8 (d+e x)^7}+\frac{c^3 (7 B d-A e)}{e^8 (d+e x)}+\frac{B c^3 \log (d+e x)}{e^8} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^8,x]

[Out]

((B*d - A*e)*(c*d^2 + a*e^2)^3)/(7*e^8*(d + e*x)^7) - ((c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(6
*e^8*(d + e*x)^6) + (3*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(5*e^8*(d + e*x)^5
) + (c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4)))/(4*e^8*(d + e*x)^4) + (c
^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3))/(3*e^8*(d + e*x)^3) - (3*c^2*(7*B*c*d^2 - 2*A*c*d*e
+ a*B*e^2))/(2*e^8*(d + e*x)^2) + (c^3*(7*B*d - A*e))/(e^8*(d + e*x)) + (B*c^3*Log[d + e*x])/e^8

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^3}{(d+e x)^8} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^3}{e^7 (d+e x)^8}+\frac{\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{e^7 (d+e x)^7}+\frac{3 c \left (c d^2+a e^2\right ) \left (-7 B c d^3+5 A c d^2 e-3 a B d e^2+a A e^3\right )}{e^7 (d+e x)^6}-\frac{c \left (-35 B c^2 d^4+20 A c^2 d^3 e-30 a B c d^2 e^2+12 a A c d e^3-3 a^2 B e^4\right )}{e^7 (d+e x)^5}+\frac{c^2 \left (-35 B c d^3+15 A c d^2 e-15 a B d e^2+3 a A e^3\right )}{e^7 (d+e x)^4}-\frac{3 c^2 \left (-7 B c d^2+2 A c d e-a B e^2\right )}{e^7 (d+e x)^3}+\frac{c^3 (-7 B d+A e)}{e^7 (d+e x)^2}+\frac{B c^3}{e^7 (d+e x)}\right ) \, dx\\ &=\frac{(B d-A e) \left (c d^2+a e^2\right )^3}{7 e^8 (d+e x)^7}-\frac{\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{6 e^8 (d+e x)^6}+\frac{3 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right )}{5 e^8 (d+e x)^5}+\frac{c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right )}{4 e^8 (d+e x)^4}+\frac{c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right )}{3 e^8 (d+e x)^3}-\frac{3 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right )}{2 e^8 (d+e x)^2}+\frac{c^3 (7 B d-A e)}{e^8 (d+e x)}+\frac{B c^3 \log (d+e x)}{e^8}\\ \end{align*}

Mathematica [A]  time = 0.225227, size = 366, normalized size = 1.12 \[ \frac{-12 A e \left (a^2 c e^4 \left (d^2+7 d e x+21 e^2 x^2\right )+5 a^3 e^6+a c^2 e^2 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )+5 c^3 \left (21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+7 d^5 e x+d^6+21 d e^5 x^5+7 e^6 x^6\right )\right )+B \left (-9 a^2 c e^4 \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )-10 a^3 e^6 (d+7 e x)-30 a c^2 e^2 \left (21 d^3 e^2 x^2+35 d^2 e^3 x^3+7 d^4 e x+d^5+35 d e^4 x^4+21 e^5 x^5\right )+c^3 d \left (20139 d^4 e^2 x^2+30625 d^3 e^3 x^3+26950 d^2 e^4 x^4+7203 d^5 e x+1089 d^6+13230 d e^5 x^5+2940 e^6 x^6\right )\right )+420 B c^3 (d+e x)^7 \log (d+e x)}{420 e^8 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^8,x]

[Out]

(-12*A*e*(5*a^3*e^6 + a^2*c*e^4*(d^2 + 7*d*e*x + 21*e^2*x^2) + a*c^2*e^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 3
5*d*e^3*x^3 + 35*e^4*x^4) + 5*c^3*(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*(-10*a^3*e^6*(d + 7*e*x) - 9*a^2*c*e^4*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3)
- 30*a*c^2*e^2*(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) + c^3*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*B*c^3*(d + e*x)^7*Log[d + e*x])/(420*e^8*(d + e*x)^7)

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Maple [B]  time = 0.009, size = 662, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^8,x)

[Out]

3/7/e^6/(e*x+d)^7*B*a*c^2*d^5+5*c^2/e^6/(e*x+d)^3*a*B*d+3/7/e^4/(e*x+d)^7*B*a^2*c*d^3-3/7/e^5/(e*x+d)^7*A*d^4*
a*c^2-3/7/e^3/(e*x+d)^7*A*d^2*a^2*c-15/2*c^2/e^6/(e*x+d)^4*B*a*d^2+2/e^5/(e*x+d)^6*A*d^3*a*c^2-3/2/e^4/(e*x+d)
^6*B*a^2*c*d^2-5/2/e^6/(e*x+d)^6*B*a*c^2*d^4+9/5*c/e^4/(e*x+d)^5*B*a^2*d+6*c^2/e^6/(e*x+d)^5*a*B*d^3-18/5*c^2/
e^5/(e*x+d)^5*A*d^2*a+3*c^2/e^5/(e*x+d)^4*A*d*a+1/e^3/(e*x+d)^6*A*a^2*c*d-1/6/e^2/(e*x+d)^6*B*a^3-c^3/e^7/(e*x
+d)*A-1/7/e/(e*x+d)^7*A*a^3+B*c^3*ln(e*x+d)/e^8+7*c^3/e^8/(e*x+d)*B*d+3*c^3/e^7/(e*x+d)^2*A*d-3/2*c^2/e^6/(e*x
+d)^2*a*B-5*c^3/e^7/(e*x+d)^3*A*d^2+35/3*c^3/e^8/(e*x+d)^3*B*d^3-1/7/e^7/(e*x+d)^7*A*d^6*c^3+1/7/e^2/(e*x+d)^7
*B*a^3*d+1/7/e^8/(e*x+d)^7*B*c^3*d^7+1/e^7/(e*x+d)^6*A*d^5*c^3+5*c^3/e^7/(e*x+d)^4*A*d^3-3/4*c/e^4/(e*x+d)^4*B
*a^2-35/4*c^3/e^8/(e*x+d)^4*B*d^4-21/2*c^3/e^8/(e*x+d)^2*B*d^2-c^2/e^5/(e*x+d)^3*a*A-3*c^3/e^7/(e*x+d)^5*A*d^4
+21/5*c^3/e^8/(e*x+d)^5*B*d^5-7/6/e^8/(e*x+d)^6*B*c^3*d^6-3/5*c/e^3/(e*x+d)^5*A*a^2

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Maxima [A]  time = 1.156, size = 711, normalized size = 2.17 \begin{align*} \frac{1089 \, B c^{3} d^{7} - 60 \, A c^{3} d^{6} e - 30 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} - 9 \, B a^{2} c d^{3} e^{4} - 12 \, A a^{2} c d^{2} e^{5} - 10 \, B a^{3} d e^{6} - 60 \, A a^{3} e^{7} + 420 \,{\left (7 \, B c^{3} d e^{6} - A c^{3} e^{7}\right )} x^{6} + 630 \,{\left (21 \, B c^{3} d^{2} e^{5} - 2 \, A c^{3} d e^{6} - B a c^{2} e^{7}\right )} x^{5} + 70 \,{\left (385 \, B c^{3} d^{3} e^{4} - 30 \, A c^{3} d^{2} e^{5} - 15 \, B a c^{2} d e^{6} - 6 \, A a c^{2} e^{7}\right )} x^{4} + 35 \,{\left (875 \, B c^{3} d^{4} e^{3} - 60 \, A c^{3} d^{3} e^{4} - 30 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} - 9 \, B a^{2} c e^{7}\right )} x^{3} + 21 \,{\left (959 \, B c^{3} d^{5} e^{2} - 60 \, A c^{3} d^{4} e^{3} - 30 \, B a c^{2} d^{3} e^{4} - 12 \, A a c^{2} d^{2} e^{5} - 9 \, B a^{2} c d e^{6} - 12 \, A a^{2} c e^{7}\right )} x^{2} + 7 \,{\left (1029 \, B c^{3} d^{6} e - 60 \, A c^{3} d^{5} e^{2} - 30 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} - 9 \, B a^{2} c d^{2} e^{5} - 12 \, A a^{2} c d e^{6} - 10 \, B a^{3} e^{7}\right )} x}{420 \,{\left (e^{15} x^{7} + 7 \, d e^{14} x^{6} + 21 \, d^{2} e^{13} x^{5} + 35 \, d^{3} e^{12} x^{4} + 35 \, d^{4} e^{11} x^{3} + 21 \, d^{5} e^{10} x^{2} + 7 \, d^{6} e^{9} x + d^{7} e^{8}\right )}} + \frac{B c^{3} \log \left (e x + d\right )}{e^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^8,x, algorithm="maxima")

[Out]

1/420*(1089*B*c^3*d^7 - 60*A*c^3*d^6*e - 30*B*a*c^2*d^5*e^2 - 12*A*a*c^2*d^4*e^3 - 9*B*a^2*c*d^3*e^4 - 12*A*a^
2*c*d^2*e^5 - 10*B*a^3*d*e^6 - 60*A*a^3*e^7 + 420*(7*B*c^3*d*e^6 - A*c^3*e^7)*x^6 + 630*(21*B*c^3*d^2*e^5 - 2*
A*c^3*d*e^6 - B*a*c^2*e^7)*x^5 + 70*(385*B*c^3*d^3*e^4 - 30*A*c^3*d^2*e^5 - 15*B*a*c^2*d*e^6 - 6*A*a*c^2*e^7)*
x^4 + 35*(875*B*c^3*d^4*e^3 - 60*A*c^3*d^3*e^4 - 30*B*a*c^2*d^2*e^5 - 12*A*a*c^2*d*e^6 - 9*B*a^2*c*e^7)*x^3 +
21*(959*B*c^3*d^5*e^2 - 60*A*c^3*d^4*e^3 - 30*B*a*c^2*d^3*e^4 - 12*A*a*c^2*d^2*e^5 - 9*B*a^2*c*d*e^6 - 12*A*a^
2*c*e^7)*x^2 + 7*(1029*B*c^3*d^6*e - 60*A*c^3*d^5*e^2 - 30*B*a*c^2*d^4*e^3 - 12*A*a*c^2*d^3*e^4 - 9*B*a^2*c*d^
2*e^5 - 12*A*a^2*c*d*e^6 - 10*B*a^3*e^7)*x)/(e^15*x^7 + 7*d*e^14*x^6 + 21*d^2*e^13*x^5 + 35*d^3*e^12*x^4 + 35*
d^4*e^11*x^3 + 21*d^5*e^10*x^2 + 7*d^6*e^9*x + d^7*e^8) + B*c^3*log(e*x + d)/e^8

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Fricas [A]  time = 1.87546, size = 1355, normalized size = 4.14 \begin{align*} \frac{1089 \, B c^{3} d^{7} - 60 \, A c^{3} d^{6} e - 30 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} - 9 \, B a^{2} c d^{3} e^{4} - 12 \, A a^{2} c d^{2} e^{5} - 10 \, B a^{3} d e^{6} - 60 \, A a^{3} e^{7} + 420 \,{\left (7 \, B c^{3} d e^{6} - A c^{3} e^{7}\right )} x^{6} + 630 \,{\left (21 \, B c^{3} d^{2} e^{5} - 2 \, A c^{3} d e^{6} - B a c^{2} e^{7}\right )} x^{5} + 70 \,{\left (385 \, B c^{3} d^{3} e^{4} - 30 \, A c^{3} d^{2} e^{5} - 15 \, B a c^{2} d e^{6} - 6 \, A a c^{2} e^{7}\right )} x^{4} + 35 \,{\left (875 \, B c^{3} d^{4} e^{3} - 60 \, A c^{3} d^{3} e^{4} - 30 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} - 9 \, B a^{2} c e^{7}\right )} x^{3} + 21 \,{\left (959 \, B c^{3} d^{5} e^{2} - 60 \, A c^{3} d^{4} e^{3} - 30 \, B a c^{2} d^{3} e^{4} - 12 \, A a c^{2} d^{2} e^{5} - 9 \, B a^{2} c d e^{6} - 12 \, A a^{2} c e^{7}\right )} x^{2} + 7 \,{\left (1029 \, B c^{3} d^{6} e - 60 \, A c^{3} d^{5} e^{2} - 30 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} - 9 \, B a^{2} c d^{2} e^{5} - 12 \, A a^{2} c d e^{6} - 10 \, B a^{3} e^{7}\right )} x + 420 \,{\left (B c^{3} e^{7} x^{7} + 7 \, B c^{3} d e^{6} x^{6} + 21 \, B c^{3} d^{2} e^{5} x^{5} + 35 \, B c^{3} d^{3} e^{4} x^{4} + 35 \, B c^{3} d^{4} e^{3} x^{3} + 21 \, B c^{3} d^{5} e^{2} x^{2} + 7 \, B c^{3} d^{6} e x + B c^{3} d^{7}\right )} \log \left (e x + d\right )}{420 \,{\left (e^{15} x^{7} + 7 \, d e^{14} x^{6} + 21 \, d^{2} e^{13} x^{5} + 35 \, d^{3} e^{12} x^{4} + 35 \, d^{4} e^{11} x^{3} + 21 \, d^{5} e^{10} x^{2} + 7 \, d^{6} e^{9} x + d^{7} e^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^8,x, algorithm="fricas")

[Out]

1/420*(1089*B*c^3*d^7 - 60*A*c^3*d^6*e - 30*B*a*c^2*d^5*e^2 - 12*A*a*c^2*d^4*e^3 - 9*B*a^2*c*d^3*e^4 - 12*A*a^
2*c*d^2*e^5 - 10*B*a^3*d*e^6 - 60*A*a^3*e^7 + 420*(7*B*c^3*d*e^6 - A*c^3*e^7)*x^6 + 630*(21*B*c^3*d^2*e^5 - 2*
A*c^3*d*e^6 - B*a*c^2*e^7)*x^5 + 70*(385*B*c^3*d^3*e^4 - 30*A*c^3*d^2*e^5 - 15*B*a*c^2*d*e^6 - 6*A*a*c^2*e^7)*
x^4 + 35*(875*B*c^3*d^4*e^3 - 60*A*c^3*d^3*e^4 - 30*B*a*c^2*d^2*e^5 - 12*A*a*c^2*d*e^6 - 9*B*a^2*c*e^7)*x^3 +
21*(959*B*c^3*d^5*e^2 - 60*A*c^3*d^4*e^3 - 30*B*a*c^2*d^3*e^4 - 12*A*a*c^2*d^2*e^5 - 9*B*a^2*c*d*e^6 - 12*A*a^
2*c*e^7)*x^2 + 7*(1029*B*c^3*d^6*e - 60*A*c^3*d^5*e^2 - 30*B*a*c^2*d^4*e^3 - 12*A*a*c^2*d^3*e^4 - 9*B*a^2*c*d^
2*e^5 - 12*A*a^2*c*d*e^6 - 10*B*a^3*e^7)*x + 420*(B*c^3*e^7*x^7 + 7*B*c^3*d*e^6*x^6 + 21*B*c^3*d^2*e^5*x^5 + 3
5*B*c^3*d^3*e^4*x^4 + 35*B*c^3*d^4*e^3*x^3 + 21*B*c^3*d^5*e^2*x^2 + 7*B*c^3*d^6*e*x + B*c^3*d^7)*log(e*x + d))
/(e^15*x^7 + 7*d*e^14*x^6 + 21*d^2*e^13*x^5 + 35*d^3*e^12*x^4 + 35*d^4*e^11*x^3 + 21*d^5*e^10*x^2 + 7*d^6*e^9*
x + d^7*e^8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**8,x)

[Out]

Timed out

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Giac [A]  time = 1.18301, size = 582, normalized size = 1.78 \begin{align*} B c^{3} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{{\left (420 \,{\left (7 \, B c^{3} d e^{5} - A c^{3} e^{6}\right )} x^{6} + 630 \,{\left (21 \, B c^{3} d^{2} e^{4} - 2 \, A c^{3} d e^{5} - B a c^{2} e^{6}\right )} x^{5} + 70 \,{\left (385 \, B c^{3} d^{3} e^{3} - 30 \, A c^{3} d^{2} e^{4} - 15 \, B a c^{2} d e^{5} - 6 \, A a c^{2} e^{6}\right )} x^{4} + 35 \,{\left (875 \, B c^{3} d^{4} e^{2} - 60 \, A c^{3} d^{3} e^{3} - 30 \, B a c^{2} d^{2} e^{4} - 12 \, A a c^{2} d e^{5} - 9 \, B a^{2} c e^{6}\right )} x^{3} + 21 \,{\left (959 \, B c^{3} d^{5} e - 60 \, A c^{3} d^{4} e^{2} - 30 \, B a c^{2} d^{3} e^{3} - 12 \, A a c^{2} d^{2} e^{4} - 9 \, B a^{2} c d e^{5} - 12 \, A a^{2} c e^{6}\right )} x^{2} + 7 \,{\left (1029 \, B c^{3} d^{6} - 60 \, A c^{3} d^{5} e - 30 \, B a c^{2} d^{4} e^{2} - 12 \, A a c^{2} d^{3} e^{3} - 9 \, B a^{2} c d^{2} e^{4} - 12 \, A a^{2} c d e^{5} - 10 \, B a^{3} e^{6}\right )} x +{\left (1089 \, B c^{3} d^{7} - 60 \, A c^{3} d^{6} e - 30 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} - 9 \, B a^{2} c d^{3} e^{4} - 12 \, A a^{2} c d^{2} e^{5} - 10 \, B a^{3} d e^{6} - 60 \, A a^{3} e^{7}\right )} e^{\left (-1\right )}\right )} e^{\left (-7\right )}}{420 \,{\left (x e + d\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^8,x, algorithm="giac")

[Out]

B*c^3*e^(-8)*log(abs(x*e + d)) + 1/420*(420*(7*B*c^3*d*e^5 - A*c^3*e^6)*x^6 + 630*(21*B*c^3*d^2*e^4 - 2*A*c^3*
d*e^5 - B*a*c^2*e^6)*x^5 + 70*(385*B*c^3*d^3*e^3 - 30*A*c^3*d^2*e^4 - 15*B*a*c^2*d*e^5 - 6*A*a*c^2*e^6)*x^4 +
35*(875*B*c^3*d^4*e^2 - 60*A*c^3*d^3*e^3 - 30*B*a*c^2*d^2*e^4 - 12*A*a*c^2*d*e^5 - 9*B*a^2*c*e^6)*x^3 + 21*(95
9*B*c^3*d^5*e - 60*A*c^3*d^4*e^2 - 30*B*a*c^2*d^3*e^3 - 12*A*a*c^2*d^2*e^4 - 9*B*a^2*c*d*e^5 - 12*A*a^2*c*e^6)
*x^2 + 7*(1029*B*c^3*d^6 - 60*A*c^3*d^5*e - 30*B*a*c^2*d^4*e^2 - 12*A*a*c^2*d^3*e^3 - 9*B*a^2*c*d^2*e^4 - 12*A
*a^2*c*d*e^5 - 10*B*a^3*e^6)*x + (1089*B*c^3*d^7 - 60*A*c^3*d^6*e - 30*B*a*c^2*d^5*e^2 - 12*A*a*c^2*d^4*e^3 -
9*B*a^2*c*d^3*e^4 - 12*A*a^2*c*d^2*e^5 - 10*B*a^3*d*e^6 - 60*A*a^3*e^7)*e^(-1))*e^(-7)/(x*e + d)^7