3.1324 \(\int \frac{(A+B x) (a+c x^2)^3}{(d+e x)^6} \, dx\)

Optimal. Leaf size=313 \[ \frac{c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{2 e^8 (d+e x)^2}+\frac{c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8 (d+e x)}+\frac{3 c^2 \log (d+e x) \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8}+\frac{c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 (d+e x)^3}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{4 e^8 (d+e x)^4}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{5 e^8 (d+e x)^5}-\frac{c^3 x (6 B d-A e)}{e^7}+\frac{B c^3 x^2}{2 e^6} \]

[Out]

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

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Rubi [A]  time = 0.344875, antiderivative size = 313, 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} \[ \frac{c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{2 e^8 (d+e x)^2}+\frac{c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8 (d+e x)}+\frac{3 c^2 \log (d+e x) \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8}+\frac{c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 (d+e x)^3}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{4 e^8 (d+e x)^4}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{5 e^8 (d+e x)^5}-\frac{c^3 x (6 B d-A e)}{e^7}+\frac{B c^3 x^2}{2 e^6} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^6,x]

[Out]

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

Mathematica [A]  time = 0.211431, size = 388, normalized size = 1.24 \[ \frac{-2 A e \left (a^2 c e^4 \left (d^2+5 d e x+10 e^2 x^2\right )+2 a^3 e^6+6 a c^2 e^2 \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )+c^3 \left (600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4+375 d^5 e x+87 d^6-50 d e^5 x^5-10 e^6 x^6\right )\right )+B \left (-3 a^2 c e^4 \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )-a^3 e^6 (d+5 e x)+a c^2 d e^2 \left (1100 d^2 e^2 x^2+625 d^3 e x+137 d^4+900 d e^3 x^3+300 e^4 x^4\right )+c^3 \left (2700 d^5 e^2 x^2+1300 d^4 e^3 x^3-400 d^3 e^4 x^4-500 d^2 e^5 x^5+1875 d^6 e x+459 d^7-70 d e^6 x^6+10 e^7 x^7\right )\right )+60 c^2 (d+e x)^5 \log (d+e x) \left (a B e^2-2 A c d e+7 B c d^2\right )}{20 e^8 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^6,x]

[Out]

(-2*A*e*(2*a^3*e^6 + a^2*c*e^4*(d^2 + 5*d*e*x + 10*e^2*x^2) + 6*a*c^2*e^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 +
10*d*e^3*x^3 + 5*e^4*x^4) + c^3*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 5
0*d*e^5*x^5 - 10*e^6*x^6)) + B*(-(a^3*e^6*(d + 5*e*x)) - 3*a^2*c*e^4*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*
x^3) + a*c^2*d*e^2*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4) + c^3*(459*d^7 + 1
875*d^6*e*x + 2700*d^5*e^2*x^2 + 1300*d^4*e^3*x^3 - 400*d^3*e^4*x^4 - 500*d^2*e^5*x^5 - 70*d*e^6*x^6 + 10*e^7*
x^7)) + 60*c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^5*Log[d + e*x])/(20*e^8*(d + e*x)^5)

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Maple [B]  time = 0.013, size = 646, normalized size = 2.1 \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)^6,x)

[Out]

1/2*B*c^3*x^2/e^6+15*c^2/e^6/(e*x+d)*a*B*d+3/e^5/(e*x+d)^4*A*a*c^2*d^3+3/2/e^3/(e*x+d)^4*A*a^2*c*d-15*c^2/e^6/
(e*x+d)^2*B*a*d^2-6*c^2/e^5/(e*x+d)^3*A*d^2*a+3*c/e^4/(e*x+d)^3*B*a^2*d+10*c^2/e^6/(e*x+d)^3*a*B*d^3+6*c^2/e^5
/(e*x+d)^2*A*d*a+3/5/e^4/(e*x+d)^5*B*d^3*a^2*c+3/5/e^6/(e*x+d)^5*B*d^5*a*c^2-9/4/e^4/(e*x+d)^4*B*a^2*c*d^2-15/
4/e^6/(e*x+d)^4*B*a*c^2*d^4-3/5/e^3/(e*x+d)^5*A*d^2*a^2*c-3/5/e^5/(e*x+d)^5*A*d^4*a*c^2-1/5/e/(e*x+d)^5*A*a^3-
1/4/e^2/(e*x+d)^4*B*a^3+c^3/e^6*A*x-3*c^2/e^5/(e*x+d)*a*A-15*c^3/e^7/(e*x+d)*A*d^2+35*c^3/e^8/(e*x+d)*B*d^3+10
*c^3/e^7/(e*x+d)^2*A*d^3-3/2*c/e^4/(e*x+d)^2*B*a^2-35/2*c^3/e^8/(e*x+d)^2*B*d^4-6*c^3/e^7*B*d*x-1/5/e^7/(e*x+d
)^5*A*c^3*d^6-6*c^3/e^7*ln(e*x+d)*A*d+3*c^2/e^6*ln(e*x+d)*a*B+21*c^3/e^8*ln(e*x+d)*B*d^2+3/2/e^7/(e*x+d)^4*A*c
^3*d^5-c/e^3/(e*x+d)^3*A*a^2-5*c^3/e^7/(e*x+d)^3*A*d^4+7*c^3/e^8/(e*x+d)^3*B*d^5-7/4/e^8/(e*x+d)^4*B*c^3*d^6+1
/5/e^8/(e*x+d)^5*B*c^3*d^7+1/5/e^2/(e*x+d)^5*B*d*a^3

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Maxima [A]  time = 1.17171, size = 674, normalized size = 2.15 \begin{align*} \frac{459 \, B c^{3} d^{7} - 174 \, A c^{3} d^{6} e + 137 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} - 3 \, B a^{2} c d^{3} e^{4} - 2 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 4 \, A a^{3} e^{7} + 20 \,{\left (35 \, B c^{3} d^{3} e^{4} - 15 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} + 10 \,{\left (245 \, B c^{3} d^{4} e^{3} - 100 \, A c^{3} d^{3} e^{4} + 90 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} - 3 \, B a^{2} c e^{7}\right )} x^{3} + 10 \,{\left (329 \, B c^{3} d^{5} e^{2} - 130 \, A c^{3} d^{4} e^{3} + 110 \, B a c^{2} d^{3} e^{4} - 12 \, A a c^{2} d^{2} e^{5} - 3 \, B a^{2} c d e^{6} - 2 \, A a^{2} c e^{7}\right )} x^{2} + 5 \,{\left (399 \, B c^{3} d^{6} e - 154 \, A c^{3} d^{5} e^{2} + 125 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} - 3 \, B a^{2} c d^{2} e^{5} - 2 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} x}{20 \,{\left (e^{13} x^{5} + 5 \, d e^{12} x^{4} + 10 \, d^{2} e^{11} x^{3} + 10 \, d^{3} e^{10} x^{2} + 5 \, d^{4} e^{9} x + d^{5} e^{8}\right )}} + \frac{B c^{3} e x^{2} - 2 \,{\left (6 \, B c^{3} d - A c^{3} e\right )} x}{2 \, e^{7}} + \frac{3 \,{\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )} \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)^6,x, algorithm="maxima")

[Out]

1/20*(459*B*c^3*d^7 - 174*A*c^3*d^6*e + 137*B*a*c^2*d^5*e^2 - 12*A*a*c^2*d^4*e^3 - 3*B*a^2*c*d^3*e^4 - 2*A*a^2
*c*d^2*e^5 - B*a^3*d*e^6 - 4*A*a^3*e^7 + 20*(35*B*c^3*d^3*e^4 - 15*A*c^3*d^2*e^5 + 15*B*a*c^2*d*e^6 - 3*A*a*c^
2*e^7)*x^4 + 10*(245*B*c^3*d^4*e^3 - 100*A*c^3*d^3*e^4 + 90*B*a*c^2*d^2*e^5 - 12*A*a*c^2*d*e^6 - 3*B*a^2*c*e^7
)*x^3 + 10*(329*B*c^3*d^5*e^2 - 130*A*c^3*d^4*e^3 + 110*B*a*c^2*d^3*e^4 - 12*A*a*c^2*d^2*e^5 - 3*B*a^2*c*d*e^6
- 2*A*a^2*c*e^7)*x^2 + 5*(399*B*c^3*d^6*e - 154*A*c^3*d^5*e^2 + 125*B*a*c^2*d^4*e^3 - 12*A*a*c^2*d^3*e^4 - 3*
B*a^2*c*d^2*e^5 - 2*A*a^2*c*d*e^6 - B*a^3*e^7)*x)/(e^13*x^5 + 5*d*e^12*x^4 + 10*d^2*e^11*x^3 + 10*d^3*e^10*x^2
+ 5*d^4*e^9*x + d^5*e^8) + 1/2*(B*c^3*e*x^2 - 2*(6*B*c^3*d - A*c^3*e)*x)/e^7 + 3*(7*B*c^3*d^2 - 2*A*c^3*d*e +
B*a*c^2*e^2)*log(e*x + d)/e^8

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Fricas [B]  time = 1.91983, size = 1551, normalized size = 4.96 \begin{align*} \frac{10 \, B c^{3} e^{7} x^{7} + 459 \, B c^{3} d^{7} - 174 \, A c^{3} d^{6} e + 137 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} - 3 \, B a^{2} c d^{3} e^{4} - 2 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 4 \, A a^{3} e^{7} - 10 \,{\left (7 \, B c^{3} d e^{6} - 2 \, A c^{3} e^{7}\right )} x^{6} - 100 \,{\left (5 \, B c^{3} d^{2} e^{5} - A c^{3} d e^{6}\right )} x^{5} - 20 \,{\left (20 \, B c^{3} d^{3} e^{4} + 5 \, A c^{3} d^{2} e^{5} - 15 \, B a c^{2} d e^{6} + 3 \, A a c^{2} e^{7}\right )} x^{4} + 10 \,{\left (130 \, B c^{3} d^{4} e^{3} - 80 \, A c^{3} d^{3} e^{4} + 90 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} - 3 \, B a^{2} c e^{7}\right )} x^{3} + 10 \,{\left (270 \, B c^{3} d^{5} e^{2} - 120 \, A c^{3} d^{4} e^{3} + 110 \, B a c^{2} d^{3} e^{4} - 12 \, A a c^{2} d^{2} e^{5} - 3 \, B a^{2} c d e^{6} - 2 \, A a^{2} c e^{7}\right )} x^{2} + 5 \,{\left (375 \, B c^{3} d^{6} e - 150 \, A c^{3} d^{5} e^{2} + 125 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} - 3 \, B a^{2} c d^{2} e^{5} - 2 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} x + 60 \,{\left (7 \, B c^{3} d^{7} - 2 \, A c^{3} d^{6} e + B a c^{2} d^{5} e^{2} +{\left (7 \, B c^{3} d^{2} e^{5} - 2 \, A c^{3} d e^{6} + B a c^{2} e^{7}\right )} x^{5} + 5 \,{\left (7 \, B c^{3} d^{3} e^{4} - 2 \, A c^{3} d^{2} e^{5} + B a c^{2} d e^{6}\right )} x^{4} + 10 \,{\left (7 \, B c^{3} d^{4} e^{3} - 2 \, A c^{3} d^{3} e^{4} + B a c^{2} d^{2} e^{5}\right )} x^{3} + 10 \,{\left (7 \, B c^{3} d^{5} e^{2} - 2 \, A c^{3} d^{4} e^{3} + B a c^{2} d^{3} e^{4}\right )} x^{2} + 5 \,{\left (7 \, B c^{3} d^{6} e - 2 \, A c^{3} d^{5} e^{2} + B a c^{2} d^{4} e^{3}\right )} x\right )} \log \left (e x + d\right )}{20 \,{\left (e^{13} x^{5} + 5 \, d e^{12} x^{4} + 10 \, d^{2} e^{11} x^{3} + 10 \, d^{3} e^{10} x^{2} + 5 \, d^{4} e^{9} x + d^{5} 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)^6,x, algorithm="fricas")

[Out]

1/20*(10*B*c^3*e^7*x^7 + 459*B*c^3*d^7 - 174*A*c^3*d^6*e + 137*B*a*c^2*d^5*e^2 - 12*A*a*c^2*d^4*e^3 - 3*B*a^2*
c*d^3*e^4 - 2*A*a^2*c*d^2*e^5 - B*a^3*d*e^6 - 4*A*a^3*e^7 - 10*(7*B*c^3*d*e^6 - 2*A*c^3*e^7)*x^6 - 100*(5*B*c^
3*d^2*e^5 - A*c^3*d*e^6)*x^5 - 20*(20*B*c^3*d^3*e^4 + 5*A*c^3*d^2*e^5 - 15*B*a*c^2*d*e^6 + 3*A*a*c^2*e^7)*x^4
+ 10*(130*B*c^3*d^4*e^3 - 80*A*c^3*d^3*e^4 + 90*B*a*c^2*d^2*e^5 - 12*A*a*c^2*d*e^6 - 3*B*a^2*c*e^7)*x^3 + 10*(
270*B*c^3*d^5*e^2 - 120*A*c^3*d^4*e^3 + 110*B*a*c^2*d^3*e^4 - 12*A*a*c^2*d^2*e^5 - 3*B*a^2*c*d*e^6 - 2*A*a^2*c
*e^7)*x^2 + 5*(375*B*c^3*d^6*e - 150*A*c^3*d^5*e^2 + 125*B*a*c^2*d^4*e^3 - 12*A*a*c^2*d^3*e^4 - 3*B*a^2*c*d^2*
e^5 - 2*A*a^2*c*d*e^6 - B*a^3*e^7)*x + 60*(7*B*c^3*d^7 - 2*A*c^3*d^6*e + B*a*c^2*d^5*e^2 + (7*B*c^3*d^2*e^5 -
2*A*c^3*d*e^6 + B*a*c^2*e^7)*x^5 + 5*(7*B*c^3*d^3*e^4 - 2*A*c^3*d^2*e^5 + B*a*c^2*d*e^6)*x^4 + 10*(7*B*c^3*d^4
*e^3 - 2*A*c^3*d^3*e^4 + B*a*c^2*d^2*e^5)*x^3 + 10*(7*B*c^3*d^5*e^2 - 2*A*c^3*d^4*e^3 + B*a*c^2*d^3*e^4)*x^2 +
5*(7*B*c^3*d^6*e - 2*A*c^3*d^5*e^2 + B*a*c^2*d^4*e^3)*x)*log(e*x + d))/(e^13*x^5 + 5*d*e^12*x^4 + 10*d^2*e^11
*x^3 + 10*d^3*e^10*x^2 + 5*d^4*e^9*x + d^5*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)**6,x)

[Out]

Timed out

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Giac [A]  time = 1.16716, size = 579, normalized size = 1.85 \begin{align*} 3 \,{\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (B c^{3} x^{2} e^{6} - 12 \, B c^{3} d x e^{5} + 2 \, A c^{3} x e^{6}\right )} e^{\left (-12\right )} + \frac{{\left (459 \, B c^{3} d^{7} - 174 \, A c^{3} d^{6} e + 137 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} - 3 \, B a^{2} c d^{3} e^{4} - 2 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} + 20 \,{\left (35 \, B c^{3} d^{3} e^{4} - 15 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} - 3 \, A a c^{2} e^{7}\right )} x^{4} - 4 \, A a^{3} e^{7} + 10 \,{\left (245 \, B c^{3} d^{4} e^{3} - 100 \, A c^{3} d^{3} e^{4} + 90 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} - 3 \, B a^{2} c e^{7}\right )} x^{3} + 10 \,{\left (329 \, B c^{3} d^{5} e^{2} - 130 \, A c^{3} d^{4} e^{3} + 110 \, B a c^{2} d^{3} e^{4} - 12 \, A a c^{2} d^{2} e^{5} - 3 \, B a^{2} c d e^{6} - 2 \, A a^{2} c e^{7}\right )} x^{2} + 5 \,{\left (399 \, B c^{3} d^{6} e - 154 \, A c^{3} d^{5} e^{2} + 125 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} - 3 \, B a^{2} c d^{2} e^{5} - 2 \, A a^{2} c d e^{6} - B a^{3} e^{7}\right )} x\right )} e^{\left (-8\right )}}{20 \,{\left (x e + d\right )}^{5}} \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)^6,x, algorithm="giac")

[Out]

3*(7*B*c^3*d^2 - 2*A*c^3*d*e + B*a*c^2*e^2)*e^(-8)*log(abs(x*e + d)) + 1/2*(B*c^3*x^2*e^6 - 12*B*c^3*d*x*e^5 +
2*A*c^3*x*e^6)*e^(-12) + 1/20*(459*B*c^3*d^7 - 174*A*c^3*d^6*e + 137*B*a*c^2*d^5*e^2 - 12*A*a*c^2*d^4*e^3 - 3
*B*a^2*c*d^3*e^4 - 2*A*a^2*c*d^2*e^5 - B*a^3*d*e^6 + 20*(35*B*c^3*d^3*e^4 - 15*A*c^3*d^2*e^5 + 15*B*a*c^2*d*e^
6 - 3*A*a*c^2*e^7)*x^4 - 4*A*a^3*e^7 + 10*(245*B*c^3*d^4*e^3 - 100*A*c^3*d^3*e^4 + 90*B*a*c^2*d^2*e^5 - 12*A*a
*c^2*d*e^6 - 3*B*a^2*c*e^7)*x^3 + 10*(329*B*c^3*d^5*e^2 - 130*A*c^3*d^4*e^3 + 110*B*a*c^2*d^3*e^4 - 12*A*a*c^2
*d^2*e^5 - 3*B*a^2*c*d*e^6 - 2*A*a^2*c*e^7)*x^2 + 5*(399*B*c^3*d^6*e - 154*A*c^3*d^5*e^2 + 125*B*a*c^2*d^4*e^3
- 12*A*a*c^2*d^3*e^4 - 3*B*a^2*c*d^2*e^5 - 2*A*a^2*c*d*e^6 - B*a^3*e^7)*x)*e^(-8)/(x*e + d)^5