3.1134 \(\int \frac{(A+B x) (b x+c x^2)^3}{(d+e x)^2} \, dx\)

Optimal. Leaf size=287 \[ -\frac{c^2 x^5 (-A c e-3 b B e+2 B c d)}{5 e^3}+\frac{d^3 (B d-A e) (c d-b e)^3}{e^8 (d+e x)}+\frac{d^2 (c d-b e)^2 \log (d+e x) (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{e^8}-\frac{c x^4 \left (A c e (2 c d-3 b e)-3 B (c d-b e)^2\right )}{4 e^4}-\frac{x^3 (c d-b e)^2 (-3 A c e-b B e+4 B c d)}{3 e^5}+\frac{x^2 (c d-b e)^2 (B d (5 c d-2 b e)-A e (4 c d-b e))}{2 e^6}+\frac{d x (c d-b e)^2 (A e (5 c d-2 b e)-3 B d (2 c d-b e))}{e^7}+\frac{B c^3 x^6}{6 e^2} \]

[Out]

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

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Rubi [A]  time = 0.545404, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ -\frac{c^2 x^5 (-A c e-3 b B e+2 B c d)}{5 e^3}+\frac{d^3 (B d-A e) (c d-b e)^3}{e^8 (d+e x)}+\frac{d^2 (c d-b e)^2 \log (d+e x) (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{e^8}-\frac{c x^4 \left (A c e (2 c d-3 b e)-3 B (c d-b e)^2\right )}{4 e^4}-\frac{x^3 (c d-b e)^2 (-3 A c e-b B e+4 B c d)}{3 e^5}+\frac{x^2 (c d-b e)^2 (B d (5 c d-2 b e)-A e (4 c d-b e))}{2 e^6}+\frac{d x (c d-b e)^2 (A e (5 c d-2 b e)-3 B d (2 c d-b e))}{e^7}+\frac{B c^3 x^6}{6 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.133692, size = 274, normalized size = 0.95 \[ \frac{12 c^2 e^5 x^5 (A c e+3 b B e-2 B c d)+\frac{60 d^3 (B d-A e) (c d-b e)^3}{d+e x}+60 d^2 (c d-b e)^2 \log (d+e x) (3 A e (b e-2 c d)+B d (7 c d-4 b e))-15 c e^4 x^4 \left (A c e (2 c d-3 b e)-3 B (c d-b e)^2\right )+20 e^3 x^3 (c d-b e)^2 (3 A c e+b B e-4 B c d)+30 e^2 x^2 (c d-b e)^2 (A e (b e-4 c d)+B d (5 c d-2 b e))-60 d e x (c d-b e)^2 (A e (2 b e-5 c d)+3 B d (2 c d-b e))+10 B c^3 e^6 x^6}{60 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-60*d*e*(c*d - b*e)^2*(3*B*d*(2*c*d - b*e) + A*e*(-5*c*d + 2*b*e))*x + 30*e^2*(c*d - b*e)^2*(B*d*(5*c*d - 2*b
*e) + A*e*(-4*c*d + b*e))*x^2 + 20*e^3*(c*d - b*e)^2*(-4*B*c*d + b*B*e + 3*A*c*e)*x^3 - 15*c*e^4*(A*c*e*(2*c*d
- 3*b*e) - 3*B*(c*d - b*e)^2)*x^4 + 12*c^2*e^5*(-2*B*c*d + 3*b*B*e + A*c*e)*x^5 + 10*B*c^3*e^6*x^6 + (60*d^3*
(B*d - A*e)*(c*d - b*e)^3)/(d + e*x) + 60*d^2*(c*d - b*e)^2*(B*d*(7*c*d - 4*b*e) + 3*A*e*(-2*c*d + b*e))*Log[d
+ e*x])/(60*e^8)

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Maple [B]  time = 0.013, size = 742, normalized size = 2.6 \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+b*x)^3/(e*x+d)^2,x)

[Out]

1/6*B*c^3*x^6/e^2-12*d^3/e^5*ln(e*x+d)*A*b^2*c+15*d^4/e^6*ln(e*x+d)*A*b*c^2+15*d^4/e^6*ln(e*x+d)*B*b^2*c-18*d^
5/e^7*ln(e*x+d)*B*b*c^2-3*d^4/e^5/(e*x+d)*A*b^2*c+3*d^5/e^6/(e*x+d)*A*b*c^2+3*d^5/e^6/(e*x+d)*B*b^2*c-3*d^6/e^
7/(e*x+d)*B*b*c^2-2/e^3*B*x^3*b^2*c*d+3/e^4*B*x^3*b*c^2*d^2-3/e^3*A*x^2*b^2*c*d+9/2/e^4*A*x^2*b*c^2*d^2+9/2/e^
4*B*x^2*b^2*c*d^2-6/e^5*B*x^2*b*c^2*d^3+9/e^4*A*b^2*c*d^2*x-12/e^5*A*b*c^2*d^3*x-12/e^5*B*b^2*c*d^3*x+15/e^6*B
*b*c^2*d^4*x-3/2/e^3*B*x^4*b*c^2*d-2/e^3*A*x^3*b*c^2*d+1/3/e^2*B*x^3*b^3+1/5/e^2*A*x^5*c^3+1/2/e^2*A*x^2*b^3-6
/e^7*B*c^3*d^5*x+1/e^2*A*x^3*b^2*c+1/e^4*A*x^3*c^3*d^2-2/e^3*A*b^3*d*x+5/2/e^6*B*x^2*c^3*d^4-1/e^3*B*x^2*b^3*d
-2/e^5*A*x^2*c^3*d^3+3/4/e^2*B*x^4*b^2*c+3/4/e^4*B*x^4*c^3*d^2-4/3/e^5*B*x^3*c^3*d^3-1/2/e^3*A*x^4*c^3*d+3/5/e
^2*B*x^5*b*c^2-2/5/e^3*B*x^5*c^3*d+3/4/e^2*A*x^4*b*c^2+3*d^2/e^4*ln(e*x+d)*A*b^3-6*d^5/e^7*ln(e*x+d)*A*c^3-4*d
^3/e^5*ln(e*x+d)*B*b^3+7*d^6/e^8*ln(e*x+d)*B*c^3+d^3/e^4/(e*x+d)*A*b^3-d^6/e^7/(e*x+d)*A*c^3+5/e^6*A*c^3*d^4*x
+3/e^4*B*b^3*d^2*x+d^7/e^8/(e*x+d)*B*c^3-d^4/e^5/(e*x+d)*B*b^3

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Maxima [A]  time = 1.0682, size = 730, normalized size = 2.54 \begin{align*} \frac{B c^{3} d^{7} + A b^{3} d^{3} e^{4} -{\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 3 \,{\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} -{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3}}{e^{9} x + d e^{8}} + \frac{10 \, B c^{3} e^{5} x^{6} - 12 \,{\left (2 \, B c^{3} d e^{4} -{\left (3 \, B b c^{2} + A c^{3}\right )} e^{5}\right )} x^{5} + 15 \,{\left (3 \, B c^{3} d^{2} e^{3} - 2 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d e^{4} + 3 \,{\left (B b^{2} c + A b c^{2}\right )} e^{5}\right )} x^{4} - 20 \,{\left (4 \, B c^{3} d^{3} e^{2} - 3 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{3} + 6 \,{\left (B b^{2} c + A b c^{2}\right )} d e^{4} -{\left (B b^{3} + 3 \, A b^{2} c\right )} e^{5}\right )} x^{3} + 30 \,{\left (5 \, B c^{3} d^{4} e + A b^{3} e^{5} - 4 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{2} + 9 \,{\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{3} - 2 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{4}\right )} x^{2} - 60 \,{\left (6 \, B c^{3} d^{5} + 2 \, A b^{3} d e^{4} - 5 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e + 12 \,{\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{2} - 3 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{3}\right )} x}{60 \, e^{7}} + \frac{{\left (7 \, B c^{3} d^{6} + 3 \, A b^{3} d^{2} e^{4} - 6 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e + 15 \,{\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{2} - 4 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{3}\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+b*x)^3/(e*x+d)^2,x, algorithm="maxima")

[Out]

(B*c^3*d^7 + A*b^3*d^3*e^4 - (3*B*b*c^2 + A*c^3)*d^6*e + 3*(B*b^2*c + A*b*c^2)*d^5*e^2 - (B*b^3 + 3*A*b^2*c)*d
^4*e^3)/(e^9*x + d*e^8) + 1/60*(10*B*c^3*e^5*x^6 - 12*(2*B*c^3*d*e^4 - (3*B*b*c^2 + A*c^3)*e^5)*x^5 + 15*(3*B*
c^3*d^2*e^3 - 2*(3*B*b*c^2 + A*c^3)*d*e^4 + 3*(B*b^2*c + A*b*c^2)*e^5)*x^4 - 20*(4*B*c^3*d^3*e^2 - 3*(3*B*b*c^
2 + A*c^3)*d^2*e^3 + 6*(B*b^2*c + A*b*c^2)*d*e^4 - (B*b^3 + 3*A*b^2*c)*e^5)*x^3 + 30*(5*B*c^3*d^4*e + A*b^3*e^
5 - 4*(3*B*b*c^2 + A*c^3)*d^3*e^2 + 9*(B*b^2*c + A*b*c^2)*d^2*e^3 - 2*(B*b^3 + 3*A*b^2*c)*d*e^4)*x^2 - 60*(6*B
*c^3*d^5 + 2*A*b^3*d*e^4 - 5*(3*B*b*c^2 + A*c^3)*d^4*e + 12*(B*b^2*c + A*b*c^2)*d^3*e^2 - 3*(B*b^3 + 3*A*b^2*c
)*d^2*e^3)*x)/e^7 + (7*B*c^3*d^6 + 3*A*b^3*d^2*e^4 - 6*(3*B*b*c^2 + A*c^3)*d^5*e + 15*(B*b^2*c + A*b*c^2)*d^4*
e^2 - 4*(B*b^3 + 3*A*b^2*c)*d^3*e^3)*log(e*x + d)/e^8

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Fricas [B]  time = 1.65918, size = 1513, normalized size = 5.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(10*B*c^3*e^7*x^7 + 60*B*c^3*d^7 + 60*A*b^3*d^3*e^4 - 60*(3*B*b*c^2 + A*c^3)*d^6*e + 180*(B*b^2*c + A*b*c
^2)*d^5*e^2 - 60*(B*b^3 + 3*A*b^2*c)*d^4*e^3 - 2*(7*B*c^3*d*e^6 - 6*(3*B*b*c^2 + A*c^3)*e^7)*x^6 + 3*(7*B*c^3*
d^2*e^5 - 6*(3*B*b*c^2 + A*c^3)*d*e^6 + 15*(B*b^2*c + A*b*c^2)*e^7)*x^5 - 5*(7*B*c^3*d^3*e^4 - 6*(3*B*b*c^2 +
A*c^3)*d^2*e^5 + 15*(B*b^2*c + A*b*c^2)*d*e^6 - 4*(B*b^3 + 3*A*b^2*c)*e^7)*x^4 + 10*(7*B*c^3*d^4*e^3 + 3*A*b^3
*e^7 - 6*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 15*(B*b^2*c + A*b*c^2)*d^2*e^5 - 4*(B*b^3 + 3*A*b^2*c)*d*e^6)*x^3 - 30*
(7*B*c^3*d^5*e^2 + 3*A*b^3*d*e^6 - 6*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 15*(B*b^2*c + A*b*c^2)*d^3*e^4 - 4*(B*b^3 +
3*A*b^2*c)*d^2*e^5)*x^2 - 60*(6*B*c^3*d^6*e + 2*A*b^3*d^2*e^5 - 5*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 12*(B*b^2*c +
A*b*c^2)*d^4*e^3 - 3*(B*b^3 + 3*A*b^2*c)*d^3*e^4)*x + 60*(7*B*c^3*d^7 + 3*A*b^3*d^3*e^4 - 6*(3*B*b*c^2 + A*c^
3)*d^6*e + 15*(B*b^2*c + A*b*c^2)*d^5*e^2 - 4*(B*b^3 + 3*A*b^2*c)*d^4*e^3 + (7*B*c^3*d^6*e + 3*A*b^3*d^2*e^5 -
6*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 15*(B*b^2*c + A*b*c^2)*d^4*e^3 - 4*(B*b^3 + 3*A*b^2*c)*d^3*e^4)*x)*log(e*x +
d))/(e^9*x + d*e^8)

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Sympy [B]  time = 3.9072, size = 592, normalized size = 2.06 \begin{align*} \frac{B c^{3} x^{6}}{6 e^{2}} - \frac{d^{2} \left (b e - c d\right )^{2} \left (- 3 A b e^{2} + 6 A c d e + 4 B b d e - 7 B c d^{2}\right ) \log{\left (d + e x \right )}}{e^{8}} - \frac{- A b^{3} d^{3} e^{4} + 3 A b^{2} c d^{4} e^{3} - 3 A b c^{2} d^{5} e^{2} + A c^{3} d^{6} e + B b^{3} d^{4} e^{3} - 3 B b^{2} c d^{5} e^{2} + 3 B b c^{2} d^{6} e - B c^{3} d^{7}}{d e^{8} + e^{9} x} + \frac{x^{5} \left (A c^{3} e + 3 B b c^{2} e - 2 B c^{3} d\right )}{5 e^{3}} + \frac{x^{4} \left (3 A b c^{2} e^{2} - 2 A c^{3} d e + 3 B b^{2} c e^{2} - 6 B b c^{2} d e + 3 B c^{3} d^{2}\right )}{4 e^{4}} + \frac{x^{3} \left (3 A b^{2} c e^{3} - 6 A b c^{2} d e^{2} + 3 A c^{3} d^{2} e + B b^{3} e^{3} - 6 B b^{2} c d e^{2} + 9 B b c^{2} d^{2} e - 4 B c^{3} d^{3}\right )}{3 e^{5}} - \frac{x^{2} \left (- A b^{3} e^{4} + 6 A b^{2} c d e^{3} - 9 A b c^{2} d^{2} e^{2} + 4 A c^{3} d^{3} e + 2 B b^{3} d e^{3} - 9 B b^{2} c d^{2} e^{2} + 12 B b c^{2} d^{3} e - 5 B c^{3} d^{4}\right )}{2 e^{6}} + \frac{x \left (- 2 A b^{3} d e^{4} + 9 A b^{2} c d^{2} e^{3} - 12 A b c^{2} d^{3} e^{2} + 5 A c^{3} d^{4} e + 3 B b^{3} d^{2} e^{3} - 12 B b^{2} c d^{3} e^{2} + 15 B b c^{2} d^{4} e - 6 B c^{3} d^{5}\right )}{e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*c**3*x**6/(6*e**2) - d**2*(b*e - c*d)**2*(-3*A*b*e**2 + 6*A*c*d*e + 4*B*b*d*e - 7*B*c*d**2)*log(d + e*x)/e**
8 - (-A*b**3*d**3*e**4 + 3*A*b**2*c*d**4*e**3 - 3*A*b*c**2*d**5*e**2 + A*c**3*d**6*e + B*b**3*d**4*e**3 - 3*B*
b**2*c*d**5*e**2 + 3*B*b*c**2*d**6*e - B*c**3*d**7)/(d*e**8 + e**9*x) + x**5*(A*c**3*e + 3*B*b*c**2*e - 2*B*c*
*3*d)/(5*e**3) + x**4*(3*A*b*c**2*e**2 - 2*A*c**3*d*e + 3*B*b**2*c*e**2 - 6*B*b*c**2*d*e + 3*B*c**3*d**2)/(4*e
**4) + x**3*(3*A*b**2*c*e**3 - 6*A*b*c**2*d*e**2 + 3*A*c**3*d**2*e + B*b**3*e**3 - 6*B*b**2*c*d*e**2 + 9*B*b*c
**2*d**2*e - 4*B*c**3*d**3)/(3*e**5) - x**2*(-A*b**3*e**4 + 6*A*b**2*c*d*e**3 - 9*A*b*c**2*d**2*e**2 + 4*A*c**
3*d**3*e + 2*B*b**3*d*e**3 - 9*B*b**2*c*d**2*e**2 + 12*B*b*c**2*d**3*e - 5*B*c**3*d**4)/(2*e**6) + x*(-2*A*b**
3*d*e**4 + 9*A*b**2*c*d**2*e**3 - 12*A*b*c**2*d**3*e**2 + 5*A*c**3*d**4*e + 3*B*b**3*d**2*e**3 - 12*B*b**2*c*d
**3*e**2 + 15*B*b*c**2*d**4*e - 6*B*c**3*d**5)/e**7

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Giac [B]  time = 1.43602, size = 909, normalized size = 3.17 \begin{align*} \frac{1}{60} \,{\left (10 \, B c^{3} - \frac{12 \,{\left (7 \, B c^{3} d e - 3 \, B b c^{2} e^{2} - A c^{3} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{45 \,{\left (7 \, B c^{3} d^{2} e^{2} - 6 \, B b c^{2} d e^{3} - 2 \, A c^{3} d e^{3} + B b^{2} c e^{4} + A b c^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{20 \,{\left (35 \, B c^{3} d^{3} e^{3} - 45 \, B b c^{2} d^{2} e^{4} - 15 \, A c^{3} d^{2} e^{4} + 15 \, B b^{2} c d e^{5} + 15 \, A b c^{2} d e^{5} - B b^{3} e^{6} - 3 \, A b^{2} c e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac{30 \,{\left (35 \, B c^{3} d^{4} e^{4} - 60 \, B b c^{2} d^{3} e^{5} - 20 \, A c^{3} d^{3} e^{5} + 30 \, B b^{2} c d^{2} e^{6} + 30 \, A b c^{2} d^{2} e^{6} - 4 \, B b^{3} d e^{7} - 12 \, A b^{2} c d e^{7} + A b^{3} e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}} - \frac{180 \,{\left (7 \, B c^{3} d^{5} e^{5} - 15 \, B b c^{2} d^{4} e^{6} - 5 \, A c^{3} d^{4} e^{6} + 10 \, B b^{2} c d^{3} e^{7} + 10 \, A b c^{2} d^{3} e^{7} - 2 \, B b^{3} d^{2} e^{8} - 6 \, A b^{2} c d^{2} e^{8} + A b^{3} d e^{9}\right )} e^{\left (-5\right )}}{{\left (x e + d\right )}^{5}}\right )}{\left (x e + d\right )}^{6} e^{\left (-8\right )} -{\left (7 \, B c^{3} d^{6} - 18 \, B b c^{2} d^{5} e - 6 \, A c^{3} d^{5} e + 15 \, B b^{2} c d^{4} e^{2} + 15 \, A b c^{2} d^{4} e^{2} - 4 \, B b^{3} d^{3} e^{3} - 12 \, A b^{2} c d^{3} e^{3} + 3 \, A b^{3} d^{2} e^{4}\right )} e^{\left (-8\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (\frac{B c^{3} d^{7} e^{6}}{x e + d} - \frac{3 \, B b c^{2} d^{6} e^{7}}{x e + d} - \frac{A c^{3} d^{6} e^{7}}{x e + d} + \frac{3 \, B b^{2} c d^{5} e^{8}}{x e + d} + \frac{3 \, A b c^{2} d^{5} e^{8}}{x e + d} - \frac{B b^{3} d^{4} e^{9}}{x e + d} - \frac{3 \, A b^{2} c d^{4} e^{9}}{x e + d} + \frac{A b^{3} d^{3} e^{10}}{x e + d}\right )} e^{\left (-14\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(10*B*c^3 - 12*(7*B*c^3*d*e - 3*B*b*c^2*e^2 - A*c^3*e^2)*e^(-1)/(x*e + d) + 45*(7*B*c^3*d^2*e^2 - 6*B*b*c
^2*d*e^3 - 2*A*c^3*d*e^3 + B*b^2*c*e^4 + A*b*c^2*e^4)*e^(-2)/(x*e + d)^2 - 20*(35*B*c^3*d^3*e^3 - 45*B*b*c^2*d
^2*e^4 - 15*A*c^3*d^2*e^4 + 15*B*b^2*c*d*e^5 + 15*A*b*c^2*d*e^5 - B*b^3*e^6 - 3*A*b^2*c*e^6)*e^(-3)/(x*e + d)^
3 + 30*(35*B*c^3*d^4*e^4 - 60*B*b*c^2*d^3*e^5 - 20*A*c^3*d^3*e^5 + 30*B*b^2*c*d^2*e^6 + 30*A*b*c^2*d^2*e^6 - 4
*B*b^3*d*e^7 - 12*A*b^2*c*d*e^7 + A*b^3*e^8)*e^(-4)/(x*e + d)^4 - 180*(7*B*c^3*d^5*e^5 - 15*B*b*c^2*d^4*e^6 -
5*A*c^3*d^4*e^6 + 10*B*b^2*c*d^3*e^7 + 10*A*b*c^2*d^3*e^7 - 2*B*b^3*d^2*e^8 - 6*A*b^2*c*d^2*e^8 + A*b^3*d*e^9)
*e^(-5)/(x*e + d)^5)*(x*e + d)^6*e^(-8) - (7*B*c^3*d^6 - 18*B*b*c^2*d^5*e - 6*A*c^3*d^5*e + 15*B*b^2*c*d^4*e^2
+ 15*A*b*c^2*d^4*e^2 - 4*B*b^3*d^3*e^3 - 12*A*b^2*c*d^3*e^3 + 3*A*b^3*d^2*e^4)*e^(-8)*log(abs(x*e + d)*e^(-1)
/(x*e + d)^2) + (B*c^3*d^7*e^6/(x*e + d) - 3*B*b*c^2*d^6*e^7/(x*e + d) - A*c^3*d^6*e^7/(x*e + d) + 3*B*b^2*c*d
^5*e^8/(x*e + d) + 3*A*b*c^2*d^5*e^8/(x*e + d) - B*b^3*d^4*e^9/(x*e + d) - 3*A*b^2*c*d^4*e^9/(x*e + d) + A*b^3
*d^3*e^10/(x*e + d))*e^(-14)