3.1156 \(\int \frac{(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=185 \[ -\frac{3 d \log (x) (c d-b e) (A b e-2 A c d+b B d)}{b^5}+\frac{3 d (c d-b e) \log (b+c x) (A b e-2 A c d+b B d)}{b^5}-\frac{d^2 (3 A b e-3 A c d+b B d)}{b^4 x}-\frac{(b B-A c) (c d-b e)^3}{2 b^3 c^2 (b+c x)^2}-\frac{A d^3}{2 b^3 x^2}-\frac{(c d-b e)^2 \left (-3 A c^2 d+b^2 B e+2 b B c d\right )}{b^4 c^2 (b+c x)} \]

[Out]

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

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Rubi [A]  time = 0.521062, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{3 d \log (x) (c d-b e) (A b e-2 A c d+b B d)}{b^5}+\frac{3 d (c d-b e) \log (b+c x) (A b e-2 A c d+b B d)}{b^5}-\frac{d^2 (3 A b e-3 A c d+b B d)}{b^4 x}-\frac{(b B-A c) (c d-b e)^3}{2 b^3 c^2 (b+c x)^2}-\frac{A d^3}{2 b^3 x^2}-\frac{(c d-b e)^2 \left (-3 A c^2 d+b^2 B e+2 b B c d\right )}{b^4 c^2 (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 64.3687, size = 185, normalized size = 1. \[ - \frac{A d^{3}}{2 b^{3} x^{2}} - \frac{\left (A c - B b\right ) \left (b e - c d\right )^{3}}{2 b^{3} c^{2} \left (b + c x\right )^{2}} - \frac{d^{2} \left (3 A b e - 3 A c d + B b d\right )}{b^{4} x} + \frac{\left (b e - c d\right )^{2} \left (3 A c^{2} d - B b^{2} e - 2 B b c d\right )}{b^{4} c^{2} \left (b + c x\right )} + \frac{3 d \left (b e - c d\right ) \left (A b e - 2 A c d + B b d\right ) \log{\left (x \right )}}{b^{5}} - \frac{3 d \left (b e - c d\right ) \left (A b e - 2 A c d + B b d\right ) \log{\left (b + c x \right )}}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*d**3/(2*b**3*x**2) - (A*c - B*b)*(b*e - c*d)**3/(2*b**3*c**2*(b + c*x)**2) -
d**2*(3*A*b*e - 3*A*c*d + B*b*d)/(b**4*x) + (b*e - c*d)**2*(3*A*c**2*d - B*b**2*
e - 2*B*b*c*d)/(b**4*c**2*(b + c*x)) + 3*d*(b*e - c*d)*(A*b*e - 2*A*c*d + B*b*d)
*log(x)/b**5 - 3*d*(b*e - c*d)*(A*b*e - 2*A*c*d + B*b*d)*log(b + c*x)/b**5

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Mathematica [A]  time = 0.215964, size = 177, normalized size = 0.96 \[ -\frac{\frac{2 b (c d-b e)^2 \left (-3 A c^2 d+b^2 B e+2 b B c d\right )}{c^2 (b+c x)}-\frac{b^2 (b B-A c) (b e-c d)^3}{c^2 (b+c x)^2}+\frac{A b^2 d^3}{x^2}+\frac{2 b d^2 (3 A b e-3 A c d+b B d)}{x}-6 d \log (x) (b e-c d) (A b e-2 A c d+b B d)+6 d (b e-c d) \log (b+c x) (A b e-2 A c d+b B d)}{2 b^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.021, size = 440, normalized size = 2.4 \[ -{\frac{A{d}^{3}}{2\,{b}^{3}{x}^{2}}}+{\frac{B{e}^{3}b}{2\,{c}^{2} \left ( cx+b \right ) ^{2}}}-{\frac{Bc{d}^{3}}{2\,{b}^{2} \left ( cx+b \right ) ^{2}}}-3\,{\frac{d\ln \left ( cx+b \right ) A{e}^{2}}{{b}^{3}}}-6\,{\frac{{d}^{3}\ln \left ( cx+b \right ) A{c}^{2}}{{b}^{5}}}-3\,{\frac{{d}^{2}\ln \left ( cx+b \right ) Be}{{b}^{3}}}+3\,{\frac{{d}^{3}\ln \left ( cx+b \right ) Bc}{{b}^{4}}}-3\,{\frac{A{d}^{2}e}{{b}^{3}x}}+3\,{\frac{A{d}^{3}c}{{b}^{4}x}}+3\,{\frac{d\ln \left ( x \right ) A{e}^{2}}{{b}^{3}}}+6\,{\frac{{d}^{3}\ln \left ( x \right ) A{c}^{2}}{{b}^{5}}}+3\,{\frac{{d}^{2}\ln \left ( x \right ) Be}{{b}^{3}}}-3\,{\frac{{d}^{3}\ln \left ( x \right ) Bc}{{b}^{4}}}+3\,{\frac{A{d}^{3}{c}^{2}}{{b}^{4} \left ( cx+b \right ) }}-2\,{\frac{Bc{d}^{3}}{{b}^{3} \left ( cx+b \right ) }}-{\frac{A{e}^{3}}{2\,c \left ( cx+b \right ) ^{2}}}-{\frac{B{e}^{3}}{{c}^{2} \left ( cx+b \right ) }}-{\frac{B{d}^{3}}{{b}^{3}x}}-9\,{\frac{{d}^{2}\ln \left ( x \right ) Ace}{{b}^{4}}}-6\,{\frac{Ac{d}^{2}e}{{b}^{3} \left ( cx+b \right ) }}-{\frac{3\,Ac{d}^{2}e}{2\,{b}^{2} \left ( cx+b \right ) ^{2}}}+9\,{\frac{{d}^{2}\ln \left ( cx+b \right ) Ace}{{b}^{4}}}+{\frac{3\,Ad{e}^{2}}{2\,b \left ( cx+b \right ) ^{2}}}-{\frac{3\,Bd{e}^{2}}{2\,c \left ( cx+b \right ) ^{2}}}+{\frac{3\,B{d}^{2}e}{2\,b \left ( cx+b \right ) ^{2}}}+{\frac{A{d}^{3}{c}^{2}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}}+3\,{\frac{Ad{e}^{2}}{{b}^{2} \left ( cx+b \right ) }}+3\,{\frac{B{d}^{2}e}{{b}^{2} \left ( cx+b \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*A*d^3/b^3/x^2+1/2/c^2*b/(c*x+b)^2*B*e^3-1/2*c/b^2/(c*x+b)^2*B*d^3-3*d/b^3*l
n(c*x+b)*A*e^2-6*d^3/b^5*ln(c*x+b)*A*c^2-3*d^2/b^3*ln(c*x+b)*B*e+3*d^3/b^4*ln(c*
x+b)*B*c-3*d^2/b^3/x*A*e+3*d^3/b^4/x*A*c+3*d/b^3*ln(x)*A*e^2+6*d^3/b^5*ln(x)*A*c
^2+3*d^2/b^3*ln(x)*B*e-3*d^3/b^4*ln(x)*B*c+3/b^4*c^2/(c*x+b)*A*d^3-2/b^3*c/(c*x+
b)*B*d^3-1/2/c/(c*x+b)^2*A*e^3-1/c^2/(c*x+b)*B*e^3-d^3/b^3/x*B-9*d^2/b^4*ln(x)*A
*c*e-6/b^3*c/(c*x+b)*A*d^2*e-3/2*c/b^2/(c*x+b)^2*A*d^2*e+9*d^2/b^4*ln(c*x+b)*A*c
*e+3/2/b/(c*x+b)^2*A*d*e^2-3/2/c/(c*x+b)^2*B*d*e^2+3/2/b/(c*x+b)^2*B*d^2*e+1/2*c
^2/b^3/(c*x+b)^2*A*d^3+3/b^2/(c*x+b)*A*d*e^2+3/b^2/(c*x+b)*B*d^2*e

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Maxima [A]  time = 0.703802, size = 468, normalized size = 2.53 \[ -\frac{A b^{3} c^{2} d^{3} - 2 \,{\left (3 \, A b^{2} c^{3} d e^{2} - B b^{4} c e^{3} - 3 \,{\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} + 3 \,{\left (B b^{2} c^{3} - 3 \, A b c^{4}\right )} d^{2} e\right )} x^{3} +{\left (9 \,{\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} - 9 \,{\left (B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} d^{2} e + 3 \,{\left (B b^{4} c - 3 \, A b^{3} c^{2}\right )} d e^{2} +{\left (B b^{5} + A b^{4} c\right )} e^{3}\right )} x^{2} + 2 \,{\left (3 \, A b^{3} c^{2} d^{2} e +{\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{3}\right )} x}{2 \,{\left (b^{4} c^{4} x^{4} + 2 \, b^{5} c^{3} x^{3} + b^{6} c^{2} x^{2}\right )}} - \frac{3 \,{\left (A b^{2} d e^{2} -{\left (B b c - 2 \, A c^{2}\right )} d^{3} +{\left (B b^{2} - 3 \, A b c\right )} d^{2} e\right )} \log \left (c x + b\right )}{b^{5}} + \frac{3 \,{\left (A b^{2} d e^{2} -{\left (B b c - 2 \, A c^{2}\right )} d^{3} +{\left (B b^{2} - 3 \, A b c\right )} d^{2} e\right )} \log \left (x\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(A*b^3*c^2*d^3 - 2*(3*A*b^2*c^3*d*e^2 - B*b^4*c*e^3 - 3*(B*b*c^4 - 2*A*c^5)
*d^3 + 3*(B*b^2*c^3 - 3*A*b*c^4)*d^2*e)*x^3 + (9*(B*b^2*c^3 - 2*A*b*c^4)*d^3 - 9
*(B*b^3*c^2 - 3*A*b^2*c^3)*d^2*e + 3*(B*b^4*c - 3*A*b^3*c^2)*d*e^2 + (B*b^5 + A*
b^4*c)*e^3)*x^2 + 2*(3*A*b^3*c^2*d^2*e + (B*b^3*c^2 - 2*A*b^2*c^3)*d^3)*x)/(b^4*
c^4*x^4 + 2*b^5*c^3*x^3 + b^6*c^2*x^2) - 3*(A*b^2*d*e^2 - (B*b*c - 2*A*c^2)*d^3
+ (B*b^2 - 3*A*b*c)*d^2*e)*log(c*x + b)/b^5 + 3*(A*b^2*d*e^2 - (B*b*c - 2*A*c^2)
*d^3 + (B*b^2 - 3*A*b*c)*d^2*e)*log(x)/b^5

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Fricas [A]  time = 0.302077, size = 846, normalized size = 4.57 \[ -\frac{A b^{4} c^{2} d^{3} - 2 \,{\left (3 \, A b^{3} c^{3} d e^{2} - B b^{5} c e^{3} - 3 \,{\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} + 3 \,{\left (B b^{3} c^{3} - 3 \, A b^{2} c^{4}\right )} d^{2} e\right )} x^{3} +{\left (9 \,{\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3} - 9 \,{\left (B b^{4} c^{2} - 3 \, A b^{3} c^{3}\right )} d^{2} e + 3 \,{\left (B b^{5} c - 3 \, A b^{4} c^{2}\right )} d e^{2} +{\left (B b^{6} + A b^{5} c\right )} e^{3}\right )} x^{2} + 2 \,{\left (3 \, A b^{4} c^{2} d^{2} e +{\left (B b^{4} c^{2} - 2 \, A b^{3} c^{3}\right )} d^{3}\right )} x + 6 \,{\left ({\left (A b^{2} c^{4} d e^{2} -{\left (B b c^{5} - 2 \, A c^{6}\right )} d^{3} +{\left (B b^{2} c^{4} - 3 \, A b c^{5}\right )} d^{2} e\right )} x^{4} + 2 \,{\left (A b^{3} c^{3} d e^{2} -{\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} +{\left (B b^{3} c^{3} - 3 \, A b^{2} c^{4}\right )} d^{2} e\right )} x^{3} +{\left (A b^{4} c^{2} d e^{2} -{\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3} +{\left (B b^{4} c^{2} - 3 \, A b^{3} c^{3}\right )} d^{2} e\right )} x^{2}\right )} \log \left (c x + b\right ) - 6 \,{\left ({\left (A b^{2} c^{4} d e^{2} -{\left (B b c^{5} - 2 \, A c^{6}\right )} d^{3} +{\left (B b^{2} c^{4} - 3 \, A b c^{5}\right )} d^{2} e\right )} x^{4} + 2 \,{\left (A b^{3} c^{3} d e^{2} -{\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} +{\left (B b^{3} c^{3} - 3 \, A b^{2} c^{4}\right )} d^{2} e\right )} x^{3} +{\left (A b^{4} c^{2} d e^{2} -{\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3} +{\left (B b^{4} c^{2} - 3 \, A b^{3} c^{3}\right )} d^{2} e\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{5} c^{4} x^{4} + 2 \, b^{6} c^{3} x^{3} + b^{7} c^{2} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(A*b^4*c^2*d^3 - 2*(3*A*b^3*c^3*d*e^2 - B*b^5*c*e^3 - 3*(B*b^2*c^4 - 2*A*b*
c^5)*d^3 + 3*(B*b^3*c^3 - 3*A*b^2*c^4)*d^2*e)*x^3 + (9*(B*b^3*c^3 - 2*A*b^2*c^4)
*d^3 - 9*(B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e + 3*(B*b^5*c - 3*A*b^4*c^2)*d*e^2 + (B*
b^6 + A*b^5*c)*e^3)*x^2 + 2*(3*A*b^4*c^2*d^2*e + (B*b^4*c^2 - 2*A*b^3*c^3)*d^3)*
x + 6*((A*b^2*c^4*d*e^2 - (B*b*c^5 - 2*A*c^6)*d^3 + (B*b^2*c^4 - 3*A*b*c^5)*d^2*
e)*x^4 + 2*(A*b^3*c^3*d*e^2 - (B*b^2*c^4 - 2*A*b*c^5)*d^3 + (B*b^3*c^3 - 3*A*b^2
*c^4)*d^2*e)*x^3 + (A*b^4*c^2*d*e^2 - (B*b^3*c^3 - 2*A*b^2*c^4)*d^3 + (B*b^4*c^2
- 3*A*b^3*c^3)*d^2*e)*x^2)*log(c*x + b) - 6*((A*b^2*c^4*d*e^2 - (B*b*c^5 - 2*A*
c^6)*d^3 + (B*b^2*c^4 - 3*A*b*c^5)*d^2*e)*x^4 + 2*(A*b^3*c^3*d*e^2 - (B*b^2*c^4
- 2*A*b*c^5)*d^3 + (B*b^3*c^3 - 3*A*b^2*c^4)*d^2*e)*x^3 + (A*b^4*c^2*d*e^2 - (B*
b^3*c^3 - 2*A*b^2*c^4)*d^3 + (B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e)*x^2)*log(x))/(b^5*
c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2)

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Sympy [A]  time = 125.384, size = 653, normalized size = 3.53 \[ - \frac{A b^{3} c^{2} d^{3} + x^{3} \left (- 6 A b^{2} c^{3} d e^{2} + 18 A b c^{4} d^{2} e - 12 A c^{5} d^{3} + 2 B b^{4} c e^{3} - 6 B b^{2} c^{3} d^{2} e + 6 B b c^{4} d^{3}\right ) + x^{2} \left (A b^{4} c e^{3} - 9 A b^{3} c^{2} d e^{2} + 27 A b^{2} c^{3} d^{2} e - 18 A b c^{4} d^{3} + B b^{5} e^{3} + 3 B b^{4} c d e^{2} - 9 B b^{3} c^{2} d^{2} e + 9 B b^{2} c^{3} d^{3}\right ) + x \left (6 A b^{3} c^{2} d^{2} e - 4 A b^{2} c^{3} d^{3} + 2 B b^{3} c^{2} d^{3}\right )}{2 b^{6} c^{2} x^{2} + 4 b^{5} c^{3} x^{3} + 2 b^{4} c^{4} x^{4}} + \frac{3 d \left (b e - c d\right ) \left (A b e - 2 A c d + B b d\right ) \log{\left (x + \frac{3 A b^{3} d e^{2} - 9 A b^{2} c d^{2} e + 6 A b c^{2} d^{3} + 3 B b^{3} d^{2} e - 3 B b^{2} c d^{3} - 3 b d \left (b e - c d\right ) \left (A b e - 2 A c d + B b d\right )}{6 A b^{2} c d e^{2} - 18 A b c^{2} d^{2} e + 12 A c^{3} d^{3} + 6 B b^{2} c d^{2} e - 6 B b c^{2} d^{3}} \right )}}{b^{5}} - \frac{3 d \left (b e - c d\right ) \left (A b e - 2 A c d + B b d\right ) \log{\left (x + \frac{3 A b^{3} d e^{2} - 9 A b^{2} c d^{2} e + 6 A b c^{2} d^{3} + 3 B b^{3} d^{2} e - 3 B b^{2} c d^{3} + 3 b d \left (b e - c d\right ) \left (A b e - 2 A c d + B b d\right )}{6 A b^{2} c d e^{2} - 18 A b c^{2} d^{2} e + 12 A c^{3} d^{3} + 6 B b^{2} c d^{2} e - 6 B b c^{2} d^{3}} \right )}}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(A*b**3*c**2*d**3 + x**3*(-6*A*b**2*c**3*d*e**2 + 18*A*b*c**4*d**2*e - 12*A*c**
5*d**3 + 2*B*b**4*c*e**3 - 6*B*b**2*c**3*d**2*e + 6*B*b*c**4*d**3) + x**2*(A*b**
4*c*e**3 - 9*A*b**3*c**2*d*e**2 + 27*A*b**2*c**3*d**2*e - 18*A*b*c**4*d**3 + B*b
**5*e**3 + 3*B*b**4*c*d*e**2 - 9*B*b**3*c**2*d**2*e + 9*B*b**2*c**3*d**3) + x*(6
*A*b**3*c**2*d**2*e - 4*A*b**2*c**3*d**3 + 2*B*b**3*c**2*d**3))/(2*b**6*c**2*x**
2 + 4*b**5*c**3*x**3 + 2*b**4*c**4*x**4) + 3*d*(b*e - c*d)*(A*b*e - 2*A*c*d + B*
b*d)*log(x + (3*A*b**3*d*e**2 - 9*A*b**2*c*d**2*e + 6*A*b*c**2*d**3 + 3*B*b**3*d
**2*e - 3*B*b**2*c*d**3 - 3*b*d*(b*e - c*d)*(A*b*e - 2*A*c*d + B*b*d))/(6*A*b**2
*c*d*e**2 - 18*A*b*c**2*d**2*e + 12*A*c**3*d**3 + 6*B*b**2*c*d**2*e - 6*B*b*c**2
*d**3))/b**5 - 3*d*(b*e - c*d)*(A*b*e - 2*A*c*d + B*b*d)*log(x + (3*A*b**3*d*e**
2 - 9*A*b**2*c*d**2*e + 6*A*b*c**2*d**3 + 3*B*b**3*d**2*e - 3*B*b**2*c*d**3 + 3*
b*d*(b*e - c*d)*(A*b*e - 2*A*c*d + B*b*d))/(6*A*b**2*c*d*e**2 - 18*A*b*c**2*d**2
*e + 12*A*c**3*d**3 + 6*B*b**2*c*d**2*e - 6*B*b*c**2*d**3))/b**5

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GIAC/XCAS [A]  time = 0.276934, size = 528, normalized size = 2.85 \[ -\frac{3 \,{\left (B b c d^{3} - 2 \, A c^{2} d^{3} - B b^{2} d^{2} e + 3 \, A b c d^{2} e - A b^{2} d e^{2}\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{5}} + \frac{3 \,{\left (B b c^{2} d^{3} - 2 \, A c^{3} d^{3} - B b^{2} c d^{2} e + 3 \, A b c^{2} d^{2} e - A b^{2} c d e^{2}\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{5} c} - \frac{6 \, B b c^{4} d^{3} x^{3} - 12 \, A c^{5} d^{3} x^{3} - 6 \, B b^{2} c^{3} d^{2} x^{3} e + 18 \, A b c^{4} d^{2} x^{3} e + 9 \, B b^{2} c^{3} d^{3} x^{2} - 18 \, A b c^{4} d^{3} x^{2} - 6 \, A b^{2} c^{3} d x^{3} e^{2} - 9 \, B b^{3} c^{2} d^{2} x^{2} e + 27 \, A b^{2} c^{3} d^{2} x^{2} e + 2 \, B b^{3} c^{2} d^{3} x - 4 \, A b^{2} c^{3} d^{3} x + 2 \, B b^{4} c x^{3} e^{3} + 3 \, B b^{4} c d x^{2} e^{2} - 9 \, A b^{3} c^{2} d x^{2} e^{2} + 6 \, A b^{3} c^{2} d^{2} x e + A b^{3} c^{2} d^{3} + B b^{5} x^{2} e^{3} + A b^{4} c x^{2} e^{3}}{2 \,{\left (c x^{2} + b x\right )}^{2} b^{4} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(B*b*c*d^3 - 2*A*c^2*d^3 - B*b^2*d^2*e + 3*A*b*c*d^2*e - A*b^2*d*e^2)*ln(abs(
x))/b^5 + 3*(B*b*c^2*d^3 - 2*A*c^3*d^3 - B*b^2*c*d^2*e + 3*A*b*c^2*d^2*e - A*b^2
*c*d*e^2)*ln(abs(c*x + b))/(b^5*c) - 1/2*(6*B*b*c^4*d^3*x^3 - 12*A*c^5*d^3*x^3 -
6*B*b^2*c^3*d^2*x^3*e + 18*A*b*c^4*d^2*x^3*e + 9*B*b^2*c^3*d^3*x^2 - 18*A*b*c^4
*d^3*x^2 - 6*A*b^2*c^3*d*x^3*e^2 - 9*B*b^3*c^2*d^2*x^2*e + 27*A*b^2*c^3*d^2*x^2*
e + 2*B*b^3*c^2*d^3*x - 4*A*b^2*c^3*d^3*x + 2*B*b^4*c*x^3*e^3 + 3*B*b^4*c*d*x^2*
e^2 - 9*A*b^3*c^2*d*x^2*e^2 + 6*A*b^3*c^2*d^2*x*e + A*b^3*c^2*d^3 + B*b^5*x^2*e^
3 + A*b^4*c*x^2*e^3)/((c*x^2 + b*x)^2*b^4*c^2)