3.1146 \(\int \frac{(A+B x) (d+e x)^4}{\left (b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=156 \[ \frac{d^3 \log (x) (4 A b e-2 A c d+b B d)}{b^3}+\frac{(b B-A c) (c d-b e)^4}{b^2 c^4 (b+c x)}-\frac{A d^4}{b^2 x}+\frac{(c d-b e)^3 \log (b+c x) \left (-b c (B d-2 A e)+2 A c^2 d-3 b^2 B e\right )}{b^3 c^4}+\frac{e^3 x (A c e-2 b B e+4 B c d)}{c^3}+\frac{B e^4 x^2}{2 c^2} \]

[Out]

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

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Rubi [A]  time = 0.472983, antiderivative size = 156, 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 \[ \frac{d^3 \log (x) (4 A b e-2 A c d+b B d)}{b^3}+\frac{(b B-A c) (c d-b e)^4}{b^2 c^4 (b+c x)}-\frac{A d^4}{b^2 x}+\frac{(c d-b e)^3 \log (b+c x) \left (-b c (B d-2 A e)+2 A c^2 d-3 b^2 B e\right )}{b^3 c^4}+\frac{e^3 x (A c e-2 b B e+4 B c d)}{c^3}+\frac{B e^4 x^2}{2 c^2} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^4)/(b*x + c*x^2)^2,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A d^{4}}{b^{2} x} + \frac{B e^{4} \int x\, dx}{c^{2}} + e^{3} \left (A c e - 2 B b e + 4 B c d\right ) \int \frac{1}{c^{3}}\, dx - \frac{\left (A c - B b\right ) \left (b e - c d\right )^{4}}{b^{2} c^{4} \left (b + c x\right )} + \frac{d^{3} \left (4 A b e - 2 A c d + B b d\right ) \log{\left (x \right )}}{b^{3}} - \frac{\left (b e - c d\right )^{3} \left (2 A b c e + 2 A c^{2} d - 3 B b^{2} e - B b c d\right ) \log{\left (b + c x \right )}}{b^{3} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**4/(c*x**2+b*x)**2,x)

[Out]

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

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Mathematica [A]  time = 0.15492, size = 155, normalized size = 0.99 \[ \frac{d^3 \log (x) (4 A b e-2 A c d+b B d)}{b^3}+\frac{(b B-A c) (c d-b e)^4}{b^2 c^4 (b+c x)}-\frac{A d^4}{b^2 x}+\frac{(b e-c d)^3 \log (b+c x) \left (b c (B d-2 A e)-2 A c^2 d+3 b^2 B e\right )}{b^3 c^4}+\frac{e^3 x (A c e-2 b B e+4 B c d)}{c^3}+\frac{B e^4 x^2}{2 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^4)/(b*x + c*x^2)^2,x]

[Out]

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

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Maple [B]  time = 0.026, size = 403, normalized size = 2.6 \[ -8\,{\frac{b\ln \left ( cx+b \right ) Bd{e}^{3}}{{c}^{3}}}+4\,{\frac{Adb{e}^{3}}{{c}^{2} \left ( cx+b \right ) }}-4\,{\frac{Bd{b}^{2}{e}^{3}}{{c}^{3} \left ( cx+b \right ) }}+6\,{\frac{Bb{d}^{2}{e}^{2}}{{c}^{2} \left ( cx+b \right ) }}-2\,{\frac{b\ln \left ( cx+b \right ) A{e}^{4}}{{c}^{3}}}+4\,{\frac{\ln \left ( cx+b \right ) Ad{e}^{3}}{{c}^{2}}}-4\,{\frac{\ln \left ( cx+b \right ) A{d}^{3}e}{{b}^{2}}}+2\,{\frac{c\ln \left ( cx+b \right ) A{d}^{4}}{{b}^{3}}}+3\,{\frac{{b}^{2}\ln \left ( cx+b \right ) B{e}^{4}}{{c}^{4}}}+6\,{\frac{\ln \left ( cx+b \right ) B{d}^{2}{e}^{2}}{{c}^{2}}}-{\frac{{b}^{2}A{e}^{4}}{{c}^{3} \left ( cx+b \right ) }}-{\frac{A{d}^{4}c}{{b}^{2} \left ( cx+b \right ) }}+{\frac{B{e}^{4}{b}^{3}}{{c}^{4} \left ( cx+b \right ) }}-2\,{\frac{B{e}^{4}bx}{{c}^{3}}}+4\,{\frac{{e}^{3}Bdx}{{c}^{2}}}+4\,{\frac{{d}^{3}\ln \left ( x \right ) Ae}{{b}^{2}}}-2\,{\frac{{d}^{4}\ln \left ( x \right ) Ac}{{b}^{3}}}-6\,{\frac{A{d}^{2}{e}^{2}}{c \left ( cx+b \right ) }}+4\,{\frac{A{d}^{3}e}{b \left ( cx+b \right ) }}-4\,{\frac{B{d}^{3}e}{c \left ( cx+b \right ) }}-{\frac{\ln \left ( cx+b \right ) B{d}^{4}}{{b}^{2}}}-{\frac{A{d}^{4}}{{b}^{2}x}}+{\frac{B{e}^{4}{x}^{2}}{2\,{c}^{2}}}+{\frac{B{d}^{4}}{b \left ( cx+b \right ) }}+{\frac{{e}^{4}Ax}{{c}^{2}}}+{\frac{{d}^{4}\ln \left ( x \right ) B}{{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^4/(c*x^2+b*x)^2,x)

[Out]

-8/c^3*b*ln(c*x+b)*B*d*e^3+4/c^2*b/(c*x+b)*A*d*e^3-4/c^3*b^2/(c*x+b)*B*d*e^3+6/c
^2*b/(c*x+b)*B*d^2*e^2-2/c^3*b*ln(c*x+b)*A*e^4+4/c^2*ln(c*x+b)*A*d*e^3-4/b^2*ln(
c*x+b)*A*d^3*e+2*c/b^3*ln(c*x+b)*A*d^4+3/c^4*b^2*ln(c*x+b)*B*e^4+6/c^2*ln(c*x+b)
*B*d^2*e^2-1/c^3*b^2/(c*x+b)*A*e^4-c/b^2/(c*x+b)*A*d^4+1/c^4*b^3/(c*x+b)*B*e^4-2
*e^4/c^3*b*B*x+4*e^3/c^2*B*d*x+4*d^3/b^2*ln(x)*A*e-2*d^4/b^3*ln(x)*A*c-6/c/(c*x+
b)*A*d^2*e^2+4/b/(c*x+b)*A*d^3*e-4/c/(c*x+b)*B*d^3*e-1/b^2*ln(c*x+b)*B*d^4-A*d^4
/b^2/x+1/2*B*e^4*x^2/c^2+1/b/(c*x+b)*B*d^4+e^4/c^2*A*x+d^4/b^2*ln(x)*B

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Maxima [A]  time = 0.728461, size = 419, normalized size = 2.69 \[ -\frac{A b c^{4} d^{4} -{\left ({\left (B b c^{4} - 2 \, A c^{5}\right )} d^{4} - 4 \,{\left (B b^{2} c^{3} - A b c^{4}\right )} d^{3} e + 6 \,{\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} d^{2} e^{2} - 4 \,{\left (B b^{4} c - A b^{3} c^{2}\right )} d e^{3} +{\left (B b^{5} - A b^{4} c\right )} e^{4}\right )} x}{b^{2} c^{5} x^{2} + b^{3} c^{4} x} + \frac{{\left (4 \, A b d^{3} e +{\left (B b - 2 \, A c\right )} d^{4}\right )} \log \left (x\right )}{b^{3}} + \frac{B c e^{4} x^{2} + 2 \,{\left (4 \, B c d e^{3} -{\left (2 \, B b - A c\right )} e^{4}\right )} x}{2 \, c^{3}} - \frac{{\left (4 \, A b c^{4} d^{3} e - 6 \, B b^{3} c^{2} d^{2} e^{2} +{\left (B b c^{4} - 2 \, A c^{5}\right )} d^{4} + 4 \,{\left (2 \, B b^{4} c - A b^{3} c^{2}\right )} d e^{3} -{\left (3 \, B b^{5} - 2 \, A b^{4} c\right )} e^{4}\right )} \log \left (c x + b\right )}{b^{3} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^4/(c*x^2 + b*x)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.300094, size = 698, normalized size = 4.47 \[ \frac{B b^{3} c^{3} e^{4} x^{4} - 2 \, A b^{2} c^{4} d^{4} +{\left (8 \, B b^{3} c^{3} d e^{3} -{\left (3 \, B b^{4} c^{2} - 2 \, A b^{3} c^{3}\right )} e^{4}\right )} x^{3} + 2 \,{\left (4 \, B b^{4} c^{2} d e^{3} -{\left (2 \, B b^{5} c - A b^{4} c^{2}\right )} e^{4}\right )} x^{2} + 2 \,{\left ({\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{4} - 4 \,{\left (B b^{3} c^{3} - A b^{2} c^{4}\right )} d^{3} e + 6 \,{\left (B b^{4} c^{2} - A b^{3} c^{3}\right )} d^{2} e^{2} - 4 \,{\left (B b^{5} c - A b^{4} c^{2}\right )} d e^{3} +{\left (B b^{6} - A b^{5} c\right )} e^{4}\right )} x - 2 \,{\left ({\left (4 \, A b c^{5} d^{3} e - 6 \, B b^{3} c^{3} d^{2} e^{2} +{\left (B b c^{5} - 2 \, A c^{6}\right )} d^{4} + 4 \,{\left (2 \, B b^{4} c^{2} - A b^{3} c^{3}\right )} d e^{3} -{\left (3 \, B b^{5} c - 2 \, A b^{4} c^{2}\right )} e^{4}\right )} x^{2} +{\left (4 \, A b^{2} c^{4} d^{3} e - 6 \, B b^{4} c^{2} d^{2} e^{2} +{\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{4} + 4 \,{\left (2 \, B b^{5} c - A b^{4} c^{2}\right )} d e^{3} -{\left (3 \, B b^{6} - 2 \, A b^{5} c\right )} e^{4}\right )} x\right )} \log \left (c x + b\right ) + 2 \,{\left ({\left (4 \, A b c^{5} d^{3} e +{\left (B b c^{5} - 2 \, A c^{6}\right )} d^{4}\right )} x^{2} +{\left (4 \, A b^{2} c^{4} d^{3} e +{\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{4}\right )} x\right )} \log \left (x\right )}{2 \,{\left (b^{3} c^{5} x^{2} + b^{4} c^{4} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^4/(c*x^2 + b*x)^2,x, algorithm="fricas")

[Out]

1/2*(B*b^3*c^3*e^4*x^4 - 2*A*b^2*c^4*d^4 + (8*B*b^3*c^3*d*e^3 - (3*B*b^4*c^2 - 2
*A*b^3*c^3)*e^4)*x^3 + 2*(4*B*b^4*c^2*d*e^3 - (2*B*b^5*c - A*b^4*c^2)*e^4)*x^2 +
2*((B*b^2*c^4 - 2*A*b*c^5)*d^4 - 4*(B*b^3*c^3 - A*b^2*c^4)*d^3*e + 6*(B*b^4*c^2
- A*b^3*c^3)*d^2*e^2 - 4*(B*b^5*c - A*b^4*c^2)*d*e^3 + (B*b^6 - A*b^5*c)*e^4)*x
- 2*((4*A*b*c^5*d^3*e - 6*B*b^3*c^3*d^2*e^2 + (B*b*c^5 - 2*A*c^6)*d^4 + 4*(2*B*
b^4*c^2 - A*b^3*c^3)*d*e^3 - (3*B*b^5*c - 2*A*b^4*c^2)*e^4)*x^2 + (4*A*b^2*c^4*d
^3*e - 6*B*b^4*c^2*d^2*e^2 + (B*b^2*c^4 - 2*A*b*c^5)*d^4 + 4*(2*B*b^5*c - A*b^4*
c^2)*d*e^3 - (3*B*b^6 - 2*A*b^5*c)*e^4)*x)*log(c*x + b) + 2*((4*A*b*c^5*d^3*e +
(B*b*c^5 - 2*A*c^6)*d^4)*x^2 + (4*A*b^2*c^4*d^3*e + (B*b^2*c^4 - 2*A*b*c^5)*d^4)
*x)*log(x))/(b^3*c^5*x^2 + b^4*c^4*x)

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Sympy [A]  time = 64.1063, size = 641, normalized size = 4.11 \[ \frac{B e^{4} x^{2}}{2 c^{2}} + \frac{- A b c^{4} d^{4} + x \left (- A b^{4} c e^{4} + 4 A b^{3} c^{2} d e^{3} - 6 A b^{2} c^{3} d^{2} e^{2} + 4 A b c^{4} d^{3} e - 2 A c^{5} d^{4} + B b^{5} e^{4} - 4 B b^{4} c d e^{3} + 6 B b^{3} c^{2} d^{2} e^{2} - 4 B b^{2} c^{3} d^{3} e + B b c^{4} d^{4}\right )}{b^{3} c^{4} x + b^{2} c^{5} x^{2}} - \frac{x \left (- A c e^{4} + 2 B b e^{4} - 4 B c d e^{3}\right )}{c^{3}} + \frac{d^{3} \left (4 A b e - 2 A c d + B b d\right ) \log{\left (x + \frac{- 4 A b^{2} c^{3} d^{3} e + 2 A b c^{4} d^{4} - B b^{2} c^{3} d^{4} + b c^{3} d^{3} \left (4 A b e - 2 A c d + B b d\right )}{- 2 A b^{4} c e^{4} + 4 A b^{3} c^{2} d e^{3} - 8 A b c^{4} d^{3} e + 4 A c^{5} d^{4} + 3 B b^{5} e^{4} - 8 B b^{4} c d e^{3} + 6 B b^{3} c^{2} d^{2} e^{2} - 2 B b c^{4} d^{4}} \right )}}{b^{3}} + \frac{\left (b e - c d\right )^{3} \left (- 2 A b c e - 2 A c^{2} d + 3 B b^{2} e + B b c d\right ) \log{\left (x + \frac{- 4 A b^{2} c^{3} d^{3} e + 2 A b c^{4} d^{4} - B b^{2} c^{3} d^{4} + \frac{b \left (b e - c d\right )^{3} \left (- 2 A b c e - 2 A c^{2} d + 3 B b^{2} e + B b c d\right )}{c}}{- 2 A b^{4} c e^{4} + 4 A b^{3} c^{2} d e^{3} - 8 A b c^{4} d^{3} e + 4 A c^{5} d^{4} + 3 B b^{5} e^{4} - 8 B b^{4} c d e^{3} + 6 B b^{3} c^{2} d^{2} e^{2} - 2 B b c^{4} d^{4}} \right )}}{b^{3} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**4/(c*x**2+b*x)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.281587, size = 425, normalized size = 2.72 \[ \frac{{\left (B b d^{4} - 2 \, A c d^{4} + 4 \, A b d^{3} e\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} + \frac{B c^{2} x^{2} e^{4} + 8 \, B c^{2} d x e^{3} - 4 \, B b c x e^{4} + 2 \, A c^{2} x e^{4}}{2 \, c^{4}} - \frac{{\left (B b c^{4} d^{4} - 2 \, A c^{5} d^{4} + 4 \, A b c^{4} d^{3} e - 6 \, B b^{3} c^{2} d^{2} e^{2} + 8 \, B b^{4} c d e^{3} - 4 \, A b^{3} c^{2} d e^{3} - 3 \, B b^{5} e^{4} + 2 \, A b^{4} c e^{4}\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{3} c^{4}} - \frac{A b c^{4} d^{4} -{\left (B b c^{4} d^{4} - 2 \, A c^{5} d^{4} - 4 \, B b^{2} c^{3} d^{3} e + 4 \, A b c^{4} d^{3} e + 6 \, B b^{3} c^{2} d^{2} e^{2} - 6 \, A b^{2} c^{3} d^{2} e^{2} - 4 \, B b^{4} c d e^{3} + 4 \, A b^{3} c^{2} d e^{3} + B b^{5} e^{4} - A b^{4} c e^{4}\right )} x}{{\left (c x + b\right )} b^{2} c^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^4/(c*x^2 + b*x)^2,x, algorithm="giac")

[Out]

(B*b*d^4 - 2*A*c*d^4 + 4*A*b*d^3*e)*ln(abs(x))/b^3 + 1/2*(B*c^2*x^2*e^4 + 8*B*c^
2*d*x*e^3 - 4*B*b*c*x*e^4 + 2*A*c^2*x*e^4)/c^4 - (B*b*c^4*d^4 - 2*A*c^5*d^4 + 4*
A*b*c^4*d^3*e - 6*B*b^3*c^2*d^2*e^2 + 8*B*b^4*c*d*e^3 - 4*A*b^3*c^2*d*e^3 - 3*B*
b^5*e^4 + 2*A*b^4*c*e^4)*ln(abs(c*x + b))/(b^3*c^4) - (A*b*c^4*d^4 - (B*b*c^4*d^
4 - 2*A*c^5*d^4 - 4*B*b^2*c^3*d^3*e + 4*A*b*c^4*d^3*e + 6*B*b^3*c^2*d^2*e^2 - 6*
A*b^2*c^3*d^2*e^2 - 4*B*b^4*c*d*e^3 + 4*A*b^3*c^2*d*e^3 + B*b^5*e^4 - A*b^4*c*e^
4)*x)/((c*x + b)*b^2*c^4*x)