3.1147 \(\int \frac{(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=128 \[ \frac{d^2 \log (x) (3 A b e-2 A c d+b B d)}{b^3}+\frac{(b B-A c) (c d-b e)^3}{b^2 c^3 (b+c x)}-\frac{A d^3}{b^2 x}+\frac{(c d-b e)^2 \log (b+c x) \left (-b c (B d-A e)+2 A c^2 d-2 b^2 B e\right )}{b^3 c^3}+\frac{B e^3 x}{c^2} \]

[Out]

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

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Rubi [A]  time = 0.369879, antiderivative size = 128, 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 \[ \frac{d^2 \log (x) (3 A b e-2 A c d+b B d)}{b^3}+\frac{(b B-A c) (c d-b e)^3}{b^2 c^3 (b+c x)}-\frac{A d^3}{b^2 x}-\frac{(c d-b e)^2 \log (b+c x) \left (-A b c e-2 A c^2 d+2 b^2 B e+b B c d\right )}{b^3 c^3}+\frac{B e^3 x}{c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

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)**2,x)

[Out]

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

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Mathematica [A]  time = 0.12737, size = 128, normalized size = 1. \[ \frac{d^2 \log (x) (3 A b e-2 A c d+b B d)}{b^3}-\frac{(b B-A c) (b e-c d)^3}{b^2 c^3 (b+c x)}-\frac{A d^3}{b^2 x}+\frac{(c d-b e)^2 \log (b+c x) \left (A b c e+2 A c^2 d-2 b^2 B e-b B c d\right )}{b^3 c^3}+\frac{B e^3 x}{c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.021, size = 286, normalized size = 2.2 \[{\frac{B{e}^{3}x}{{c}^{2}}}-{\frac{A{d}^{3}}{{b}^{2}x}}+3\,{\frac{{d}^{2}\ln \left ( x \right ) Ae}{{b}^{2}}}-2\,{\frac{{d}^{3}\ln \left ( x \right ) Ac}{{b}^{3}}}+{\frac{{d}^{3}\ln \left ( x \right ) B}{{b}^{2}}}+{\frac{\ln \left ( cx+b \right ) A{e}^{3}}{{c}^{2}}}-3\,{\frac{\ln \left ( cx+b \right ) A{d}^{2}e}{{b}^{2}}}+2\,{\frac{c\ln \left ( cx+b \right ) A{d}^{3}}{{b}^{3}}}-2\,{\frac{b\ln \left ( cx+b \right ) B{e}^{3}}{{c}^{3}}}+3\,{\frac{\ln \left ( cx+b \right ) Bd{e}^{2}}{{c}^{2}}}-{\frac{\ln \left ( cx+b \right ) B{d}^{3}}{{b}^{2}}}+{\frac{Ab{e}^{3}}{{c}^{2} \left ( cx+b \right ) }}-3\,{\frac{dA{e}^{2}}{c \left ( cx+b \right ) }}+3\,{\frac{A{d}^{2}e}{b \left ( cx+b \right ) }}-{\frac{A{d}^{3}c}{{b}^{2} \left ( cx+b \right ) }}-{\frac{B{e}^{3}{b}^{2}}{{c}^{3} \left ( cx+b \right ) }}+3\,{\frac{Bbd{e}^{2}}{{c}^{2} \left ( cx+b \right ) }}-3\,{\frac{B{d}^{2}e}{c \left ( cx+b \right ) }}+{\frac{B{d}^{3}}{b \left ( cx+b \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.71803, size = 304, normalized size = 2.38 \[ \frac{B e^{3} x}{c^{2}} - \frac{A b c^{3} d^{3} -{\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - 3 \,{\left (B b^{2} c^{2} - A b c^{3}\right )} d^{2} e + 3 \,{\left (B b^{3} c - A b^{2} c^{2}\right )} d e^{2} -{\left (B b^{4} - A b^{3} c\right )} e^{3}\right )} x}{b^{2} c^{4} x^{2} + b^{3} c^{3} x} + \frac{{\left (3 \, A b d^{2} e +{\left (B b - 2 \, A c\right )} d^{3}\right )} \log \left (x\right )}{b^{3}} - \frac{{\left (3 \, A b c^{3} d^{2} e - 3 \, B b^{3} c d e^{2} +{\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} +{\left (2 \, B b^{4} - A b^{3} c\right )} e^{3}\right )} \log \left (c x + b\right )}{b^{3} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.306357, size = 489, normalized size = 3.82 \[ \frac{B b^{3} c^{2} e^{3} x^{3} + B b^{4} c e^{3} x^{2} - A b^{2} c^{3} d^{3} +{\left ({\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} - 3 \,{\left (B b^{3} c^{2} - A b^{2} c^{3}\right )} d^{2} e + 3 \,{\left (B b^{4} c - A b^{3} c^{2}\right )} d e^{2} -{\left (B b^{5} - A b^{4} c\right )} e^{3}\right )} x -{\left ({\left (3 \, A b c^{4} d^{2} e - 3 \, B b^{3} c^{2} d e^{2} +{\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} +{\left (2 \, B b^{4} c - A b^{3} c^{2}\right )} e^{3}\right )} x^{2} +{\left (3 \, A b^{2} c^{3} d^{2} e - 3 \, B b^{4} c d e^{2} +{\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} +{\left (2 \, B b^{5} - A b^{4} c\right )} e^{3}\right )} x\right )} \log \left (c x + b\right ) +{\left ({\left (3 \, A b c^{4} d^{2} e +{\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3}\right )} x^{2} +{\left (3 \, A b^{2} c^{3} d^{2} e +{\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3}\right )} x\right )} \log \left (x\right )}{b^{3} c^{4} x^{2} + b^{4} c^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 39.1027, size = 502, normalized size = 3.92 \[ \frac{B e^{3} x}{c^{2}} - \frac{A b c^{3} d^{3} + x \left (- A b^{3} c e^{3} + 3 A b^{2} c^{2} d e^{2} - 3 A b c^{3} d^{2} e + 2 A c^{4} d^{3} + B b^{4} e^{3} - 3 B b^{3} c d e^{2} + 3 B b^{2} c^{2} d^{2} e - B b c^{3} d^{3}\right )}{b^{3} c^{3} x + b^{2} c^{4} x^{2}} + \frac{d^{2} \left (3 A b e - 2 A c d + B b d\right ) \log{\left (x + \frac{3 A b^{2} c^{2} d^{2} e - 2 A b c^{3} d^{3} + B b^{2} c^{2} d^{3} - b c^{2} d^{2} \left (3 A b e - 2 A c d + B b d\right )}{- A b^{3} c e^{3} + 6 A b c^{3} d^{2} e - 4 A c^{4} d^{3} + 2 B b^{4} e^{3} - 3 B b^{3} c d e^{2} + 2 B b c^{3} d^{3}} \right )}}{b^{3}} - \frac{\left (b e - c d\right )^{2} \left (- A b c e - 2 A c^{2} d + 2 B b^{2} e + B b c d\right ) \log{\left (x + \frac{3 A b^{2} c^{2} d^{2} e - 2 A b c^{3} d^{3} + B b^{2} c^{2} d^{3} + \frac{b \left (b e - c d\right )^{2} \left (- A b c e - 2 A c^{2} d + 2 B b^{2} e + B b c d\right )}{c}}{- A b^{3} c e^{3} + 6 A b c^{3} d^{2} e - 4 A c^{4} d^{3} + 2 B b^{4} e^{3} - 3 B b^{3} c d e^{2} + 2 B b c^{3} d^{3}} \right )}}{b^{3} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.280868, size = 309, normalized size = 2.41 \[ \frac{B x e^{3}}{c^{2}} + \frac{{\left (B b d^{3} - 2 \, A c d^{3} + 3 \, A b d^{2} e\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} - \frac{{\left (B b c^{3} d^{3} - 2 \, A c^{4} d^{3} + 3 \, A b c^{3} d^{2} e - 3 \, B b^{3} c d e^{2} + 2 \, B b^{4} e^{3} - A b^{3} c e^{3}\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{3} c^{3}} - \frac{A b c^{2} d^{3} - \frac{{\left (B b c^{3} d^{3} - 2 \, A c^{4} d^{3} - 3 \, B b^{2} c^{2} d^{2} e + 3 \, A b c^{3} d^{2} e + 3 \, B b^{3} c d e^{2} - 3 \, A b^{2} c^{2} d e^{2} - B b^{4} e^{3} + A b^{3} c e^{3}\right )} x}{c}}{{\left (c x + b\right )} b^{2} c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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