Optimal. Leaf size=144 \[ \frac{2 c^3 (c d-2 b e) \log (b+c x)}{b^3 (c d-b e)^3}-\frac{2 \log (x) (b e+c d)}{b^3 d^3}-\frac{c^3}{b^2 (b+c x) (c d-b e)^2}-\frac{1}{b^2 d^2 x}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3}-\frac{e^3}{d^2 (d+e x) (c d-b e)^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.393192, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 c^3 (c d-2 b e) \log (b+c x)}{b^3 (c d-b e)^3}-\frac{2 \log (x) (b e+c d)}{b^3 d^3}-\frac{c^3}{b^2 (b+c x) (c d-b e)^2}-\frac{1}{b^2 d^2 x}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3}-\frac{e^3}{d^2 (d+e x) (c d-b e)^2} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^2*(b*x + c*x^2)^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 54.4629, size = 133, normalized size = 0.92 \[ - \frac{e^{3}}{d^{2} \left (d + e x\right ) \left (b e - c d\right )^{2}} + \frac{2 e^{3} \left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{d^{3} \left (b e - c d\right )^{3}} - \frac{c^{3}}{b^{2} \left (b + c x\right ) \left (b e - c d\right )^{2}} - \frac{1}{b^{2} d^{2} x} + \frac{2 c^{3} \left (2 b e - c d\right ) \log{\left (b + c x \right )}}{b^{3} \left (b e - c d\right )^{3}} - \frac{2 \left (b e + c d\right ) \log{\left (x \right )}}{b^{3} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**2/(c*x**2+b*x)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.300735, size = 145, normalized size = 1.01 \[ \frac{2 c^3 (2 b e-c d) \log (b+c x)}{b^3 (b e-c d)^3}-\frac{2 \log (x) (b e+c d)}{b^3 d^3}-\frac{c^3}{b^2 (b+c x) (c d-b e)^2}-\frac{1}{b^2 d^2 x}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{d^3 (c d-b e)^3}-\frac{e^3}{d^2 (d+e x) (c d-b e)^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^2*(b*x + c*x^2)^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.026, size = 185, normalized size = 1.3 \[ -{\frac{1}{{b}^{2}{d}^{2}x}}-2\,{\frac{\ln \left ( x \right ) e}{{d}^{3}{b}^{2}}}-2\,{\frac{\ln \left ( x \right ) c}{{d}^{2}{b}^{3}}}-{\frac{{c}^{3}}{ \left ( be-cd \right ) ^{2}{b}^{2} \left ( cx+b \right ) }}+4\,{\frac{{c}^{3}\ln \left ( cx+b \right ) e}{ \left ( be-cd \right ) ^{3}{b}^{2}}}-2\,{\frac{{c}^{4}\ln \left ( cx+b \right ) d}{ \left ( be-cd \right ) ^{3}{b}^{3}}}-{\frac{{e}^{3}}{{d}^{2} \left ( be-cd \right ) ^{2} \left ( ex+d \right ) }}+2\,{\frac{{e}^{4}\ln \left ( ex+d \right ) b}{{d}^{3} \left ( be-cd \right ) ^{3}}}-4\,{\frac{{e}^{3}\ln \left ( ex+d \right ) c}{{d}^{2} \left ( be-cd \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^2/(c*x^2+b*x)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.713247, size = 504, normalized size = 3.5 \[ \frac{2 \,{\left (c^{4} d - 2 \, b c^{3} e\right )} \log \left (c x + b\right )}{b^{3} c^{3} d^{3} - 3 \, b^{4} c^{2} d^{2} e + 3 \, b^{5} c d e^{2} - b^{6} e^{3}} + \frac{2 \,{\left (2 \, c d e^{3} - b e^{4}\right )} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac{b c^{2} d^{3} - 2 \, b^{2} c d^{2} e + b^{3} d e^{2} + 2 \,{\left (c^{3} d^{2} e - b c^{2} d e^{2} + b^{2} c e^{3}\right )} x^{2} +{\left (2 \, c^{3} d^{3} - b c^{2} d^{2} e - b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} x}{{\left (b^{2} c^{3} d^{4} e - 2 \, b^{3} c^{2} d^{3} e^{2} + b^{4} c d^{2} e^{3}\right )} x^{3} +{\left (b^{2} c^{3} d^{5} - b^{3} c^{2} d^{4} e - b^{4} c d^{3} e^{2} + b^{5} d^{2} e^{3}\right )} x^{2} +{\left (b^{3} c^{2} d^{5} - 2 \, b^{4} c d^{4} e + b^{5} d^{3} e^{2}\right )} x} - \frac{2 \,{\left (c d + b e\right )} \log \left (x\right )}{b^{3} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x)^2*(e*x + d)^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 15.2738, size = 882, normalized size = 6.12 \[ -\frac{b^{2} c^{3} d^{5} - 3 \, b^{3} c^{2} d^{4} e + 3 \, b^{4} c d^{3} e^{2} - b^{5} d^{2} e^{3} + 2 \,{\left (b c^{4} d^{4} e - 2 \, b^{2} c^{3} d^{3} e^{2} + 2 \, b^{3} c^{2} d^{2} e^{3} - b^{4} c d e^{4}\right )} x^{2} +{\left (2 \, b c^{4} d^{5} - 3 \, b^{2} c^{3} d^{4} e + 3 \, b^{4} c d^{2} e^{3} - 2 \, b^{5} d e^{4}\right )} x - 2 \,{\left ({\left (c^{5} d^{4} e - 2 \, b c^{4} d^{3} e^{2}\right )} x^{3} +{\left (c^{5} d^{5} - b c^{4} d^{4} e - 2 \, b^{2} c^{3} d^{3} e^{2}\right )} x^{2} +{\left (b c^{4} d^{5} - 2 \, b^{2} c^{3} d^{4} e\right )} x\right )} \log \left (c x + b\right ) - 2 \,{\left ({\left (2 \, b^{3} c^{2} d e^{4} - b^{4} c e^{5}\right )} x^{3} +{\left (2 \, b^{3} c^{2} d^{2} e^{3} + b^{4} c d e^{4} - b^{5} e^{5}\right )} x^{2} +{\left (2 \, b^{4} c d^{2} e^{3} - b^{5} d e^{4}\right )} x\right )} \log \left (e x + d\right ) + 2 \,{\left ({\left (c^{5} d^{4} e - 2 \, b c^{4} d^{3} e^{2} + 2 \, b^{3} c^{2} d e^{4} - b^{4} c e^{5}\right )} x^{3} +{\left (c^{5} d^{5} - b c^{4} d^{4} e - 2 \, b^{2} c^{3} d^{3} e^{2} + 2 \, b^{3} c^{2} d^{2} e^{3} + b^{4} c d e^{4} - b^{5} e^{5}\right )} x^{2} +{\left (b c^{4} d^{5} - 2 \, b^{2} c^{3} d^{4} e + 2 \, b^{4} c d^{2} e^{3} - b^{5} d e^{4}\right )} x\right )} \log \left (x\right )}{{\left (b^{3} c^{4} d^{6} e - 3 \, b^{4} c^{3} d^{5} e^{2} + 3 \, b^{5} c^{2} d^{4} e^{3} - b^{6} c d^{3} e^{4}\right )} x^{3} +{\left (b^{3} c^{4} d^{7} - 2 \, b^{4} c^{3} d^{6} e + 2 \, b^{6} c d^{4} e^{3} - b^{7} d^{3} e^{4}\right )} x^{2} +{\left (b^{4} c^{3} d^{7} - 3 \, b^{5} c^{2} d^{6} e + 3 \, b^{6} c d^{5} e^{2} - b^{7} d^{4} e^{3}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x)^2*(e*x + d)^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)**2/(c*x**2+b*x)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.236203, size = 749, normalized size = 5.2 \[ \frac{{\left (2 \, c^{4} d^{4} e^{2} - 4 \, b c^{3} d^{3} e^{3} + 2 \, b^{3} c d e^{5} - b^{4} e^{6}\right )} e^{\left (-2\right )}{\rm ln}\left (\frac{{\left | -2 \, c d e + \frac{2 \, c d^{2} e}{x e + d} + b e^{2} - \frac{2 \, b d e^{2}}{x e + d} -{\left | b \right |} e^{2} \right |}}{{\left | -2 \, c d e + \frac{2 \, c d^{2} e}{x e + d} + b e^{2} - \frac{2 \, b d e^{2}}{x e + d} +{\left | b \right |} e^{2} \right |}}\right )}{{\left (b^{2} c^{3} d^{6} - 3 \, b^{3} c^{2} d^{5} e + 3 \, b^{4} c d^{4} e^{2} - b^{5} d^{3} e^{3}\right )}{\left | b \right |}} - \frac{{\left (2 \, c d e^{3} - b e^{4}\right )}{\rm ln}\left ({\left | -c + \frac{2 \, c d}{x e + d} - \frac{c d^{2}}{{\left (x e + d\right )}^{2}} - \frac{b e}{x e + d} + \frac{b d e}{{\left (x e + d\right )}^{2}} \right |}\right )}{c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}} - \frac{e^{7}}{{\left (c^{2} d^{4} e^{4} - 2 \, b c d^{3} e^{5} + b^{2} d^{2} e^{6}\right )}{\left (x e + d\right )}} - \frac{\frac{2 \, c^{4} d^{3} e - 3 \, b c^{3} d^{2} e^{2} + 3 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}}{c d^{2} - b d e} - \frac{{\left (2 \, c^{4} d^{4} e^{2} - 4 \, b c^{3} d^{3} e^{3} + 6 \, b^{2} c^{2} d^{2} e^{4} - 4 \, b^{3} c d e^{5} + b^{4} e^{6}\right )} e^{\left (-1\right )}}{{\left (c d^{2} - b d e\right )}{\left (x e + d\right )}}}{{\left (c d - b e\right )}^{2} b^{2}{\left (c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{b e}{x e + d} - \frac{b d e}{{\left (x e + d\right )}^{2}}\right )} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x)^2*(e*x + d)^2),x, algorithm="giac")
[Out]