Optimal. Leaf size=110 \[ \frac{c^2 (2 c d-3 b e) \log (b+c x)}{b^3 (c d-b e)^2}-\frac{\log (x) (b e+2 c d)}{b^3 d^2}-\frac{c^2}{b^2 (b+c x) (c d-b e)}-\frac{1}{b^2 d x}+\frac{e^3 \log (d+e x)}{d^2 (c d-b e)^2} \]
[Out]
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Rubi [A] time = 0.281047, antiderivative size = 110, 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 \[ \frac{c^2 (2 c d-3 b e) \log (b+c x)}{b^3 (c d-b e)^2}-\frac{\log (x) (b e+2 c d)}{b^3 d^2}-\frac{c^2}{b^2 (b+c x) (c d-b e)}-\frac{1}{b^2 d x}+\frac{e^3 \log (d+e x)}{d^2 (c d-b e)^2} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)*(b*x + c*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 34.4933, size = 99, normalized size = 0.9 \[ \frac{e^{3} \log{\left (d + e x \right )}}{d^{2} \left (b e - c d\right )^{2}} + \frac{c^{2}}{b^{2} \left (b + c x\right ) \left (b e - c d\right )} - \frac{1}{b^{2} d x} - \frac{c^{2} \left (3 b e - 2 c d\right ) \log{\left (b + c x \right )}}{b^{3} \left (b e - c d\right )^{2}} - \frac{\left (b e + 2 c d\right ) \log{\left (x \right )}}{b^{3} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)/(c*x**2+b*x)**2,x)
[Out]
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Mathematica [A] time = 0.162733, size = 111, normalized size = 1.01 \[ \frac{\left (2 c^3 d-3 b c^2 e\right ) \log (b+c x)}{b^3 (b e-c d)^2}+\frac{\log (x) (-b e-2 c d)}{b^3 d^2}+\frac{c^2}{b^2 (b+c x) (b e-c d)}-\frac{1}{b^2 d x}+\frac{e^3 \log (d+e x)}{d^2 (c d-b e)^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)*(b*x + c*x^2)^2),x]
[Out]
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Maple [A] time = 0.03, size = 132, normalized size = 1.2 \[ -{\frac{1}{{b}^{2}dx}}-{\frac{\ln \left ( x \right ) e}{{b}^{2}{d}^{2}}}-2\,{\frac{\ln \left ( x \right ) c}{d{b}^{3}}}+{\frac{{c}^{2}}{ \left ( be-cd \right ){b}^{2} \left ( cx+b \right ) }}-3\,{\frac{{c}^{2}\ln \left ( cx+b \right ) e}{ \left ( be-cd \right ) ^{2}{b}^{2}}}+2\,{\frac{{c}^{3}\ln \left ( cx+b \right ) d}{ \left ( be-cd \right ) ^{2}{b}^{3}}}+{\frac{{e}^{3}\ln \left ( ex+d \right ) }{{d}^{2} \left ( be-cd \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)/(c*x^2+b*x)^2,x)
[Out]
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Maxima [A] time = 0.697288, size = 239, normalized size = 2.17 \[ \frac{e^{3} \log \left (e x + d\right )}{c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}} + \frac{{\left (2 \, c^{3} d - 3 \, b c^{2} e\right )} \log \left (c x + b\right )}{b^{3} c^{2} d^{2} - 2 \, b^{4} c d e + b^{5} e^{2}} - \frac{b c d - b^{2} e +{\left (2 \, c^{2} d - b c e\right )} x}{{\left (b^{2} c^{2} d^{2} - b^{3} c d e\right )} x^{2} +{\left (b^{3} c d^{2} - b^{4} d e\right )} x} - \frac{{\left (2 \, c d + b e\right )} \log \left (x\right )}{b^{3} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x)^2*(e*x + d)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 4.48143, size = 389, normalized size = 3.54 \[ -\frac{b^{2} c^{2} d^{3} - 2 \, b^{3} c d^{2} e + b^{4} d e^{2} +{\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + b^{3} c d e^{2}\right )} x -{\left ({\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e\right )} x^{2} +{\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e\right )} x\right )} \log \left (c x + b\right ) -{\left (b^{3} c e^{3} x^{2} + b^{4} e^{3} x\right )} \log \left (e x + d\right ) +{\left ({\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e + b^{3} c e^{3}\right )} x^{2} +{\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + b^{4} e^{3}\right )} x\right )} \log \left (x\right )}{{\left (b^{3} c^{3} d^{4} - 2 \, b^{4} c^{2} d^{3} e + b^{5} c d^{2} e^{2}\right )} x^{2} +{\left (b^{4} c^{2} d^{4} - 2 \, b^{5} c d^{3} e + b^{6} d^{2} e^{2}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x)^2*(e*x + d)),x, algorithm="fricas")
[Out]
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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)/(c*x**2+b*x)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.210172, size = 273, normalized size = 2.48 \[ \frac{{\left (2 \, c^{4} d - 3 \, b c^{3} e\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{3} c^{3} d^{2} - 2 \, b^{4} c^{2} d e + b^{5} c e^{2}} + \frac{e^{4}{\rm ln}\left ({\left | x e + d \right |}\right )}{c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3}} - \frac{{\left (2 \, c d + b e\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{3} d^{2}} - \frac{b c^{2} d^{3} - 2 \, b^{2} c d^{2} e + b^{3} d e^{2} +{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + b^{2} c d e^{2}\right )} x}{{\left (c d - b e\right )}^{2}{\left (c x + b\right )} b^{2} d^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x)^2*(e*x + d)),x, algorithm="giac")
[Out]