Optimal. Leaf size=87 \[ -\frac{c^2 \log (b+c x)}{b (c d-b e)^2}+\frac{e (2 c d-b e) \log (d+e x)}{d^2 (c d-b e)^2}-\frac{e}{d (d+e x) (c d-b e)}+\frac{\log (x)}{b d^2} \]
[Out]
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Rubi [A] time = 0.192499, antiderivative size = 87, 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 \[ -\frac{c^2 \log (b+c x)}{b (c d-b e)^2}+\frac{e (2 c d-b e) \log (d+e x)}{d^2 (c d-b e)^2}-\frac{e}{d (d+e x) (c d-b e)}+\frac{\log (x)}{b d^2} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^2*(b*x + c*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 28.7541, size = 71, normalized size = 0.82 \[ \frac{e}{d \left (d + e x\right ) \left (b e - c d\right )} - \frac{e \left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{d^{2} \left (b e - c d\right )^{2}} - \frac{c^{2} \log{\left (b + c x \right )}}{b \left (b e - c d\right )^{2}} + \frac{\log{\left (x \right )}}{b d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**2/(c*x**2+b*x),x)
[Out]
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Mathematica [A] time = 0.169706, size = 83, normalized size = 0.95 \[ \frac{\frac{b e ((d+e x) (2 c d-b e) \log (d+e x)+d (b e-c d))-c^2 d^2 (d+e x) \log (b+c x)}{(d+e x) (c d-b e)^2}+\log (x)}{b d^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^2*(b*x + c*x^2)),x]
[Out]
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Maple [A] time = 0.027, size = 105, normalized size = 1.2 \[{\frac{\ln \left ( x \right ) }{{d}^{2}b}}-{\frac{{c}^{2}\ln \left ( cx+b \right ) }{ \left ( be-cd \right ) ^{2}b}}+{\frac{e}{d \left ( be-cd \right ) \left ( ex+d \right ) }}-{\frac{{e}^{2}\ln \left ( ex+d \right ) b}{{d}^{2} \left ( be-cd \right ) ^{2}}}+2\,{\frac{e\ln \left ( ex+d \right ) c}{d \left ( be-cd \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^2/(c*x^2+b*x),x)
[Out]
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Maxima [A] time = 0.701009, size = 173, normalized size = 1.99 \[ -\frac{c^{2} \log \left (c x + b\right )}{b c^{2} d^{2} - 2 \, b^{2} c d e + b^{3} e^{2}} + \frac{{\left (2 \, c d e - b e^{2}\right )} \log \left (e x + d\right )}{c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}} - \frac{e}{c d^{3} - b d^{2} e +{\left (c d^{2} e - b d e^{2}\right )} x} + \frac{\log \left (x\right )}{b d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x)*(e*x + d)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 1.03931, size = 279, normalized size = 3.21 \[ -\frac{b c d^{2} e - b^{2} d e^{2} +{\left (c^{2} d^{2} e x + c^{2} d^{3}\right )} \log \left (c x + b\right ) -{\left (2 \, b c d^{2} e - b^{2} d e^{2} +{\left (2 \, b c d e^{2} - b^{2} e^{3}\right )} x\right )} \log \left (e x + d\right ) -{\left (c^{2} d^{3} - 2 \, b c d^{2} e + b^{2} d e^{2} +{\left (c^{2} d^{2} e - 2 \, b c d e^{2} + b^{2} e^{3}\right )} x\right )} \log \left (x\right )}{b c^{2} d^{5} - 2 \, b^{2} c d^{4} e + b^{3} d^{3} e^{2} +{\left (b c^{2} d^{4} e - 2 \, b^{2} c d^{3} e^{2} + b^{3} d^{2} e^{3}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x)*(e*x + d)^2),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)**2/(c*x**2+b*x),x)
[Out]
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GIAC/XCAS [A] time = 0.228496, size = 389, normalized size = 4.47 \[ -\frac{{\left (2 \, c^{2} d^{2} e^{2} - 2 \, b c d e^{3} + b^{2} e^{4}\right )} e^{\left (-2\right )}{\rm ln}\left (\frac{{\left | -2 \, c d e + \frac{2 \, c d^{2} e}{x e + d} + b e^{2} - \frac{2 \, b d e^{2}}{x e + d} -{\left | b \right |} e^{2} \right |}}{{\left | -2 \, c d e + \frac{2 \, c d^{2} e}{x e + d} + b e^{2} - \frac{2 \, b d e^{2}}{x e + d} +{\left | b \right |} e^{2} \right |}}\right )}{2 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )}{\left | b \right |}} - \frac{{\left (2 \, c d e - b e^{2}\right )}{\rm ln}\left ({\left | -c + \frac{2 \, c d}{x e + d} - \frac{c d^{2}}{{\left (x e + d\right )}^{2}} - \frac{b e}{x e + d} + \frac{b d e}{{\left (x e + d\right )}^{2}} \right |}\right )}{2 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )}} - \frac{e^{3}}{{\left (c d^{2} e^{2} - b d e^{3}\right )}{\left (x e + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2 + b*x)*(e*x + d)^2),x, algorithm="giac")
[Out]