Optimal. Leaf size=65 \[ -\frac{\log (x) (2 c d-b e)}{b^3}+\frac{(2 c d-b e) \log (b+c x)}{b^3}-\frac{c d-b e}{b^2 (b+c x)}-\frac{d}{b^2 x} \]
[Out]
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Rubi [A] time = 0.140696, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{\log (x) (2 c d-b e)}{b^3}+\frac{(2 c d-b e) \log (b+c x)}{b^3}-\frac{c d-b e}{b^2 (b+c x)}-\frac{d}{b^2 x} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 16.0259, size = 54, normalized size = 0.83 \[ - \frac{d}{b^{2} x} + \frac{b e - c d}{b^{2} \left (b + c x\right )} + \frac{\left (b e - 2 c d\right ) \log{\left (x \right )}}{b^{3}} - \frac{\left (b e - 2 c d\right ) \log{\left (b + c x \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/(c*x**2+b*x)**2,x)
[Out]
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Mathematica [A] time = 0.0692546, size = 56, normalized size = 0.86 \[ \frac{\frac{b (b e-c d)}{b+c x}+\log (x) (b e-2 c d)+(2 c d-b e) \log (b+c x)-\frac{b d}{x}}{b^3} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(b*x + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.015, size = 78, normalized size = 1.2 \[ -{\frac{d}{{b}^{2}x}}+{\frac{\ln \left ( x \right ) e}{{b}^{2}}}-2\,{\frac{\ln \left ( x \right ) cd}{{b}^{3}}}-{\frac{\ln \left ( cx+b \right ) e}{{b}^{2}}}+2\,{\frac{\ln \left ( cx+b \right ) cd}{{b}^{3}}}+{\frac{e}{b \left ( cx+b \right ) }}-{\frac{cd}{{b}^{2} \left ( cx+b \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/(c*x^2+b*x)^2,x)
[Out]
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Maxima [A] time = 0.694941, size = 93, normalized size = 1.43 \[ -\frac{b d +{\left (2 \, c d - b e\right )} x}{b^{2} c x^{2} + b^{3} x} + \frac{{\left (2 \, c d - b e\right )} \log \left (c x + b\right )}{b^{3}} - \frac{{\left (2 \, c d - b e\right )} \log \left (x\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*x^2 + b*x)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227875, size = 150, normalized size = 2.31 \[ -\frac{b^{2} d +{\left (2 \, b c d - b^{2} e\right )} x -{\left ({\left (2 \, c^{2} d - b c e\right )} x^{2} +{\left (2 \, b c d - b^{2} e\right )} x\right )} \log \left (c x + b\right ) +{\left ({\left (2 \, c^{2} d - b c e\right )} x^{2} +{\left (2 \, b c d - b^{2} e\right )} x\right )} \log \left (x\right )}{b^{3} c x^{2} + b^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*x^2 + b*x)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.54772, size = 128, normalized size = 1.97 \[ \frac{- b d + x \left (b e - 2 c d\right )}{b^{3} x + b^{2} c x^{2}} + \frac{\left (b e - 2 c d\right ) \log{\left (x + \frac{b^{2} e - 2 b c d - b \left (b e - 2 c d\right )}{2 b c e - 4 c^{2} d} \right )}}{b^{3}} - \frac{\left (b e - 2 c d\right ) \log{\left (x + \frac{b^{2} e - 2 b c d + b \left (b e - 2 c d\right )}{2 b c e - 4 c^{2} d} \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/(c*x**2+b*x)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.209313, size = 104, normalized size = 1.6 \[ -\frac{{\left (2 \, c d - b e\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} - \frac{2 \, c d x - b x e + b d}{{\left (c x^{2} + b x\right )} b^{2}} + \frac{{\left (2 \, c^{2} d - b c e\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(c*x^2 + b*x)^2,x, algorithm="giac")
[Out]