Optimal. Leaf size=62 \[ \frac{\log (x) (b B-2 A c)}{b^3}-\frac{(b B-2 A c) \log (b+c x)}{b^3}+\frac{b B-A c}{b^2 (b+c x)}-\frac{A}{b^2 x} \]
[Out]
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Rubi [A] time = 0.118902, antiderivative size = 62, 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 \[ \frac{\log (x) (b B-2 A c)}{b^3}-\frac{(b B-2 A c) \log (b+c x)}{b^3}+\frac{b B-A c}{b^2 (b+c x)}-\frac{A}{b^2 x} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 15.9132, size = 54, normalized size = 0.87 \[ - \frac{A}{b^{2} x} - \frac{A c - B b}{b^{2} \left (b + c x\right )} - \frac{\left (2 A c - B b\right ) \log{\left (x \right )}}{b^{3}} + \frac{\left (2 A c - B b\right ) \log{\left (b + c x \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(c*x**2+b*x)**2,x)
[Out]
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Mathematica [A] time = 0.0828449, size = 56, normalized size = 0.9 \[ \frac{\frac{b (b B-A c)}{b+c x}+\log (x) (b B-2 A c)+(2 A c-b B) \log (b+c x)-\frac{A b}{x}}{b^3} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(b*x + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.016, size = 78, normalized size = 1.3 \[ -{\frac{A}{{b}^{2}x}}-2\,{\frac{Ac\ln \left ( x \right ) }{{b}^{3}}}+{\frac{\ln \left ( x \right ) B}{{b}^{2}}}-{\frac{Ac}{{b}^{2} \left ( cx+b \right ) }}+{\frac{B}{b \left ( cx+b \right ) }}+2\,{\frac{\ln \left ( cx+b \right ) Ac}{{b}^{3}}}-{\frac{\ln \left ( cx+b \right ) B}{{b}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(c*x^2+b*x)^2,x)
[Out]
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Maxima [A] time = 0.691733, size = 90, normalized size = 1.45 \[ -\frac{A b -{\left (B b - 2 \, A c\right )} x}{b^{2} c x^{2} + b^{3} x} - \frac{{\left (B b - 2 \, A c\right )} \log \left (c x + b\right )}{b^{3}} + \frac{{\left (B b - 2 \, A c\right )} \log \left (x\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(c*x^2 + b*x)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.279354, size = 144, normalized size = 2.32 \[ -\frac{A b^{2} -{\left (B b^{2} - 2 \, A b c\right )} x +{\left ({\left (B b c - 2 \, A c^{2}\right )} x^{2} +{\left (B b^{2} - 2 \, A b c\right )} x\right )} \log \left (c x + b\right ) -{\left ({\left (B b c - 2 \, A c^{2}\right )} x^{2} +{\left (B b^{2} - 2 \, A b c\right )} x\right )} \log \left (x\right )}{b^{3} c x^{2} + b^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(c*x^2 + b*x)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.47937, size = 128, normalized size = 2.06 \[ \frac{- A b + x \left (- 2 A c + B b\right )}{b^{3} x + b^{2} c x^{2}} + \frac{\left (- 2 A c + B b\right ) \log{\left (x + \frac{- 2 A b c + B b^{2} - b \left (- 2 A c + B b\right )}{- 4 A c^{2} + 2 B b c} \right )}}{b^{3}} - \frac{\left (- 2 A c + B b\right ) \log{\left (x + \frac{- 2 A b c + B b^{2} + b \left (- 2 A c + B b\right )}{- 4 A c^{2} + 2 B b c} \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/(c*x**2+b*x)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.280418, size = 96, normalized size = 1.55 \[ \frac{{\left (B b - 2 \, A c\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} + \frac{B b x - 2 \, A c x - A b}{{\left (c x^{2} + b x\right )} b^{2}} - \frac{{\left (B b c - 2 \, A c^{2}\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(c*x^2 + b*x)^2,x, algorithm="giac")
[Out]