Optimal. Leaf size=58 \[ \frac{3 b^2 \log (x)}{a^4}-\frac{3 b^2 \log (a+b x)}{a^4}+\frac{b^2}{a^3 (a+b x)}+\frac{2 b}{a^3 x}-\frac{1}{2 a^2 x^2} \]
[Out]
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Rubi [A] time = 0.0724918, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{3 b^2 \log (x)}{a^4}-\frac{3 b^2 \log (a+b x)}{a^4}+\frac{b^2}{a^3 (a+b x)}+\frac{2 b}{a^3 x}-\frac{1}{2 a^2 x^2} \]
Antiderivative was successfully verified.
[In] Int[x/(a*x^2 + b*x^3)^2,x]
[Out]
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Rubi in Sympy [A] time = 12.9001, size = 56, normalized size = 0.97 \[ - \frac{1}{2 a^{2} x^{2}} + \frac{b^{2}}{a^{3} \left (a + b x\right )} + \frac{2 b}{a^{3} x} + \frac{3 b^{2} \log{\left (x \right )}}{a^{4}} - \frac{3 b^{2} \log{\left (a + b x \right )}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(b*x**3+a*x**2)**2,x)
[Out]
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Mathematica [A] time = 0.0942862, size = 53, normalized size = 0.91 \[ \frac{a \left (\frac{2 b^2}{a+b x}-\frac{a}{x^2}+\frac{4 b}{x}\right )-6 b^2 \log (a+b x)+6 b^2 \log (x)}{2 a^4} \]
Antiderivative was successfully verified.
[In] Integrate[x/(a*x^2 + b*x^3)^2,x]
[Out]
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Maple [A] time = 0.016, size = 57, normalized size = 1. \[ -{\frac{1}{2\,{a}^{2}{x}^{2}}}+2\,{\frac{b}{x{a}^{3}}}+{\frac{{b}^{2}}{{a}^{3} \left ( bx+a \right ) }}+3\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{4}}}-3\,{\frac{{b}^{2}\ln \left ( bx+a \right ) }{{a}^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(b*x^3+a*x^2)^2,x)
[Out]
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Maxima [A] time = 5.91187, size = 86, normalized size = 1.48 \[ \frac{6 \, b^{2} x^{2} + 3 \, a b x - a^{2}}{2 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} - \frac{3 \, b^{2} \log \left (b x + a\right )}{a^{4}} + \frac{3 \, b^{2} \log \left (x\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x^3 + a*x^2)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.216441, size = 116, normalized size = 2. \[ \frac{6 \, a b^{2} x^{2} + 3 \, a^{2} b x - a^{3} - 6 \,{\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \log \left (b x + a\right ) + 6 \,{\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{4} b x^{3} + a^{5} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x^3 + a*x^2)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.70343, size = 54, normalized size = 0.93 \[ \frac{- a^{2} + 3 a b x + 6 b^{2} x^{2}}{2 a^{4} x^{2} + 2 a^{3} b x^{3}} + \frac{3 b^{2} \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x**3+a*x**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.219615, size = 86, normalized size = 1.48 \[ -\frac{3 \, b^{2}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{4}} + \frac{3 \, b^{2}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} + \frac{6 \, a b^{2} x^{2} + 3 \, a^{2} b x - a^{3}}{2 \,{\left (b x + a\right )} a^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x^3 + a*x^2)^2,x, algorithm="giac")
[Out]