Optimal. Leaf size=44 \[ -\frac{b^2}{2 a^3 \left (a x^2+b\right )}-\frac{b \log \left (a x^2+b\right )}{a^3}+\frac{x^2}{2 a^2} \]
[Out]
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Rubi [A] time = 0.0867551, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{b^2}{2 a^3 \left (a x^2+b\right )}-\frac{b \log \left (a x^2+b\right )}{a^3}+\frac{x^2}{2 a^2} \]
Antiderivative was successfully verified.
[In] Int[x/(a + b/x^2)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\int ^{x^{2}} \frac{1}{a^{2}}\, dx}{2} - \frac{b^{2}}{2 a^{3} \left (a x^{2} + b\right )} - \frac{b \log{\left (a x^{2} + b \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(a+b/x**2)**2,x)
[Out]
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Mathematica [A] time = 0.0289089, size = 38, normalized size = 0.86 \[ \frac{-\frac{b^2}{a x^2+b}-2 b \log \left (a x^2+b\right )+a x^2}{2 a^3} \]
Antiderivative was successfully verified.
[In] Integrate[x/(a + b/x^2)^2,x]
[Out]
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Maple [A] time = 0.013, size = 41, normalized size = 0.9 \[{\frac{{x}^{2}}{2\,{a}^{2}}}-{\frac{{b}^{2}}{2\,{a}^{3} \left ( a{x}^{2}+b \right ) }}-{\frac{b\ln \left ( a{x}^{2}+b \right ) }{{a}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(a+b/x^2)^2,x)
[Out]
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Maxima [A] time = 1.43775, size = 58, normalized size = 1.32 \[ -\frac{b^{2}}{2 \,{\left (a^{4} x^{2} + a^{3} b\right )}} + \frac{x^{2}}{2 \, a^{2}} - \frac{b \log \left (a x^{2} + b\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a + b/x^2)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.225497, size = 76, normalized size = 1.73 \[ \frac{a^{2} x^{4} + a b x^{2} - b^{2} - 2 \,{\left (a b x^{2} + b^{2}\right )} \log \left (a x^{2} + b\right )}{2 \,{\left (a^{4} x^{2} + a^{3} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a + b/x^2)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.54483, size = 39, normalized size = 0.89 \[ - \frac{b^{2}}{2 a^{4} x^{2} + 2 a^{3} b} + \frac{x^{2}}{2 a^{2}} - \frac{b \log{\left (a x^{2} + b \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a+b/x**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.223274, size = 55, normalized size = 1.25 \[ \frac{x^{2}}{2 \, a^{2}} - \frac{b{\rm ln}\left ({\left | a x^{2} + b \right |}\right )}{a^{3}} - \frac{b^{2}}{2 \,{\left (a x^{2} + b\right )} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a + b/x^2)^2,x, algorithm="giac")
[Out]