Optimal. Leaf size=73 \[ -\frac{2}{3} \sqrt{\pi } b^{3/2} f^a \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )+\frac{2}{3} b x \log (f) f^{a+\frac{b}{x^2}}+\frac{1}{3} x^3 f^{a+\frac{b}{x^2}} \]
[Out]
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Rubi [A] time = 0.104748, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{2}{3} \sqrt{\pi } b^{3/2} f^a \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )+\frac{2}{3} b x \log (f) f^{a+\frac{b}{x^2}}+\frac{1}{3} x^3 f^{a+\frac{b}{x^2}} \]
Antiderivative was successfully verified.
[In] Int[f^(a + b/x^2)*x^2,x]
[Out]
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Rubi in Sympy [A] time = 8.83937, size = 70, normalized size = 0.96 \[ - \frac{2 \sqrt{\pi } b^{\frac{3}{2}} f^{a} \log{\left (f \right )}^{\frac{3}{2}} \operatorname{erfi}{\left (\frac{\sqrt{b} \sqrt{\log{\left (f \right )}}}{x} \right )}}{3} + \frac{2 b f^{a + \frac{b}{x^{2}}} x \log{\left (f \right )}}{3} + \frac{f^{a + \frac{b}{x^{2}}} x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(f**(a+b/x**2)*x**2,x)
[Out]
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Mathematica [A] time = 0.0336177, size = 60, normalized size = 0.82 \[ \frac{1}{3} f^a \left (x f^{\frac{b}{x^2}} \left (2 b \log (f)+x^2\right )-2 \sqrt{\pi } b^{3/2} \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (f)}}{x}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[f^(a + b/x^2)*x^2,x]
[Out]
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Maple [A] time = 0.029, size = 67, normalized size = 0.9 \[{\frac{{f}^{a}{x}^{3}}{3}{f}^{{\frac{b}{{x}^{2}}}}}+{\frac{2\,{f}^{a}\ln \left ( f \right ) bx}{3}{f}^{{\frac{b}{{x}^{2}}}}}-{\frac{2\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}\sqrt{\pi }}{3}{\it Erf} \left ({\frac{1}{x}\sqrt{-b\ln \left ( f \right ) }} \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(f^(a+b/x^2)*x^2,x)
[Out]
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Maxima [A] time = 0.880215, size = 92, normalized size = 1.26 \[ -\frac{2 \, \sqrt{\pi } b^{2} f^{a}{\left (\operatorname{erf}\left (\sqrt{-\frac{b \log \left (f\right )}{x^{2}}}\right ) - 1\right )} \log \left (f\right )^{2}}{3 \, x \sqrt{-\frac{b \log \left (f\right )}{x^{2}}}} + \frac{1}{3} \,{\left (f^{a} x^{3} + 2 \, b f^{a} x \log \left (f\right )\right )} f^{\frac{b}{x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^(a + b/x^2)*x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.318096, size = 93, normalized size = 1.27 \[ -\frac{2 \, \sqrt{\pi } b^{2} f^{a} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (f\right )}}{x}\right ) \log \left (f\right )^{2} -{\left (x^{3} + 2 \, b x \log \left (f\right )\right )} \sqrt{-b \log \left (f\right )} f^{\frac{a x^{2} + b}{x^{2}}}}{3 \, \sqrt{-b \log \left (f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^(a + b/x^2)*x^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int f^{a + \frac{b}{x^{2}}} x^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f**(a+b/x**2)*x**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int f^{a + \frac{b}{x^{2}}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^(a + b/x^2)*x^2,x, algorithm="giac")
[Out]