Optimal. Leaf size=46 \[ \frac{1}{2} f^a x^{m+1} \left (-\frac{b \log (f)}{x^2}\right )^{\frac{m+1}{2}} \text{Gamma}\left (\frac{1}{2} (-m-1),-\frac{b \log (f)}{x^2}\right ) \]
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Rubi [A] time = 0.0236666, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2218} \[ \frac{1}{2} f^a x^{m+1} \left (-\frac{b \log (f)}{x^2}\right )^{\frac{m+1}{2}} \text{Gamma}\left (\frac{1}{2} (-m-1),-\frac{b \log (f)}{x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int f^{a+\frac{b}{x^2}} x^m \, dx &=\frac{1}{2} f^a x^{1+m} \Gamma \left (\frac{1}{2} (-1-m),-\frac{b \log (f)}{x^2}\right ) \left (-\frac{b \log (f)}{x^2}\right )^{\frac{1+m}{2}}\\ \end{align*}
Mathematica [A] time = 0.0102872, size = 46, normalized size = 1. \[ \frac{1}{2} f^a x^{m+1} \left (-\frac{b \log (f)}{x^2}\right )^{\frac{m+1}{2}} \text{Gamma}\left (\frac{1}{2} (-m-1),-\frac{b \log (f)}{x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.038, size = 169, normalized size = 3.7 \begin{align*} -{\frac{{f}^{a}}{2} \left ( -b \right ) ^{{\frac{m}{2}}+{\frac{1}{2}}} \left ( \ln \left ( f \right ) \right ) ^{{\frac{m}{2}}+{\frac{1}{2}}} \left ( 2\,{\frac{{x}^{-1+m} \left ( -b \right ) ^{-m/2-1/2} \left ( \ln \left ( f \right ) \right ) ^{1/2-m/2}b\Gamma \left ( 1/2-m/2 \right ) }{1+m} \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{2}}} \right ) ^{-1/2+m/2}}-2\,{\frac{{x}^{1+m} \left ( -b \right ) ^{-m/2-1/2} \left ( \ln \left ( f \right ) \right ) ^{-m/2-1/2}}{1+m}{{\rm e}^{{\frac{b\ln \left ( f \right ) }{{x}^{2}}}}}}-2\,{\frac{{x}^{-1+m} \left ( -b \right ) ^{-m/2-1/2} \left ( \ln \left ( f \right ) \right ) ^{1/2-m/2}b}{1+m} \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{2}}} \right ) ^{-1/2+m/2}\Gamma \left ( 1/2-m/2,-{\frac{b\ln \left ( f \right ) }{{x}^{2}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.3925, size = 51, normalized size = 1.11 \begin{align*} \frac{1}{2} \, f^{a} x^{m + 1} \left (-\frac{b \log \left (f\right )}{x^{2}}\right )^{\frac{1}{2} \, m + \frac{1}{2}} \Gamma \left (-\frac{1}{2} \, m - \frac{1}{2}, -\frac{b \log \left (f\right )}{x^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (f^{\frac{a x^{2} + b}{x^{2}}} x^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + \frac{b}{x^{2}}} x^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{a + \frac{b}{x^{2}}} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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