Optimal. Leaf size=35 \[ f^a x^{m+1} \left (-\frac{b \log (f)}{x}\right )^{m+1} \text{Gamma}\left (-m-1,-\frac{b \log (f)}{x}\right ) \]
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Rubi [A] time = 0.0179775, antiderivative size = 35, 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} \[ f^a x^{m+1} \left (-\frac{b \log (f)}{x}\right )^{m+1} \text{Gamma}\left (-m-1,-\frac{b \log (f)}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int f^{a+\frac{b}{x}} x^m \, dx &=f^a x^{1+m} \Gamma \left (-1-m,-\frac{b \log (f)}{x}\right ) \left (-\frac{b \log (f)}{x}\right )^{1+m}\\ \end{align*}
Mathematica [A] time = 0.0071684, size = 35, normalized size = 1. \[ f^a x^{m+1} \left (-\frac{b \log (f)}{x}\right )^{m+1} \text{Gamma}\left (-m-1,-\frac{b \log (f)}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.036, size = 136, normalized size = 3.9 \begin{align*}{f}^{a} \left ( -b \right ) ^{m} \left ( \ln \left ( f \right ) \right ) ^{1+m}b \left ( -{\frac{{x}^{m} \left ( -b \right ) ^{-m} \left ( \ln \left ( f \right ) \right ) ^{-m}\Gamma \left ( -m \right ) }{1+m} \left ( -{\frac{b\ln \left ( f \right ) }{x}} \right ) ^{m}}+{\frac{{x}^{1+m} \left ( -b \right ) ^{-m} \left ( \ln \left ( f \right ) \right ) ^{-m-1}}{ \left ( 1+m \right ) b}{{\rm e}^{{\frac{b\ln \left ( f \right ) }{x}}}}}+{\frac{{x}^{m} \left ( -b \right ) ^{-m} \left ( \ln \left ( f \right ) \right ) ^{-m}}{1+m} \left ( -{\frac{b\ln \left ( f \right ) }{x}} \right ) ^{m}\Gamma \left ( -m,-{\frac{b\ln \left ( f \right ) }{x}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.27095, size = 47, normalized size = 1.34 \begin{align*} f^{a} x^{m + 1} \left (-\frac{b \log \left (f\right )}{x}\right )^{m + 1} \Gamma \left (-m - 1, -\frac{b \log \left (f\right )}{x}\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 + b}{x}} 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}} 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}} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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