Optimal. Leaf size=32 \[ \frac{1}{3} x f^a \sqrt [3]{-\frac{b \log (f)}{x^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{b \log (f)}{x^3}\right ) \]
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Rubi [A] time = 0.0047191, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2208} \[ \frac{1}{3} x f^a \sqrt [3]{-\frac{b \log (f)}{x^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{b \log (f)}{x^3}\right ) \]
Antiderivative was successfully verified.
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Rule 2208
Rubi steps
\begin{align*} \int f^{a+\frac{b}{x^3}} \, dx &=\frac{1}{3} f^a x \Gamma \left (-\frac{1}{3},-\frac{b \log (f)}{x^3}\right ) \sqrt [3]{-\frac{b \log (f)}{x^3}}\\ \end{align*}
Mathematica [A] time = 0.0029922, size = 32, normalized size = 1. \[ \frac{1}{3} x f^a \sqrt [3]{-\frac{b \log (f)}{x^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{b \log (f)}{x^3}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.029, size = 98, normalized size = 3.1 \begin{align*} -{\frac{{f}^{a}}{3}\sqrt [3]{-b}\sqrt [3]{\ln \left ( f \right ) } \left ( 3\,{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2/3}b\Gamma \left ( 2/3 \right ) }{{x}^{2}\sqrt [3]{-b}} \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) ^{-2/3}}-3\,{\frac{x}{\sqrt [3]{-b}\sqrt [3]{\ln \left ( f \right ) }}{{\rm e}^{{\frac{b\ln \left ( f \right ) }{{x}^{3}}}}}}-3\,{\frac{ \left ( \ln \left ( f \right ) \right ) ^{2/3}b}{{x}^{2}\sqrt [3]{-b}}\Gamma \left ( 2/3,-{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) \left ( -{\frac{b\ln \left ( f \right ) }{{x}^{3}}} \right ) ^{-2/3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24753, size = 35, normalized size = 1.09 \begin{align*} \frac{1}{3} \, f^{a} x \left (-\frac{b \log \left (f\right )}{x^{3}}\right )^{\frac{1}{3}} \Gamma \left (-\frac{1}{3}, -\frac{b \log \left (f\right )}{x^{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8089, size = 100, normalized size = 3.12 \begin{align*} -\left (-b \log \left (f\right )\right )^{\frac{1}{3}} f^{a} \Gamma \left (\frac{2}{3}, -\frac{b \log \left (f\right )}{x^{3}}\right ) + f^{\frac{a x^{3} + b}{x^{3}}} x \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^{3}}}\, 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^{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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