Optimal. Leaf size=34 \[ \frac {1}{3} x^2 f^a \left (-\frac {b \log (f)}{x^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {b \log (f)}{x^3}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2218} \[ \frac {1}{3} x^2 f^a \left (-\frac {b \log (f)}{x^3}\right )^{2/3} \text {Gamma}\left (-\frac {2}{3},-\frac {b \log (f)}{x^3}\right ) \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int f^{a+\frac {b}{x^3}} x \, dx &=\frac {1}{3} f^a x^2 \Gamma \left (-\frac {2}{3},-\frac {b \log (f)}{x^3}\right ) \left (-\frac {b \log (f)}{x^3}\right )^{2/3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 34, normalized size = 1.00 \[ \frac {1}{3} x^2 f^a \left (-\frac {b \log (f)}{x^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {b \log (f)}{x^3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 41, normalized size = 1.21 \[ \frac {1}{2} \, f^{\frac {a x^{3} + b}{x^{3}}} x^{2} - \frac {1}{2} \, \left (-b \log \relax (f)\right )^{\frac {2}{3}} f^{a} \Gamma \left (\frac {1}{3}, -\frac {b \log \relax (f)}{x^{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x^{3}}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 105, normalized size = 3.09 \[ -\frac {\left (-b \right )^{\frac {2}{3}} \left (-\frac {3 x^{2} {\mathrm e}^{\frac {b \ln \relax (f )}{x^{3}}}}{2 \left (-b \right )^{\frac {2}{3}} \ln \relax (f )^{\frac {2}{3}}}-\frac {3 b \Gamma \left (\frac {1}{3}, -\frac {b \ln \relax (f )}{x^{3}}\right ) \ln \relax (f )^{\frac {1}{3}}}{2 \left (-b \right )^{\frac {2}{3}} \left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {1}{3}} x}+\frac {\pi \sqrt {3}\, b \ln \relax (f )^{\frac {1}{3}}}{\left (-b \right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right ) \left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {1}{3}} x}\right ) f^{a} \ln \relax (f )^{\frac {2}{3}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 28, normalized size = 0.82 \[ \frac {1}{3} \, f^{a} x^{2} \left (-\frac {b \log \relax (f)}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (-\frac {2}{3}, -\frac {b \log \relax (f)}{x^{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.57, size = 70, normalized size = 2.06 \[ \frac {f^a\,f^{\frac {b}{x^3}}\,x^2}{2}-\frac {f^a\,x^2\,\Gamma \left (\frac {1}{3},-\frac {b\,\ln \relax (f)}{x^3}\right )\,{\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}^{2/3}}{2}+\frac {\pi \,\sqrt {3}\,f^a\,x^2\,{\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}^{2/3}}{3\,\Gamma \left (\frac {2}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x^{3}}} x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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