Optimal. Leaf size=34 \[ -\frac {f^a \left (-b x^3 \log (f)\right )^{2/3} \Gamma \left (-\frac {2}{3},-b x^3 \log (f)\right )}{3 x^2} \]
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Rubi [A] time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, 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} \[ -\frac {f^a \left (-b x^3 \log (f)\right )^{2/3} \text {Gamma}\left (-\frac {2}{3},-b x^3 \log (f)\right )}{3 x^2} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int \frac {f^{a+b x^3}}{x^3} \, dx &=-\frac {f^a \Gamma \left (-\frac {2}{3},-b x^3 \log (f)\right ) \left (-b x^3 \log (f)\right )^{2/3}}{3 x^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 34, normalized size = 1.00 \[ -\frac {f^a \left (-b x^3 \log (f)\right )^{2/3} \Gamma \left (-\frac {2}{3},-b x^3 \log (f)\right )}{3 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 41, normalized size = 1.21 \[ \frac {\left (-b \log \relax (f)\right )^{\frac {2}{3}} f^{a} x^{2} \Gamma \left (\frac {1}{3}, -b x^{3} \log \relax (f)\right ) - f^{b x^{3} + a}}{2 \, x^{2}} \]
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 \[ \int \frac {f^{b x^{3} + a}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 102, normalized size = 3.00 \[ -\frac {\left (-\frac {3 b x \Gamma \left (\frac {1}{3}, -b \,x^{3} \ln \relax (f )\right ) \ln \relax (f )^{\frac {1}{3}}}{2 \left (-b \right )^{\frac {2}{3}} \left (-b \,x^{3} \ln \relax (f )\right )^{\frac {1}{3}}}+\frac {\pi \sqrt {3}\, b x \ln \relax (f )^{\frac {1}{3}}}{\left (-b \right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right ) \left (-b \,x^{3} \ln \relax (f )\right )^{\frac {1}{3}}}-\frac {3 \,{\mathrm e}^{b \,x^{3} \ln \relax (f )}}{2 \left (-b \right )^{\frac {2}{3}} x^{2} \ln \relax (f )^{\frac {2}{3}}}\right ) b \,f^{a} \ln \relax (f )^{\frac {2}{3}}}{3 \left (-b \right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.65, size = 28, normalized size = 0.82 \[ -\frac {\left (-b x^{3} \log \relax (f)\right )^{\frac {2}{3}} f^{a} \Gamma \left (-\frac {2}{3}, -b x^{3} \log \relax (f)\right )}{3 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.58, size = 70, normalized size = 2.06 \[ \frac {f^a\,\Gamma \left (\frac {1}{3},-b\,x^3\,\ln \relax (f)\right )\,{\left (-b\,x^3\,\ln \relax (f)\right )}^{2/3}}{2\,x^2}-\frac {f^a\,f^{b\,x^3}}{2\,x^2}-\frac {\pi \,\sqrt {3}\,f^a\,{\left (-b\,x^3\,\ln \relax (f)\right )}^{2/3}}{3\,x^2\,\Gamma \left (\frac {2}{3}\right )} \]
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 \[ \int \frac {f^{a + b x^{3}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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