Optimal. Leaf size=34 \[ \frac {1}{3} x^4 f^a \left (-\frac {b \log (f)}{x^3}\right )^{4/3} \Gamma \left (-\frac {4}{3},-\frac {b \log (f)}{x^3}\right ) \]
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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} \[ \frac {1}{3} x^4 f^a \left (-\frac {b \log (f)}{x^3}\right )^{4/3} \text {Gamma}\left (-\frac {4}{3},-\frac {b \log (f)}{x^3}\right ) \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int f^{a+\frac {b}{x^3}} x^3 \, dx &=\frac {1}{3} f^a x^4 \Gamma \left (-\frac {4}{3},-\frac {b \log (f)}{x^3}\right ) \left (-\frac {b \log (f)}{x^3}\right )^{4/3}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 34, normalized size = 1.00 \[ \frac {1}{3} x^4 f^a \left (-\frac {b \log (f)}{x^3}\right )^{4/3} \Gamma \left (-\frac {4}{3},-\frac {b \log (f)}{x^3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 51, normalized size = 1.50 \[ -\frac {3}{4} \, \left (-b \log \relax (f)\right )^{\frac {1}{3}} b f^{a} \Gamma \left (\frac {2}{3}, -\frac {b \log \relax (f)}{x^{3}}\right ) \log \relax (f) + \frac {1}{4} \, {\left (x^{4} + 3 \, b x \log \relax (f)\right )} f^{\frac {a x^{3} + b}{x^{3}}} \]
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 \[ \int f^{a + \frac {b}{x^{3}}} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 115, normalized size = 3.38 \[ \frac {\left (-b \right )^{\frac {1}{3}} \left (-\frac {3 \left (\frac {3 b \ln \relax (f )}{x^{3}}+1\right ) x^{4} {\mathrm e}^{\frac {b \ln \relax (f )}{x^{3}}}}{4 \left (-b \right )^{\frac {4}{3}} \ln \relax (f )^{\frac {4}{3}}}-\frac {9 b^{2} \Gamma \left (\frac {2}{3}, -\frac {b \ln \relax (f )}{x^{3}}\right ) \ln \relax (f )^{\frac {2}{3}}}{4 \left (-b \right )^{\frac {4}{3}} \left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {2}{3}} x^{2}}+\frac {9 \Gamma \left (\frac {2}{3}\right ) b^{2} \ln \relax (f )^{\frac {2}{3}}}{4 \left (-b \right )^{\frac {4}{3}} \left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {2}{3}} x^{2}}\right ) b \,f^{a} \ln \relax (f )^{\frac {4}{3}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 28, normalized size = 0.82 \[ \frac {1}{3} \, f^{a} x^{4} \left (-\frac {b \log \relax (f)}{x^{3}}\right )^{\frac {4}{3}} \Gamma \left (-\frac {4}{3}, -\frac {b \log \relax (f)}{x^{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.59, size = 80, normalized size = 2.35 \[ \frac {f^a\,f^{\frac {b}{x^3}}\,x^4}{4}-\frac {3\,f^a\,x^4\,\Gamma \left (\frac {2}{3}\right )\,{\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}^{4/3}}{4}+\frac {3\,f^a\,x^4\,\Gamma \left (\frac {2}{3},-\frac {b\,\ln \relax (f)}{x^3}\right )\,{\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}^{4/3}}{4}+\frac {3\,b\,f^a\,f^{\frac {b}{x^3}}\,x\,\ln \relax (f)}{4} \]
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 \[ \int f^{a + \frac {b}{x^{3}}} x^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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