Optimal. Leaf size=34 \[ \frac {f^a \Gamma \left (\frac {4}{3},-\frac {b \log (f)}{x^3}\right )}{3 x^4 \left (-\frac {b \log (f)}{x^3}\right )^{4/3}} \]
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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} \[ \frac {f^a \text {Gamma}\left (\frac {4}{3},-\frac {b \log (f)}{x^3}\right )}{3 x^4 \left (-\frac {b \log (f)}{x^3}\right )^{4/3}} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int \frac {f^{a+\frac {b}{x^3}}}{x^5} \, dx &=\frac {f^a \Gamma \left (\frac {4}{3},-\frac {b \log (f)}{x^3}\right )}{3 x^4 \left (-\frac {b \log (f)}{x^3}\right )^{4/3}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 1.00 \[ \frac {f^a \Gamma \left (\frac {4}{3},-\frac {b \log (f)}{x^3}\right )}{3 x^4 \left (-\frac {b \log (f)}{x^3}\right )^{4/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 53, normalized size = 1.56 \[ \frac {\left (-b \log \relax (f)\right )^{\frac {2}{3}} f^{a} x \Gamma \left (\frac {1}{3}, -\frac {b \log \relax (f)}{x^{3}}\right ) - 3 \, b f^{\frac {a x^{3} + b}{x^{3}}} \log \relax (f)}{9 \, b^{2} x \log \relax (f)^{2}} \]
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 \[ \int \frac {f^{a + \frac {b}{x^{3}}}}{x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 112, normalized size = 3.29 \[ -\frac {\left (\frac {\left (-b \right )^{\frac {4}{3}} \Gamma \left (\frac {1}{3}, -\frac {b \ln \relax (f )}{x^{3}}\right ) \ln \relax (f )^{\frac {1}{3}}}{3 \left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {1}{3}} b x}+\frac {\left (-b \right )^{\frac {4}{3}} {\mathrm e}^{\frac {b \ln \relax (f )}{x^{3}}} \ln \relax (f )^{\frac {1}{3}}}{b x}-\frac {2 \left (-b \right )^{\frac {4}{3}} \pi \sqrt {3}\, \ln \relax (f )^{\frac {1}{3}}}{9 \Gamma \left (\frac {2}{3}\right ) \left (-\frac {b \ln \relax (f )}{x^{3}}\right )^{\frac {1}{3}} b x}\right ) f^{a}}{3 \left (-b \right )^{\frac {4}{3}} \ln \relax (f )^{\frac {4}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 28, normalized size = 0.82 \[ \frac {f^{a} \Gamma \left (\frac {4}{3}, -\frac {b \log \relax (f)}{x^{3}}\right )}{3 \, x^{4} \left (-\frac {b \log \relax (f)}{x^{3}}\right )^{\frac {4}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.54, size = 77, normalized size = 2.26 \[ \frac {f^a\,\Gamma \left (\frac {1}{3},-\frac {b\,\ln \relax (f)}{x^3}\right )}{9\,x^4\,{\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}^{4/3}}-\frac {f^a\,f^{\frac {b}{x^3}}}{3\,b\,x\,\ln \relax (f)}-\frac {2\,\pi \,\sqrt {3}\,f^a}{27\,x^4\,\Gamma \left (\frac {2}{3}\right )\,{\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}^{4/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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