Optimal. Leaf size=46 \[ -\frac {1}{3} f^a x^{m+1} \left (-b x^3 \log (f)\right )^{\frac {1}{3} (-m-1)} \Gamma \left (\frac {m+1}{3},-b x^3 \log (f)\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 46, 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} \[ -\frac {1}{3} f^a x^{m+1} \left (-b x^3 \log (f)\right )^{\frac {1}{3} (-m-1)} \text {Gamma}\left (\frac {m+1}{3},-b x^3 \log (f)\right ) \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int f^{a+b x^3} x^m \, dx &=-\frac {1}{3} f^a x^{1+m} \Gamma \left (\frac {1+m}{3},-b x^3 \log (f)\right ) \left (-b x^3 \log (f)\right )^{\frac {1}{3} (-1-m)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 1.00 \[ -\frac {1}{3} f^a x^{m+1} \left (-b x^3 \log (f)\right )^{\frac {1}{3} (-m-1)} \Gamma \left (\frac {m+1}{3},-b x^3 \log (f)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 40, normalized size = 0.87 \[ \frac {e^{\left (-\frac {1}{3} \, {\left (m - 2\right )} \log \left (-b \log \relax (f)\right ) + a \log \relax (f)\right )} \Gamma \left (\frac {1}{3} \, m + \frac {1}{3}, -b x^{3} \log \relax (f)\right )}{3 \, b \log \relax (f)} \]
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 \[ \int f^{b x^{3} + a} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 140, normalized size = 3.04 \[ \frac {\left (\frac {3 \left (\frac {m}{3}+\frac {1}{3}\right ) x^{m +1} \left (-b \right )^{\frac {m}{3}+\frac {1}{3}} \left (-b \,x^{3} \ln \relax (f )\right )^{-\frac {m}{3}-\frac {1}{3}} \ln \relax (f )^{\frac {m}{3}+\frac {1}{3}} \Gamma \left (\frac {m}{3}+\frac {1}{3}\right )}{m +1}+\frac {3 \left (-\frac {m}{3}-\frac {1}{3}\right ) x^{m +1} \left (-b \right )^{\frac {m}{3}+\frac {1}{3}} \left (-b \,x^{3} \ln \relax (f )\right )^{-\frac {m}{3}-\frac {1}{3}} \ln \relax (f )^{\frac {m}{3}+\frac {1}{3}} \Gamma \left (\frac {m}{3}+\frac {1}{3}, -b \,x^{3} \ln \relax (f )\right )}{m +1}\right ) f^{a} \left (-b \right )^{-\frac {m}{3}-\frac {1}{3}} \ln \relax (f )^{-\frac {m}{3}-\frac {1}{3}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 38, normalized size = 0.83 \[ -\frac {1}{3} \, \left (-b x^{3} \log \relax (f)\right )^{-\frac {1}{3} \, m - \frac {1}{3}} f^{a} x^{m + 1} \Gamma \left (\frac {1}{3} \, m + \frac {1}{3}, -b x^{3} \log \relax (f)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.39, size = 56, normalized size = 1.22 \[ \frac {f^a\,x^{m+1}\,{\mathrm {e}}^{\frac {b\,x^3\,\ln \relax (f)}{2}}\,{\mathrm {M}}_{\frac {1}{3}-\frac {m}{6},\frac {m}{6}+\frac {1}{6}}\left (b\,x^3\,\ln \relax (f)\right )}{\left (m+1\right )\,{\left (b\,x^3\,\ln \relax (f)\right )}^{\frac {m}{6}+\frac {2}{3}}} \]
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 \[ \int f^{a + b x^{3}} x^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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