Optimal. Leaf size=46 \[ -\frac {1}{3} f^a x^{1+m} \Gamma \left (\frac {1+m}{3},-b x^3 \log (f)\right ) \left (-b x^3 \log (f)\right )^{\frac {1}{3} (-1-m)} \]
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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 = {2250}
\begin {gather*} -\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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
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 = 0.02, size = 46, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(139\) vs.
\(2(38)=76\).
time = 0.03, size = 140, normalized size = 3.04
method | result | size |
meijerg | \(\frac {f^{a} \left (-b \right )^{-\frac {m}{3}-\frac {1}{3}} \ln \left (f \right )^{-\frac {m}{3}-\frac {1}{3}} \left (\frac {3 x^{1+m} \left (-b \right )^{\frac {1}{3}+\frac {m}{3}} \ln \left (f \right )^{\frac {1}{3}+\frac {m}{3}} \left (\frac {1}{3}+\frac {m}{3}\right ) \left (-b \,x^{3} \ln \left (f \right )\right )^{-\frac {m}{3}-\frac {1}{3}} \Gamma \left (\frac {1}{3}+\frac {m}{3}\right )}{1+m}+\frac {3 x^{1+m} \left (-b \right )^{\frac {1}{3}+\frac {m}{3}} \ln \left (f \right )^{\frac {1}{3}+\frac {m}{3}} \left (-\frac {m}{3}-\frac {1}{3}\right ) \left (-b \,x^{3} \ln \left (f \right )\right )^{-\frac {m}{3}-\frac {1}{3}} \Gamma \left (\frac {1}{3}+\frac {m}{3}, -b \,x^{3} \ln \left (f \right )\right )}{1+m}\right )}{3}\) | \(140\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.07, size = 38, normalized size = 0.83 \begin {gather*} -\frac {1}{3} \, \left (-b x^{3} \log \left (f\right )\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 \left (f\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.08, size = 40, normalized size = 0.87 \begin {gather*} \frac {e^{\left (-\frac {1}{3} \, {\left (m - 2\right )} \log \left (-b \log \left (f\right )\right ) + a \log \left (f\right )\right )} \Gamma \left (\frac {1}{3} \, m + \frac {1}{3}, -b x^{3} \log \left (f\right )\right )}{3 \, b \log \left (f\right )} \end {gather*}
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 \begin {gather*} \int f^{a + b x^{3}} x^{m}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \frac {f^a\,x^{m+1}\,{\mathrm {e}}^{\frac {b\,x^3\,\ln \left (f\right )}{2}}\,{\mathrm {M}}_{\frac {1}{3}-\frac {m}{6},\frac {m}{6}+\frac {1}{6}}\left (b\,x^3\,\ln \left (f\right )\right )}{\left (m+1\right )\,{\left (b\,x^3\,\ln \left (f\right )\right )}^{\frac {m}{6}+\frac {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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