Optimal. Leaf size=61 \[ -\frac {F^a (c+d x)^{1+m} \Gamma \left (\frac {1+m}{3},-b (c+d x)^3 \log (F)\right ) \left (-b (c+d x)^3 \log (F)\right )^{\frac {1}{3} (-1-m)}}{3 d} \]
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Rubi [A]
time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2250}
\begin {gather*} -\frac {F^a (c+d x)^{m+1} \left (-b \log (F) (c+d x)^3\right )^{\frac {1}{3} (-m-1)} \text {Gamma}\left (\frac {m+1}{3},-b \log (F) (c+d x)^3\right )}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^3} (c+d x)^m \, dx &=-\frac {F^a (c+d x)^{1+m} \Gamma \left (\frac {1+m}{3},-b (c+d x)^3 \log (F)\right ) \left (-b (c+d x)^3 \log (F)\right )^{\frac {1}{3} (-1-m)}}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 61, normalized size = 1.00 \begin {gather*} -\frac {F^a (c+d x)^{1+m} \Gamma \left (\frac {1+m}{3},-b (c+d x)^3 \log (F)\right ) \left (-b (c+d x)^3 \log (F)\right )^{\frac {1}{3} (-1-m)}}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int F^{a +b \left (d x +c \right )^{3}} \left (d x +c \right )^{m}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.10, size = 71, normalized size = 1.16 \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}, -{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right )}{3 \, b d \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 \left (c + d x\right )^{3}} \left (c + d x\right )^{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 = 3.68, size = 75, normalized size = 1.23 \begin {gather*} \frac {F^a\,{\mathrm {e}}^{\frac {b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3}{2}}\,{\left (c+d\,x\right )}^{m+1}\,{\mathrm {M}}_{\frac {1}{3}-\frac {m}{6},\frac {m}{6}+\frac {1}{6}}\left (b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )}{d\,\left (m+1\right )\,{\left (b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )}^{\frac {m}{6}+\frac {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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