Optimal. Leaf size=49 \[ -\frac {F^a \Gamma \left (-\frac {1}{3},-b (c+d x)^3 \log (F)\right ) \sqrt [3]{-b (c+d x)^3 \log (F)}}{3 d (c+d x)} \]
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Rubi [A]
time = 0.04, antiderivative size = 49, 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 \sqrt [3]{-b \log (F) (c+d x)^3} \text {Gamma}\left (-\frac {1}{3},-b \log (F) (c+d x)^3\right )}{3 d (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
Rubi steps
\begin {align*} \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^2} \, dx &=-\frac {F^a \Gamma \left (-\frac {1}{3},-b (c+d x)^3 \log (F)\right ) \sqrt [3]{-b (c+d x)^3 \log (F)}}{3 d (c+d x)}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 49, normalized size = 1.00 \begin {gather*} -\frac {F^a \Gamma \left (-\frac {1}{3},-b (c+d x)^3 \log (F)\right ) \sqrt [3]{-b (c+d x)^3 \log (F)}}{3 d (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {F^{a +b \left (d x +c \right )^{3}}}{\left (d x +c \right )^{2}}\, dx\]
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 110 vs.
\(2 (43) = 86\).
time = 0.10, size = 110, normalized size = 2.24 \begin {gather*} \frac {\left (-b d^{3} \log \left (F\right )\right )^{\frac {1}{3}} {\left (d x + c\right )} F^{a} \Gamma \left (\frac {2}{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 ) - F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a} d}{d^{3} x + c d^{2}} \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 \frac {F^{a + b \left (c + d x\right )^{3}}}{\left (c + d x\right )^{2}}\, 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 = 3.73, size = 74, normalized size = 1.51 \begin {gather*} -\frac {F^a\,\left (F^{b\,{\left (c+d\,x\right )}^3}-\Gamma \left (\frac {2}{3},-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )\,{\left (-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )}^{1/3}+\Gamma \left (\frac {2}{3}\right )\,{\left (-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )}^{1/3}\right )}{d\,\left (c+d\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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