Optimal. Leaf size=49 \[ -\frac {F^a \sqrt [3]{-b \log (F) (c+d x)^3} \Gamma \left (-\frac {1}{3},-b (c+d x)^3 \log (F)\right )}{3 d (c+d x)} \]
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Rubi [A] time = 0.06, 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 = {2218} \[ -\frac {F^a \sqrt [3]{-b \log (F) (c+d x)^3} \text {Gamma}\left (-\frac {1}{3},-b \log (F) (c+d x)^3\right )}{3 d (c+d x)} \]
Antiderivative was successfully verified.
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Rule 2218
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 = 0.02, size = 49, normalized size = 1.00 \[ -\frac {F^a \sqrt [3]{-b \log (F) (c+d x)^3} \Gamma \left (-\frac {1}{3},-b (c+d x)^3 \log (F)\right )}{3 d (c+d x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 110, normalized size = 2.24 \[ \frac {\left (-b d^{3} \log \relax (F)\right )^{\frac {1}{3}} {\left (d x + c\right )} F^{a} \Gamma \left (\frac {2}{3}, -{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \relax (F)\right ) - F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a} d}{d^{3} x + c d^{2}} \]
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 \[ \int \frac {F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {F^{a +\left (d x +c \right )^{3} b}}{\left (d x +c \right )^{2}}\, dx \]
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 \[ \int \frac {F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{2}}\,{d x} \]
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 \[ -\frac {F^a\,\left (F^{b\,{\left (c+d\,x\right )}^3}-\Gamma \left (\frac {2}{3},-b\,\ln \relax (F)\,{\left (c+d\,x\right )}^3\right )\,{\left (-b\,\ln \relax (F)\,{\left (c+d\,x\right )}^3\right )}^{1/3}+\Gamma \left (\frac {2}{3}\right )\,{\left (-b\,\ln \relax (F)\,{\left (c+d\,x\right )}^3\right )}^{1/3}\right )}{d\,\left (c+d\,x\right )} \]
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 \[ \int \frac {F^{a + b \left (c + d x\right )^{3}}}{\left (c + d x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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