Optimal. Leaf size=49 \[ -\frac{F^a \sqrt [3]{-b \log (F) (c+d x)^3} \text{Gamma}\left (-\frac{1}{3},-b \log (F) (c+d x)^3\right )}{3 d (c+d x)} \]
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Rubi [A] time = 0.0643983, antiderivative size = 49, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0159363, size = 49, normalized size = 1. \[ -\frac{F^a \sqrt [3]{-b \log (F) (c+d x)^3} \text{Gamma}\left (-\frac{1}{3},-b \log (F) (c+d x)^3\right )}{3 d (c+d x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{a+b \left ( dx+c \right ) ^{3}}}{ \left ( dx+c \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.54923, size = 250, normalized size = 5.1 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + b \left (c + d x\right )^{3}}}{\left (c + d x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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