Optimal. Leaf size=50 \[ \frac{F^a (c+d x)^{m+1} \left (-\frac{b \log (F)}{c+d x}\right )^{m+1} \text{Gamma}\left (-m-1,-\frac{b \log (F)}{c+d x}\right )}{d} \]
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Rubi [A] time = 0.0437885, antiderivative size = 50, 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} \[ \frac{F^a (c+d x)^{m+1} \left (-\frac{b \log (F)}{c+d x}\right )^{m+1} \text{Gamma}\left (-m-1,-\frac{b \log (F)}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin{align*} \int F^{a+\frac{b}{c+d x}} (c+d x)^m \, dx &=\frac{F^a (c+d x)^{1+m} \Gamma \left (-1-m,-\frac{b \log (F)}{c+d x}\right ) \left (-\frac{b \log (F)}{c+d x}\right )^{1+m}}{d}\\ \end{align*}
Mathematica [A] time = 0.0181963, size = 50, normalized size = 1. \[ \frac{F^a (c+d x)^{m+1} \left (-\frac{b \log (F)}{c+d x}\right )^{m+1} \text{Gamma}\left (-m-1,-\frac{b \log (F)}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+{\frac{b}{dx+c}}} \left ( dx+c \right ) ^{m}\, 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{\left (d x + c\right )}^{m} F^{a + \frac{b}{d x + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (d x + c\right )}^{m} F^{\frac{a d x + a c + b}{d x + c}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (d x + c\right )}^{m} F^{a + \frac{b}{d x + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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