Optimal. Leaf size=67 \[ \frac{(d+e x)^m F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-m} \text{Gamma}\left (m+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
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Rubi [A] time = 0.0399163, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2181} \[ \frac{(d+e x)^m F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-m} \text{Gamma}\left (m+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
Antiderivative was successfully verified.
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Rule 2181
Rubi steps
\begin{align*} \int F^{c (a+b x)} (d+e x)^m \, dx &=\frac{F^{c \left (a-\frac{b d}{e}\right )} (d+e x)^m \Gamma \left (1+m,-\frac{b c (d+e x) \log (F)}{e}\right ) \left (-\frac{b c (d+e x) \log (F)}{e}\right )^{-m}}{b c \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0382056, size = 67, normalized size = 1. \[ \frac{(d+e x)^m F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-m} \text{Gamma}\left (m+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{F}^{c \left ( bx+a \right ) } \left ( ex+d \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 (e x + d\right )}^{m} F^{{\left (b x + a\right )} c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52543, size = 153, normalized size = 2.28 \begin{align*} \frac{e^{\left (-\frac{e m \log \left (-\frac{b c \log \left (F\right )}{e}\right ) +{\left (b c d - a c e\right )} \log \left (F\right )}{e}\right )} \Gamma \left (m + 1, -\frac{{\left (b c e x + b c d\right )} \log \left (F\right )}{e}\right )}{b c \log \left (F\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 (e x + d\right )}^{m} F^{{\left (b x + a\right )} c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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