Optimal. Leaf size=72 \[ \frac{\left ((d+e x)^n\right )^m F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-m n} \text{Gamma}\left (m n+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
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Rubi [A] time = 0.0536632, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2188, 2181} \[ \frac{\left ((d+e x)^n\right )^m F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-m n} \text{Gamma}\left (m n+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
Antiderivative was successfully verified.
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Rule 2188
Rule 2181
Rubi steps
\begin{align*} \int F^{c (a+b x)} \left ((d+e x)^n\right )^m \, dx &=(d+e x)^{-m n} \left ((d+e x)^n\right )^m \int F^{a c+b c x} (d+e x)^{m n} \, dx\\ &=\frac{F^{c \left (a-\frac{b d}{e}\right )} \left ((d+e x)^n\right )^m \Gamma \left (1+m n,-\frac{b c (d+e x) \log (F)}{e}\right ) \left (-\frac{b c (d+e x) \log (F)}{e}\right )^{-m n}}{b c \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0130439, size = 72, normalized size = 1. \[ \frac{\left ((d+e x)^n\right )^m F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^{-m n} \text{Gamma}\left (m n+1,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{F}^{c \left ( bx+a \right ) } \left ( \left ( ex+d \right ) ^{n} \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 ({\left (e x + d\right )}^{n}\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 = 1.77738, size = 158, normalized size = 2.19 \begin{align*} \frac{e^{\left (-\frac{e m n \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 n + 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 ({\left (e x + d\right )}^{n}\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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