Optimal. Leaf size=69 \[ \frac{(d+e x)^{-m} F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^m \text{Gamma}\left (1-m,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.024389, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2181} \[ \frac{(d+e x)^{-m} F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^m \text{Gamma}\left (1-m,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
Antiderivative was successfully verified.
[In]
[Out]
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 = 0.0148788, size = 69, normalized size = 1. \[ \frac{(d+e x)^{-m} F^{c \left (a-\frac{b d}{e}\right )} \left (-\frac{b c \log (F) (d+e x)}{e}\right )^m \text{Gamma}\left (1-m,-\frac{b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.045, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{c \left ( bx+a \right ) }}{ \left ( ex+d \right ) ^{m}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{m}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.56557, size = 153, normalized size = 2.22 \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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{m}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]