Optimal. Leaf size=72 \[ \frac {\left ((d+e x)^n\right )^m F^{c \left (a-\frac {b d}{e}\right )} \left (-\frac {b c \log (F) (d+e x)}{e}\right )^{-m n} \Gamma \left (m n+1,-\frac {b c (d+e x) \log (F)}{e}\right )}{b c \log (F)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 72, normalized size of antiderivative = 1.00, 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.
[In]
[Out]
Rule 2181
Rule 2188
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 = 0.01, size = 72, normalized size = 1.00 \[ \frac {\left ((d+e x)^n\right )^m F^{c \left (a-\frac {b d}{e}\right )} \left (-\frac {b c \log (F) (d+e x)}{e}\right )^{-m n} \Gamma \left (m n+1,-\frac {b c (d+e x) \log (F)}{e}\right )}{b c \log (F)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 68, normalized size = 0.94 \[ \frac {e^{\left (-\frac {e m n \log \left (-\frac {b c \log \relax (F)}{e}\right ) + {\left (b c d - a c e\right )} \log \relax (F)}{e}\right )} \Gamma \left (m n + 1, -\frac {{\left (b c e x + b c d\right )} \log \relax (F)}{e}\right )}{b c \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left ({\left (e x + d\right )}^{n}\right )}^{m} F^{{\left (b x + a\right )} c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int F^{\left (b x +a \right ) c} \left (\left (e x +d \right )^{n}\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left ({\left (e x + d\right )}^{n}\right )}^{m} F^{{\left (b x + a\right )} c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int F^{c\,\left (a+b\,x\right )}\,{\left ({\left (d+e\,x\right )}^n\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{c \left (a + b x\right )} \left (\left (d + e x\right )^{n}\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________