Optimal. Leaf size=67 \[ \frac {(d+e x)^m F^{c \left (a-\frac {b d}{e}\right )} \left (-\frac {b c \log (F) (d+e x)}{e}\right )^{-m} \Gamma \left (m+1,-\frac {b c (d+e x) \log (F)}{e}\right )}{b c \log (F)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, 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.
[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 = 0.04, size = 67, normalized size = 1.00 \[ \frac {(d+e x)^m F^{c \left (a-\frac {b d}{e}\right )} \left (-\frac {b c \log (F) (d+e x)}{e}\right )^{-m} \Gamma \left (m+1,-\frac {b c (d+e x) \log (F)}{e}\right )}{b c \log (F)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.60, size = 65, normalized size = 0.97 \[ \frac {e^{\left (-\frac {e m \log \left (-\frac {b c \log \relax (F)}{e}\right ) + {\left (b c d - a c e\right )} \log \relax (F)}{e}\right )} \Gamma \left (m + 1, -\frac {{\left (b c e x + b c d\right )} \log \relax (F)}{e}\right )}{b c \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )}^{m} F^{{\left (b x + a\right )} c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int F^{\left (b x +a \right ) c} \left (e x +d \right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x + d\right )}^{m} F^{{\left (b x + a\right )} c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int F^{c\,\left (a+b\,x\right )}\,{\left (d+e\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{c \left (a + b x\right )} \left (d + e x\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________