Optimal. Leaf size=57 \[ \frac{b c \log (F) F^{c \left (a-\frac{b d}{e}\right )} \text{Ei}\left (\frac{b c (d+e x) \log (F)}{e}\right )}{e^2}-\frac{F^{c (a+b x)}}{e (d+e x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.043363, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2177, 2178} \[ \frac{b c \log (F) F^{c \left (a-\frac{b d}{e}\right )} \text{Ei}\left (\frac{b c (d+e x) \log (F)}{e}\right )}{e^2}-\frac{F^{c (a+b x)}}{e (d+e x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2177
Rule 2178
Rubi steps
\begin{align*} \int \frac{F^{c (a+b x)}}{(d+e x)^2} \, dx &=-\frac{F^{c (a+b x)}}{e (d+e x)}+\frac{(b c \log (F)) \int \frac{F^{c (a+b x)}}{d+e x} \, dx}{e}\\ &=-\frac{F^{c (a+b x)}}{e (d+e x)}+\frac{b c F^{c \left (a-\frac{b d}{e}\right )} \text{Ei}\left (\frac{b c (d+e x) \log (F)}{e}\right ) \log (F)}{e^2}\\ \end{align*}
Mathematica [A] time = 0.118915, size = 55, normalized size = 0.96 \[ \frac{F^{a c} \left (b c \log (F) F^{-\frac{b c d}{e}} \text{Ei}\left (\frac{b c (d+e x) \log (F)}{e}\right )-\frac{e F^{b c x}}{d+e x}\right )}{e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.038, size = 99, normalized size = 1.7 \begin{align*} -{\frac{bc\ln \left ( F \right ){F}^{bcx}{F}^{ac}}{{e}^{2}} \left ( bcx\ln \left ( F \right ) +{\frac{\ln \left ( F \right ) bcd}{e}} \right ) ^{-1}}-{\frac{bc\ln \left ( F \right ) }{{e}^{2}}{F}^{{\frac{c \left ( ae-bd \right ) }{e}}}{\it Ei} \left ( 1,-bcx\ln \left ( F \right ) -ac\ln \left ( F \right ) -{\frac{-\ln \left ( F \right ) ace+\ln \left ( F \right ) bcd}{e}} \right ) } \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 )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.53936, size = 159, normalized size = 2.79 \begin{align*} -\frac{F^{b c x + a c} e - \frac{{\left (b c e x + b c d\right )}{\rm Ei}\left (\frac{{\left (b c e x + b c d\right )} \log \left (F\right )}{e}\right ) \log \left (F\right )}{F^{\frac{b c d - a c e}{e}}}}{e^{3} x + d e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{c \left (a + b x\right )}}{\left (d + e x\right )^{2}}\, dx \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 )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]