Optimal. Leaf size=71 \[ \frac{F^{a-\frac{b f}{d e-c f}} \text{Ei}\left (\frac{b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{f}-\frac{F^a \text{Ei}\left (\frac{b \log (F)}{c+d x}\right )}{f} \]
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Rubi [A] time = 0.405028, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2222, 2210, 2228, 2178} \[ \frac{F^{a-\frac{b f}{d e-c f}} \text{Ei}\left (\frac{b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{f}-\frac{F^a \text{Ei}\left (\frac{b \log (F)}{c+d x}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 2222
Rule 2210
Rule 2228
Rule 2178
Rubi steps
\begin{align*} \int \frac{F^{a+\frac{b}{c+d x}}}{e+f x} \, dx &=\frac{d \int \frac{F^{a+\frac{b}{c+d x}}}{c+d x} \, dx}{f}-\frac{(d e-c f) \int \frac{F^{a+\frac{b}{c+d x}}}{(c+d x) (e+f x)} \, dx}{f}\\ &=-\frac{F^a \text{Ei}\left (\frac{b \log (F)}{c+d x}\right )}{f}+\frac{\operatorname{Subst}\left (\int \frac{F^{a-\frac{b f}{d e-c f}+\frac{b d x}{d e-c f}}}{x} \, dx,x,\frac{e+f x}{c+d x}\right )}{f}\\ &=-\frac{F^a \text{Ei}\left (\frac{b \log (F)}{c+d x}\right )}{f}+\frac{F^{a-\frac{b f}{d e-c f}} \text{Ei}\left (\frac{b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{f}\\ \end{align*}
Mathematica [A] time = 0.123814, size = 66, normalized size = 0.93 \[ \frac{F^a \left (F^{\frac{b f}{c f-d e}} \text{Ei}\left (\frac{b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )-\text{Ei}\left (\frac{b \log (F)}{c+d x}\right )\right )}{f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.162, size = 106, normalized size = 1.5 \begin{align*}{\frac{{F}^{a}}{f}{\it Ei} \left ( 1,-{\frac{b\ln \left ( F \right ) }{dx+c}} \right ) }-{\frac{1}{f}{F}^{{\frac{acf-ade+bf}{cf-de}}}{\it Ei} \left ( 1,-{\frac{b\ln \left ( F \right ) }{dx+c}}-\ln \left ( F \right ) a-{\frac{-\ln \left ( F \right ) acf+\ln \left ( F \right ) ade-\ln \left ( F \right ) bf}{cf-de}} \right ) } \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 \frac{F^{a + \frac{b}{d x + c}}}{f x + e}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57847, size = 185, normalized size = 2.61 \begin{align*} \frac{F^{\frac{a d e -{\left (a c + b\right )} f}{d e - c f}}{\rm Ei}\left (\frac{{\left (b d f x + b d e\right )} \log \left (F\right )}{c d e - c^{2} f +{\left (d^{2} e - c d f\right )} x}\right ) - F^{a}{\rm Ei}\left (\frac{b \log \left (F\right )}{d x + c}\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{c + d x}}}{e + f x}\, dx \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 \frac{F^{a + \frac{b}{d x + c}}}{f x + e}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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