Optimal. Leaf size=116 \[ -\frac{b d \log (F) F^{a-\frac{b f}{d e-c f}} \text{Ei}\left (\frac{b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{(d e-c f)^2}+\frac{d F^{a+\frac{b}{c+d x}}}{f (d e-c f)}-\frac{F^{a+\frac{b}{c+d x}}}{f (e+f x)} \]
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Rubi [A] time = 1.01075, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2223, 6742, 2209, 2210, 2222, 2228, 2178} \[ -\frac{b d \log (F) F^{a-\frac{b f}{d e-c f}} \text{Ei}\left (\frac{b d (e+f x) \log (F)}{(d e-c f) (c+d x)}\right )}{(d e-c f)^2}+\frac{d F^{a+\frac{b}{c+d x}}}{f (d e-c f)}-\frac{F^{a+\frac{b}{c+d x}}}{f (e+f x)} \]
Antiderivative was successfully verified.
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Rule 2223
Rule 6742
Rule 2209
Rule 2210
Rule 2222
Rule 2228
Rule 2178
Rubi steps
\begin{align*} \int \frac{F^{a+\frac{b}{c+d x}}}{(e+f x)^2} \, dx &=-\frac{F^{a+\frac{b}{c+d x}}}{f (e+f x)}-\frac{(b d \log (F)) \int \frac{F^{a+\frac{b}{c+d x}}}{(c+d x)^2 (e+f x)} \, dx}{f}\\ &=-\frac{F^{a+\frac{b}{c+d x}}}{f (e+f x)}-\frac{(b d \log (F)) \int \left (\frac{d F^{a+\frac{b}{c+d x}}}{(d e-c f) (c+d x)^2}-\frac{d f F^{a+\frac{b}{c+d x}}}{(d e-c f)^2 (c+d x)}+\frac{f^2 F^{a+\frac{b}{c+d x}}}{(d e-c f)^2 (e+f x)}\right ) \, dx}{f}\\ &=-\frac{F^{a+\frac{b}{c+d x}}}{f (e+f x)}+\frac{\left (b d^2 \log (F)\right ) \int \frac{F^{a+\frac{b}{c+d x}}}{c+d x} \, dx}{(d e-c f)^2}-\frac{(b d f \log (F)) \int \frac{F^{a+\frac{b}{c+d x}}}{e+f x} \, dx}{(d e-c f)^2}-\frac{\left (b d^2 \log (F)\right ) \int \frac{F^{a+\frac{b}{c+d x}}}{(c+d x)^2} \, dx}{f (d e-c f)}\\ &=\frac{d F^{a+\frac{b}{c+d x}}}{f (d e-c f)}-\frac{F^{a+\frac{b}{c+d x}}}{f (e+f x)}-\frac{b d F^a \text{Ei}\left (\frac{b \log (F)}{c+d x}\right ) \log (F)}{(d e-c f)^2}-\frac{\left (b d^2 \log (F)\right ) \int \frac{F^{a+\frac{b}{c+d x}}}{c+d x} \, dx}{(d e-c f)^2}+\frac{(b d \log (F)) \int \frac{F^{a+\frac{b}{c+d x}}}{(c+d x) (e+f x)} \, dx}{d e-c f}\\ &=\frac{d F^{a+\frac{b}{c+d x}}}{f (d e-c f)}-\frac{F^{a+\frac{b}{c+d x}}}{f (e+f x)}-\frac{(b d \log (F)) \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 )}{(d e-c f)^2}\\ &=\frac{d F^{a+\frac{b}{c+d x}}}{f (d e-c f)}-\frac{F^{a+\frac{b}{c+d x}}}{f (e+f x)}-\frac{b d 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 ) \log (F)}{(d e-c f)^2}\\ \end{align*}
Mathematica [A] time = 0.339725, size = 116, normalized size = 1. \[ -\frac{b d \log (F) F^{a+\frac{b f}{c f-d e}} \text{Ei}\left (\frac{b \log (F)}{c+d x}-\frac{b f \log (F)}{c f-d e}\right )}{(d e-c f)^2}+\frac{d F^{a+\frac{b}{c+d x}}}{f (d e-c f)}-\frac{F^{a+\frac{b}{c+d x}}}{f (e+f x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.128, size = 191, normalized size = 1.7 \begin{align*}{\frac{\ln \left ( F \right ) bd{F}^{a}}{ \left ( cf-de \right ) ^{2}}{F}^{{\frac{b}{dx+c}}} \left ({\frac{b\ln \left ( F \right ) }{dx+c}}+\ln \left ( F \right ) a-{\frac{\ln \left ( F \right ) acf}{cf-de}}+{\frac{\ln \left ( F \right ) ade}{cf-de}}-{\frac{\ln \left ( F \right ) bf}{cf-de}} \right ) ^{-1}}+{\frac{\ln \left ( F \right ) bd}{ \left ( cf-de \right ) ^{2}}{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}}}{{\left (f x + e\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6438, size = 375, normalized size = 3.23 \begin{align*} -\frac{{\left (b d f x + b d e\right )} 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 ) \log \left (F\right ) -{\left (c d e - c^{2} f +{\left (d^{2} e - c d f\right )} x\right )} F^{\frac{a d x + a c + b}{d x + c}}}{d^{2} e^{3} - 2 \, c d e^{2} f + c^{2} e f^{2} +{\left (d^{2} e^{2} f - 2 \, c d e f^{2} + c^{2} f^{3}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{d x + c}}}{{\left (f x + e\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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