Optimal. Leaf size=57 \[ \frac {b c \log (F) F^{c \left (a-\frac {b d}{e}\right )} \text {Ei}\left (\frac {b c (d+e x) \log (F)}{e}\right )}{e^2}-\frac {F^{c (a+b x)}}{e (d+e x)} \]
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Rubi [A] time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, 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.
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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*}
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Mathematica [A] time = 0.12, 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.
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fricas [A] time = 0.42, size = 77, normalized size = 1.35 \[ -\frac {F^{b c x + a c} e - \frac {{\left (b c e x + b c d\right )} {\rm Ei}\left (\frac {{\left (b c e x + b c d\right )} \log \relax (F)}{e}\right ) \log \relax (F)}{F^{\frac {b c d - a c e}{e}}}}{e^{3} x + d e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 99, normalized size = 1.74 \[ -\frac {b c \,F^{a c} F^{b c x} \ln \relax (F )}{\left (b c x \ln \relax (F )+\frac {b c d \ln \relax (F )}{e}\right ) e^{2}}-\frac {b c \,F^{\frac {\left (a e -b d \right ) c}{e}} \Ei \left (1, -b c x \ln \relax (F )-a c \ln \relax (F )-\frac {-a c e \ln \relax (F )+b c d \ln \relax (F )}{e}\right ) \ln \relax (F )}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{{\left (b x + a\right )} c}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {F^{c\,\left (a+b\,x\right )}}{{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{c \left (a + b x\right )}}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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