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 = 0.41, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, 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 2178
Rule 2210
Rule 2222
Rule 2228
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*}
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Mathematica [A] time = 0.13, 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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fricas [A] time = 0.43, size = 89, normalized size = 1.25 \[ \frac {F^{\frac {a d e - {\left (a c + b\right )} f}{d e - c f}} {\rm Ei}\left (\frac {{\left (b d f x + b d e\right )} \log \relax (F)}{c d e - c^{2} f + {\left (d^{2} e - c d f\right )} x}\right ) - F^{a} {\rm Ei}\left (\frac {b \log \relax (F)}{d x + c}\right )}{f} \]
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 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{f x + e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 106, normalized size = 1.49 \[ \frac {F^{a} \Ei \left (1, -\frac {b \ln \relax (F )}{d x +c}\right )}{f}-\frac {F^{\frac {a c f -a d e +b f}{c f -d e}} \Ei \left (1, -a \ln \relax (F )-\frac {b \ln \relax (F )}{d x +c}-\frac {-a c f \ln \relax (F )+a d e \ln \relax (F )-b f \ln \relax (F )}{c f -d e}\right )}{f} \]
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 \[ \int \frac {F^{a + \frac {b}{d x + c}}}{f x + e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {F^{a+\frac {b}{c+d\,x}}}{e+f\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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