Integrand size = 23, antiderivative size = 68 \[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{c+d x} \, dx=\frac {b F^{\left (e-\frac {c f}{d}\right ) g n-g n (e+f x)} \left (F^{e g+f g x}\right )^n \operatorname {ExpIntegralEi}\left (\frac {f g n (c+d x) \log (F)}{d}\right )}{d}+\frac {a \log (c+d x)}{d} \]
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Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2214, 2213, 2209} \[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{c+d x} \, dx=\frac {a \log (c+d x)}{d}+\frac {b \left (F^{e g+f g x}\right )^n F^{g n \left (e-\frac {c f}{d}\right )-g n (e+f x)} \operatorname {ExpIntegralEi}\left (\frac {f g n (c+d x) \log (F)}{d}\right )}{d} \]
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Rule 2209
Rule 2213
Rule 2214
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a}{c+d x}+\frac {b \left (F^{e g+f g x}\right )^n}{c+d x}\right ) \, dx \\ & = \frac {a \log (c+d x)}{d}+b \int \frac {\left (F^{e g+f g x}\right )^n}{c+d x} \, dx \\ & = \frac {a \log (c+d x)}{d}+\left (b F^{-n (e g+f g x)} \left (F^{e g+f g x}\right )^n\right ) \int \frac {F^{n (e g+f g x)}}{c+d x} \, dx \\ & = \frac {b F^{\left (e-\frac {c f}{d}\right ) g n-g n (e+f x)} \left (F^{e g+f g x}\right )^n \text {Ei}\left (\frac {f g n (c+d x) \log (F)}{d}\right )}{d}+\frac {a \log (c+d x)}{d} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{c+d x} \, dx=\frac {b F^{-\frac {f g n (c+d x)}{d}} \left (F^{g (e+f x)}\right )^n \operatorname {ExpIntegralEi}\left (\frac {f g n (c+d x) \log (F)}{d}\right )+a \log (c+d x)}{d} \]
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\[\int \frac {a +b \left (F^{g \left (f x +e \right )}\right )^{n}}{d x +c}d x\]
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none
Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.74 \[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{c+d x} \, dx=\frac {F^{\frac {{\left (d e - c f\right )} g n}{d}} b {\rm Ei}\left (\frac {{\left (d f g n x + c f g n\right )} \log \left (F\right )}{d}\right ) + a \log \left (d x + c\right )}{d} \]
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\[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{c+d x} \, dx=\int \frac {a + b \left (F^{e g + f g x}\right )^{n}}{c + d x}\, dx \]
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\[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{c+d x} \, dx=\int { \frac {{\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a}{d x + c} \,d x } \]
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\[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{c+d x} \, dx=\int { \frac {{\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a}{d x + c} \,d x } \]
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Timed out. \[ \int \frac {a+b \left (F^{g (e+f x)}\right )^n}{c+d x} \, dx=\int \frac {a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n}{c+d\,x} \,d x \]
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