Integrand size = 25, antiderivative size = 106 \[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\frac {F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \left (1+\frac {b G^{h (f+g x)}}{a}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,\frac {d e \log (F)}{g h \log (G)},1+\frac {d e \log (F)}{g h \log (G)},-\frac {b G^{h (f+g x)}}{a}\right )}{d e \log (F)} \]
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Time = 0.06 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2284, 2283} \[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\frac {F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \left (\frac {b G^{h (f+g x)}}{a}+1\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,\frac {d e \log (F)}{g h \log (G)},\frac {d e \log (F)}{g h \log (G)}+1,-\frac {b G^{h (f+g x)}}{a}\right )}{d e \log (F)} \]
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Rule 2283
Rule 2284
Rubi steps \begin{align*} \text {integral}& = \left (\left (a+b G^{h (f+g x)}\right )^n \left (1+\frac {b G^{h (f+g x)}}{a}\right )^{-n}\right ) \int F^{e (c+d x)} \left (1+\frac {b G^{h (f+g x)}}{a}\right )^n \, dx \\ & = \frac {F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \left (1+\frac {b G^{h (f+g x)}}{a}\right )^{-n} \, _2F_1\left (-n,\frac {d e \log (F)}{g h \log (G)};1+\frac {d e \log (F)}{g h \log (G)};-\frac {b G^{h (f+g x)}}{a}\right )}{d e \log (F)} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\frac {F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n+\frac {d e \log (F)}{g h \log (G)},1+\frac {d e \log (F)}{g h \log (G)},-\frac {b G^{h (f+g x)}}{a}\right )}{a d e \log (F)} \]
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\[\int F^{e \left (d x +c \right )} \left (a +b \,G^{h \left (g x +f \right )}\right )^{n}d x\]
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\[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\int { {\left (G^{{\left (g x + f\right )} h} b + a\right )}^{n} F^{{\left (d x + c\right )} e} \,d x } \]
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Timed out. \[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\text {Timed out} \]
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\[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\int { {\left (G^{{\left (g x + f\right )} h} b + a\right )}^{n} F^{{\left (d x + c\right )} e} \,d x } \]
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\[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\int { {\left (G^{{\left (g x + f\right )} h} b + a\right )}^{n} F^{{\left (d x + c\right )} e} \,d x } \]
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Timed out. \[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\int F^{e\,\left (c+d\,x\right )}\,{\left (a+G^{h\,\left (f+g\,x\right )}\,b\right )}^n \,d x \]
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