Optimal. Leaf size=106 \[ \frac {F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \left (\frac {b G^{h (f+g x)}}{a}+1\right )^{-n} \, _2F_1\left (-n,\frac {d e \log (F)}{g h \log (G)};\frac {d e \log (F)}{g h \log (G)}+1;-\frac {b G^{h (f+g x)}}{a}\right )}{d e \log (F)} \]
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Rubi [A] time = 0.09, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2252, 2251} \[ \frac {F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \left (\frac {b G^{h (f+g x)}}{a}+1\right )^{-n} \, _2F_1\left (-n,\frac {d e \log (F)}{g h \log (G)};\frac {d e \log (F)}{g h \log (G)}+1;-\frac {b G^{h (f+g x)}}{a}\right )}{d e \log (F)} \]
Antiderivative was successfully verified.
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Rule 2251
Rule 2252
Rubi steps
\begin {align*} \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx &=\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*}
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Mathematica [A] time = 0.05, size = 92, normalized size = 0.87 \[ \frac {F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^{n+1} \, _2F_1\left (1,n+\frac {d e \log (F)}{g h \log (G)}+1;\frac {d e \log (F)}{g h \log (G)}+1;-\frac {b G^{h (f+g x)}}{a}\right )}{a d e \log (F)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (G^{g h x + f h} b + a\right )}^{n} F^{d e x + c e}, x\right ) \]
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 \[ \int {\left (G^{{\left (g x + f\right )} h} b + a\right )}^{n} F^{{\left (d x + c\right )} e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int F^{\left (d x +c \right ) e} \left (b \,G^{\left (g x +f \right ) h}+a \right )^{n}\, dx \]
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 \[ \int {\left (G^{{\left (g x + f\right )} h} b + a\right )}^{n} F^{{\left (d x + c\right )} e}\,{d x} \]
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 \[ \int F^{e\,\left (c+d\,x\right )}\,{\left (a+G^{h\,\left (f+g\,x\right )}\,b\right )}^n \,d x \]
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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