Optimal. Leaf size=80 \[ \frac{\left (F^{e (c+d x)}\right )^{-n} \left (a+b \left (F^{e (c+d x)}\right )^n\right )^{p+1} \left (G^{h (f+g x)}\right )^{\frac{d e n \log (F)}{g h \log (G)}}}{b d e n (p+1) \log (F)} \]
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Rubi [A] time = 0.134639, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068, Rules used = {2247, 2246, 32} \[ \frac{\left (F^{e (c+d x)}\right )^{-n} \left (a+b \left (F^{e (c+d x)}\right )^n\right )^{p+1} \left (G^{h (f+g x)}\right )^{\frac{d e n \log (F)}{g h \log (G)}}}{b d e n (p+1) \log (F)} \]
Antiderivative was successfully verified.
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Rule 2247
Rule 2246
Rule 32
Rubi steps
\begin{align*} \int \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \left (G^{h (f+g x)}\right )^{\frac{d e n \log (F)}{g h \log (G)}} \, dx &=\left (\left (F^{e (c+d x)}\right )^{-n} \left (G^{h (f+g x)}\right )^{\frac{d e n \log (F)}{g h \log (G)}}\right ) \int \left (F^{e (c+d x)}\right )^n \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \, dx\\ &=\frac{\left (\left (F^{e (c+d x)}\right )^{-n} \left (G^{h (f+g x)}\right )^{\frac{d e n \log (F)}{g h \log (G)}}\right ) \operatorname{Subst}\left (\int (a+b x)^p \, dx,x,\left (F^{e (c+d x)}\right )^n\right )}{d e n \log (F)}\\ &=\frac{\left (F^{e (c+d x)}\right )^{-n} \left (a+b \left (F^{e (c+d x)}\right )^n\right )^{1+p} \left (G^{h (f+g x)}\right )^{\frac{d e n \log (F)}{g h \log (G)}}}{b d e n (1+p) \log (F)}\\ \end{align*}
Mathematica [F] time = 0.302933, size = 0, normalized size = 0. \[ \int \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \left (G^{h (f+g x)}\right )^{\frac{d e n \log (F)}{g h \log (G)}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.802, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ({F}^{e \left ( dx+c \right ) } \right ) ^{n} \right ) ^{p} \left ({G}^{h \left ( gx+f \right ) } \right ) ^{{\frac{nde\ln \left ( F \right ) }{gh\ln \left ( G \right ) }}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left ({\left (F^{{\left (d x + c\right )} e}\right )}^{n} b + a\right )}^{p}{\left (G^{{\left (g x + f\right )} h}\right )}^{\frac{d e n \log \left (F\right )}{g h \log \left (G\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58859, size = 186, normalized size = 2.32 \begin{align*} \frac{{\left (F^{d e n x + c e n} F^{\frac{{\left (d e f - c e g\right )} n}{g}} b + F^{\frac{{\left (d e f - c e g\right )} n}{g}} a\right )}{\left (F^{d e n x + c e n} b + a\right )}^{p}}{{\left (b d e n p + b d e n\right )} \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left ({\left (F^{{\left (d x + c\right )} e}\right )}^{n} b + a\right )}^{p}{\left (G^{{\left (g x + f\right )} h}\right )}^{\frac{d e n \log \left (F\right )}{g h \log \left (G\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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