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 = 0.13, antiderivative size = 80, normalized size of antiderivative = 1.00, 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 32
Rule 2246
Rule 2247
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*}
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Mathematica [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \left (G^{h (f+g x)}\right )^{\frac {d e n \log (F)}{g h \log (G)}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.45, size = 88, normalized size = 1.10 \[ \frac {{\left (F^{d e n x + c e n} F^{\frac {{\left (d e f - c e g\right )} n}{g}} b + F^{\frac {{\left (d e f - c e g\right )} n}{g}} a\right )} {\left (F^{d e n x + c e n} b + a\right )}^{p}}{{\left (b d e n p + b d e n\right )} \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.93, size = 156, normalized size = 1.95 \[ \frac {F^{\frac {d f n e}{g}} b e^{\left (2 \, d n x e \log \relax (F) + c n e \log \relax (F) + p \log \left (b e^{\left (d n x e \log \relax (F) + c n e \log \relax (F)\right )} + a\right )\right )} + F^{\frac {d f n e}{g}} a e^{\left (d n x e \log \relax (F) + p \log \left (b e^{\left (d n x e \log \relax (F) + c n e \log \relax (F)\right )} + a\right )\right )}}{b d n p e^{\left (d n x e \log \relax (F) + c n e \log \relax (F) + 1\right )} \log \relax (F) + b d n e^{\left (d n x e \log \relax (F) + c n e \log \relax (F) + 1\right )} \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.74, size = 0, normalized size = 0.00 \[ \int \left (G^{\left (g x +f \right ) h}\right )^{\frac {d e n \ln \relax (F )}{g h \ln \relax (G )}} \left (b \left (F^{\left (d x +c \right ) e}\right )^{n}+a \right )^{p}\, dx \]
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 \[ \int {\left ({\left (F^{{\left (d x + c\right )} e}\right )}^{n} b + a\right )}^{p} {\left (G^{{\left (g x + f\right )} h}\right )}^{\frac {d e n \log \relax (F)}{g h \log \relax (G)}}\,{d x} \]
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 \[ \int {\left (G^{h\,\left (f+g\,x\right )}\right )}^{\frac {d\,e\,n\,\ln \relax (F)}{g\,h\,\ln \relax (G)}}\,{\left (a+b\,{\left (F^{e\,\left (c+d\,x\right )}\right )}^n\right )}^p \,d x \]
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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