Optimal. Leaf size=80 \[ \frac {\left (F^{e (c+d x)}\right )^{-n} \left (a+b \left (F^{e (c+d x)}\right )^n\right )^{1+p} \left (G^{h (f+g x)}\right )^{\frac {d e n \log (F)}{g h \log (G)}}}{b d e n (1+p) \log (F)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, 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 = {2279, 2278, 32}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 2278
Rule 2279
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 ) \text {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 = 0.35, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.23, size = 0, normalized size = 0.00 \[\int \left (a +b \left (F^{e \left (d x +c \right )}\right )^{n}\right )^{p} \left (G^{h \left (g x +f \right )}\right )^{\frac {d e n \ln \left (F \right )}{g h \ln \left (G \right )}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.31, size = 86, normalized size = 1.08 \begin {gather*} \frac {{\left (F^{d e n x} F^{c e n + \frac {d e f n}{g}} b + F^{\frac {d e f n}{g}} a\right )} {\left (F^{d e n x} F^{c e n} b + a\right )}^{p}}{F^{c e n} b d e n {\left (p + 1\right )} \log \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.41, size = 90, normalized size = 1.12 \begin {gather*} \frac {{\left (F^{{\left (d n x + c n\right )} e} F^{\frac {{\left (d f - c g\right )} n e}{g}} b + F^{\frac {{\left (d f - c g\right )} n e}{g}} a\right )} {\left (F^{{\left (d n x + c n\right )} e} b + a\right )}^{p} e^{\left (-1\right )}}{{\left (b d n p + b d n\right )} \log \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 2.95, size = 156, normalized size = 1.95 \begin {gather*} \frac {F^{\frac {d f n e}{g}} b e^{\left (2 \, d n x e \log \left (F\right ) + c n e \log \left (F\right ) + p \log \left (b e^{\left (d n x e \log \left (F\right ) + c n e \log \left (F\right )\right )} + a\right )\right )} + F^{\frac {d f n e}{g}} a e^{\left (d n x e \log \left (F\right ) + p \log \left (b e^{\left (d n x e \log \left (F\right ) + c n e \log \left (F\right )\right )} + a\right )\right )}}{b d n p e^{\left (d n x e \log \left (F\right ) + c n e \log \left (F\right ) + 1\right )} \log \left (F\right ) + b d n e^{\left (d n x e \log \left (F\right ) + c n e \log \left (F\right ) + 1\right )} \log \left (F\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (G^{h\,\left (f+g\,x\right )}\right )}^{\frac {d\,e\,n\,\ln \left (F\right )}{g\,h\,\ln \left (G\right )}}\,{\left (a+b\,{\left (F^{e\,\left (c+d\,x\right )}\right )}^n\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________