Optimal. Leaf size=106 \[ \frac {F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \left (1+\frac {b G^{h (f+g x)}}{a}\right )^{-n} \, _2F_1\left (-n,\frac {d e \log (F)}{g h \log (G)};1+\frac {d e \log (F)}{g h \log (G)};-\frac {b G^{h (f+g x)}}{a}\right )}{d e \log (F)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, 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 = {2284, 2283}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2283
Rule 2284
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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 92, normalized size = 0.87 \begin {gather*} \frac {F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^{1+n} \, _2F_1\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)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int F^{e \left (d x +c \right )} \left (a +b \,G^{h \left (g x +f \right )}\right )^{n}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int F^{e \left (c + d x\right )} \left (G^{f h} G^{g h x} b + a\right )^{n}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 F^{e\,\left (c+d\,x\right )}\,{\left (a+G^{h\,\left (f+g\,x\right )}\,b\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________