Integrand size = 25, antiderivative size = 106 \[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\frac {F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \left (1+\frac {b G^{h (f+g x)}}{a}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,\frac {d e \log (F)}{g h \log (G)},1+\frac {d e \log (F)}{g h \log (G)},-\frac {b G^{h (f+g x)}}{a}\right )}{d e \log (F)} \] Output:
F^(e*(d*x+c))*(a+b*G^(h*(g*x+f)))^n*hypergeom([-n, d*e*ln(F)/g/h/ln(G)],[1 +d*e*ln(F)/g/h/ln(G)],-b*G^(h*(g*x+f))/a)/d/e/((1+b*G^(h*(g*x+f))/a)^n)/ln (F)
Time = 0.14 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\frac {F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n+\frac {d e \log (F)}{g h \log (G)},1+\frac {d e \log (F)}{g h \log (G)},-\frac {b G^{h (f+g x)}}{a}\right )}{a d e \log (F)} \] Input:
Integrate[F^(e*(c + d*x))*(a + b*G^(h*(f + g*x)))^n,x]
Output:
(F^(e*(c + d*x))*(a + b*G^(h*(f + g*x)))^(1 + n)*Hypergeometric2F1[1, 1 + n + (d*e*Log[F])/(g*h*Log[G]), 1 + (d*e*Log[F])/(g*h*Log[G]), -((b*G^(h*(f + g*x)))/a)])/(a*d*e*Log[F])
Time = 0.49 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2682, 2681}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx\) |
\(\Big \downarrow \) 2682 |
\(\displaystyle \left (a+b G^{h (f+g x)}\right )^n \left (\frac {b G^{h (f+g x)}}{a}+1\right )^{-n} \int F^{e (c+d x)} \left (\frac {b G^{h (f+g x)}}{a}+1\right )^ndx\) |
\(\Big \downarrow \) 2681 |
\(\displaystyle \frac {F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \left (\frac {b G^{h (f+g x)}}{a}+1\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,\frac {d e \log (F)}{g h \log (G)},\frac {d e \log (F)}{g h \log (G)}+1,-\frac {b G^{h (f+g x)}}{a}\right )}{d e \log (F)}\) |
Input:
Int[F^(e*(c + d*x))*(a + b*G^(h*(f + g*x)))^n,x]
Output:
(F^(e*(c + d*x))*(a + b*G^(h*(f + g*x)))^n*Hypergeometric2F1[-n, (d*e*Log[ F])/(g*h*Log[G]), 1 + (d*e*Log[F])/(g*h*Log[G]), -((b*G^(h*(f + g*x)))/a)] )/(d*e*(1 + (b*G^(h*(f + g*x)))/a)^n*Log[F])
Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_. ) + (g_.)*(x_))), x_Symbol] :> Simp[a^p*(G^(h*(f + g*x))/(g*h*Log[G]))*Hype rgeometric2F1[-p, g*h*(Log[G]/(d*e*Log[F])), g*h*(Log[G]/(d*e*Log[F])) + 1, Simplify[(-b/a)*F^(e*(c + d*x))]], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] || GtQ[a, 0])
Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_. ) + (g_.)*(x_))), x_Symbol] :> Simp[(a + b*F^(e*(c + d*x)))^p/(1 + (b/a)*F^ (e*(c + d*x)))^p Int[G^(h*(f + g*x))*(1 + (b/a)*F^(e*(c + d*x)))^p, x], x ] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && !(ILtQ[p, 0] || GtQ[a, 0])
\[\int F^{e \left (d x +c \right )} \left (a +b \,G^{h \left (g x +f \right )}\right )^{n}d x\]
Input:
int(F^(e*(d*x+c))*(a+b*G^(h*(g*x+f)))^n,x)
Output:
int(F^(e*(d*x+c))*(a+b*G^(h*(g*x+f)))^n,x)
\[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\int { {\left (G^{{\left (g x + f\right )} h} b + a\right )}^{n} F^{{\left (d x + c\right )} e} \,d x } \] Input:
integrate(F^(e*(d*x+c))*(a+b*G^(h*(g*x+f)))^n,x, algorithm="fricas")
Output:
integral((G^(g*h*x + f*h)*b + a)^n*F^(d*e*x + c*e), x)
Timed out. \[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\text {Timed out} \] Input:
integrate(F**(e*(d*x+c))*(a+b*G**(h*(g*x+f)))**n,x)
Output:
Timed out
\[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\int { {\left (G^{{\left (g x + f\right )} h} b + a\right )}^{n} F^{{\left (d x + c\right )} e} \,d x } \] Input:
integrate(F^(e*(d*x+c))*(a+b*G^(h*(g*x+f)))^n,x, algorithm="maxima")
Output:
integrate((G^((g*x + f)*h)*b + a)^n*F^((d*x + c)*e), x)
\[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\int { {\left (G^{{\left (g x + f\right )} h} b + a\right )}^{n} F^{{\left (d x + c\right )} e} \,d x } \] Input:
integrate(F^(e*(d*x+c))*(a+b*G^(h*(g*x+f)))^n,x, algorithm="giac")
Output:
integrate((G^((g*x + f)*h)*b + a)^n*F^((d*x + c)*e), x)
Timed out. \[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\int F^{e\,\left (c+d\,x\right )}\,{\left (a+G^{h\,\left (f+g\,x\right )}\,b\right )}^n \,d x \] Input:
int(F^(e*(c + d*x))*(a + G^(h*(f + g*x))*b)^n,x)
Output:
int(F^(e*(c + d*x))*(a + G^(h*(f + g*x))*b)^n, x)
\[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=f^{c e} \left (\int f^{d e x} \left (g^{g h x +f h} b +a \right )^{n}d x \right ) \] Input:
int(F^(e*(d*x+c))*(a+b*G^(h*(g*x+f)))^n,x)
Output:
f**(c*e)*int(f**(d*e*x)*(g**(f*h + g*h*x)*b + a)**n,x)