\(\int F^{e (c+d x)} (a+b G^{h (f+g x)})^n \, dx\) [66]

Optimal result

Integrand size = 25, antiderivative size = 106 \[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\frac {F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \left (1+\frac {b G^{h (f+g x)}}{a}\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,\frac {d e \log (F)}{g h \log (G)},1+\frac {d e \log (F)}{g h \log (G)},-\frac {b G^{h (f+g x)}}{a}\right )}{d e \log (F)} \]

[Out]

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)

Rubi [A] (verified)

Time = 0.06 (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 = {2284, 2283} \[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\frac {F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \left (\frac {b G^{h (f+g x)}}{a}+1\right )^{-n} \operatorname {Hypergeometric2F1}\left (-n,\frac {d e \log (F)}{g h \log (G)},\frac {d e \log (F)}{g h \log (G)}+1,-\frac {b G^{h (f+g x)}}{a}\right )}{d e \log (F)} \]

[In]

Int[F^(e*(c + d*x))*(a + b*G^(h*(f + g*x)))^n,x]

[Out]

(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])

Rule 2283

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]))*Hypergeometric2F1[-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] || G
tQ[a, 0])

Rule 2284

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Dist
[(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])

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.09 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.87 \[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\frac {F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n+\frac {d e \log (F)}{g h \log (G)},1+\frac {d e \log (F)}{g h \log (G)},-\frac {b G^{h (f+g x)}}{a}\right )}{a d e \log (F)} \]

[In]

Integrate[F^(e*(c + d*x))*(a + b*G^(h*(f + g*x)))^n,x]

[Out]

(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])

Maple [F]

\[\int F^{e \left (d x +c \right )} \left (a +b \,G^{h \left (g x +f \right )}\right )^{n}d x\]

[In]

int(F^(e*(d*x+c))*(a+b*G^(h*(g*x+f)))^n,x)

[Out]

int(F^(e*(d*x+c))*(a+b*G^(h*(g*x+f)))^n,x)

Fricas [F]

\[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\int { {\left (G^{{\left (g x + f\right )} h} b + a\right )}^{n} F^{{\left (d x + c\right )} e} \,d x } \]

[In]

integrate(F^(e*(d*x+c))*(a+b*G^(h*(g*x+f)))^n,x, algorithm="fricas")

[Out]

integral((G^(g*h*x + f*h)*b + a)^n*F^(d*e*x + c*e), x)

Sympy [F(-1)]

Timed out. \[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\text {Timed out} \]

[In]

integrate(F**(e*(d*x+c))*(a+b*G**(h*(g*x+f)))**n,x)

[Out]

Timed out

Maxima [F]

\[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\int { {\left (G^{{\left (g x + f\right )} h} b + a\right )}^{n} F^{{\left (d x + c\right )} e} \,d x } \]

[In]

integrate(F^(e*(d*x+c))*(a+b*G^(h*(g*x+f)))^n,x, algorithm="maxima")

[Out]

integrate((G^((g*x + f)*h)*b + a)^n*F^((d*x + c)*e), x)

Giac [F]

\[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\int { {\left (G^{{\left (g x + f\right )} h} b + a\right )}^{n} F^{{\left (d x + c\right )} e} \,d x } \]

[In]

integrate(F^(e*(d*x+c))*(a+b*G^(h*(g*x+f)))^n,x, algorithm="giac")

[Out]

integrate((G^((g*x + f)*h)*b + a)^n*F^((d*x + c)*e), x)

Mupad [F(-1)]

Timed out. \[ \int F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \, dx=\int F^{e\,\left (c+d\,x\right )}\,{\left (a+G^{h\,\left (f+g\,x\right )}\,b\right )}^n \,d x \]

[In]

int(F^(e*(c + d*x))*(a + G^(h*(f + g*x))*b)^n,x)

[Out]

int(F^(e*(c + d*x))*(a + G^(h*(f + g*x))*b)^n, x)