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

Optimal. Leaf size=106 \[ \frac{F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \left (\frac{b G^{h (f+g x)}}{a}+1\right )^{-n} \, _2F_1\left (-n,\frac{d e \log (F)}{g h \log (G)};\frac{d e \log (F)}{g h \log (G)}+1;-\frac{b G^{h (f+g x)}}{a}\right )}{d e \log (F)} \]

[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])

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Rubi [A]  time = 0.0930117, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2252, 2251} \[ \frac{F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^n \left (\frac{b G^{h (f+g x)}}{a}+1\right )^{-n} \, _2F_1\left (-n,\frac{d e \log (F)}{g h \log (G)};\frac{d e \log (F)}{g h \log (G)}+1;-\frac{b G^{h (f+g x)}}{a}\right )}{d e \log (F)} \]

Antiderivative was successfully verified.

[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 2252

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*F^(e*(c + d*x)))/a)^p, x
], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 2251

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Simp
[(a^p*G^(h*(f + g*x))*Hypergeometric2F1[-p, (g*h*Log[G])/(d*e*Log[F]), (g*h*Log[G])/(d*e*Log[F]) + 1, Simplify
[-((b*F^(e*(c + d*x)))/a)]])/(g*h*Log[G]), x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] ||
GtQ[a, 0])

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 = 0.0453719, size = 92, normalized size = 0.87 \[ \frac{F^{e (c+d x)} \left (a+b G^{h (f+g x)}\right )^{n+1} \, _2F_1\left (1,n+\frac{d e \log (F)}{g h \log (G)}+1;\frac{d e \log (F)}{g h \log (G)}+1;-\frac{b G^{h (f+g x)}}{a}\right )}{a d e \log (F)} \]

Warning: Unable to verify antiderivative.

[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])

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Maple [F]  time = 0.092, size = 0, normalized size = 0. \begin{align*} \int{F}^{e \left ( dx+c \right ) } \left ( a+b{G}^{h \left ( gx+f \right ) } \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (G^{{\left (g x + f\right )} h} b + a\right )}^{n} F^{{\left (d x + c\right )} e}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (G^{g h x + f h} b + a\right )}^{n} F^{d e x + c e}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (G^{{\left (g x + f\right )} h} b + a\right )}^{n} F^{{\left (d x + c\right )} e}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)