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

Optimal. Leaf size=80 \[ \frac{\left (F^{e (c+d x)}\right )^{-n} \left (a+b \left (F^{e (c+d x)}\right )^n\right )^{p+1} \left (G^{h (f+g x)}\right )^{\frac{d e n \log (F)}{g h \log (G)}}}{b d e n (p+1) \log (F)} \]

[Out]

((a + b*(F^(e*(c + d*x)))^n)^(1 + p)*(G^(h*(f + g*x)))^((d*e*n*Log[F])/(g*h*Log[G])))/(b*d*e*(F^(e*(c + d*x)))
^n*n*(1 + p)*Log[F])

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Rubi [A]  time = 0.134639, antiderivative size = 80, normalized size of antiderivative = 1., 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 = {2247, 2246, 32} \[ \frac{\left (F^{e (c+d x)}\right )^{-n} \left (a+b \left (F^{e (c+d x)}\right )^n\right )^{p+1} \left (G^{h (f+g x)}\right )^{\frac{d e n \log (F)}{g h \log (G)}}}{b d e n (p+1) \log (F)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*(F^(e*(c + d*x)))^n)^(1 + p)*(G^(h*(f + g*x)))^((d*e*n*Log[F])/(g*h*Log[G])))/(b*d*e*(F^(e*(c + d*x)))
^n*n*(1 + p)*Log[F])

Rule 2247

Int[((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.)*((G_)^((h_.)*((f_.) + (g_.)*(x_))))^(m_.),
x_Symbol] :> Dist[(G^(h*(f + g*x)))^m/(F^(e*(c + d*x)))^n, Int[(F^(e*(c + d*x)))^n*(a + b*(F^(e*(c + d*x)))^n)
^p, x], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, m, n, p}, x] && EqQ[d*e*n*Log[F], g*h*m*Log[G]]

Rule 2246

Int[((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)*((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.),
x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int[(a + b*x)^p, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b,
c, d, e, n, p}, x]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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 ) \operatorname{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 = 0.302933, size = 0, normalized size = 0. \[ \int \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \left (G^{h (f+g x)}\right )^{\frac{d e n \log (F)}{g h \log (G)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.802, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ({F}^{e \left ( dx+c \right ) } \right ) ^{n} \right ) ^{p} \left ({G}^{h \left ( gx+f \right ) } \right ) ^{{\frac{nde\ln \left ( F \right ) }{gh\ln \left ( G \right ) }}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.58859, size = 186, normalized size = 2.32 \begin{align*} \frac{{\left (F^{d e n x + c e n} F^{\frac{{\left (d e f - c e g\right )} n}{g}} b + F^{\frac{{\left (d e f - c e g\right )} n}{g}} a\right )}{\left (F^{d e n x + c e n} b + a\right )}^{p}}{{\left (b d e n p + b d e n\right )} \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(F^(d*e*n*x + c*e*n)*F^((d*e*f - c*e*g)*n/g)*b + F^((d*e*f - c*e*g)*n/g)*a)*(F^(d*e*n*x + c*e*n)*b + a)^p/((b*
d*e*n*p + b*d*e*n)*log(F))

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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