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

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])

_______________________________________________________________________________________

Rubi [A]  time = 0.231639, 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 \[ \frac{\left (F^{e (c+d x)}\right )^{-n} \left (a+b \left (F^{e (c+d x)}\right )^n\right )^{p+1} \left (G^{h (f+g x)}\right )^{\frac{d e n \log (F)}{g h \log (G)}}}{b d e n (p+1) \log (F)} \]

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])

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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

_______________________________________________________________________________________

Mathematica [A]  time = 0.280123, 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*L
og[G])), x]

_______________________________________________________________________________________

Maple [F]  time = 1.185, size = 0, normalized size = 0. \[ \int \left ( a+b \left ({F}^{e \left ( dx+c \right ) } \right ) ^{n} \right ) ^{p} \left ({G}^{h \left ( gx+f \right ) } \right ) ^{{\frac{nde\ln \left ( F \right ) }{gh\ln \left ( G \right ) }}}\, dx \]

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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, 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)

_______________________________________________________________________________________

Fricas [A]  time = 0.30548, size = 119, normalized size = 1.49 \[ \frac{{\left (F^{d e n x + c e n} F^{\frac{{\left (d e f - c e g\right )} n}{g}} b + F^{\frac{{\left (d e f - c e g\right )} n}{g}} a\right )}{\left (F^{d e n x + c e n} b + a\right )}^{p}}{{\left (b d e n p + b d e n\right )} \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, 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))

_______________________________________________________________________________________

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

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  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, 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)