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