3.749 \(\int \left (a F^{c+d x}\right )^m \left (b F^{e+f x}\right )^n \, dx\)

Optimal. Leaf size=36 \[ \frac{\left (a F^{c+d x}\right )^m \left (b F^{e+f x}\right )^n}{\log (F) (d m+f n)} \]

[Out]

((a*F^(c + d*x))^m*(b*F^(e + f*x))^n)/((d*m + f*n)*Log[F])

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Rubi [A]  time = 0.154346, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{\left (a F^{c+d x}\right )^m \left (b F^{e+f x}\right )^n}{\log (F) (d m+f n)} \]

Antiderivative was successfully verified.

[In]  Int[(a*F^(c + d*x))^m*(b*F^(e + f*x))^n,x]

[Out]

((a*F^(c + d*x))^m*(b*F^(e + f*x))^n)/((d*m + f*n)*Log[F])

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Rubi in Sympy [A]  time = 16.5993, size = 73, normalized size = 2.03 \[ \frac{F^{m \left (- c - d x\right )} F^{n \left (- e - f x\right )} \left (F^{c + d x} a\right )^{m} \left (F^{e + f x} b\right )^{n} e^{x \left (d m + f n\right ) \log{\left (F \right )} + \left (c m + e n\right ) \log{\left (F \right )}}}{\left (d m + f n\right ) \log{\left (F \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*F**(d*x+c))**m*(b*F**(f*x+e))**n,x)

[Out]

F**(m*(-c - d*x))*F**(n*(-e - f*x))*(F**(c + d*x)*a)**m*(F**(e + f*x)*b)**n*exp(
x*(d*m + f*n)*log(F) + (c*m + e*n)*log(F))/((d*m + f*n)*log(F))

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Mathematica [A]  time = 0.0652596, size = 36, normalized size = 1. \[ \frac{\left (a F^{c+d x}\right )^m \left (b F^{e+f x}\right )^n}{d m \log (F)+f n \log (F)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*F^(c + d*x))^m*(b*F^(e + f*x))^n,x]

[Out]

((a*F^(c + d*x))^m*(b*F^(e + f*x))^n)/(d*m*Log[F] + f*n*Log[F])

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Maple [A]  time = 0.009, size = 37, normalized size = 1. \[{\frac{ \left ( a{F}^{dx+c} \right ) ^{m} \left ( b{F}^{fx+e} \right ) ^{n}}{\ln \left ( F \right ) \left ( md+fn \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*F^(d*x+c))^m*(b*F^(f*x+e))^n,x)

[Out]

(a*F^(d*x+c))^m*(b*F^(f*x+e))^n/(d*m+f*n)/ln(F)

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Maxima [A]  time = 0.809535, size = 88, normalized size = 2.44 \[ \frac{{\left (F^{e}\right )}^{n} a^{m} b^{n} e^{\left (m \log \left (F^{d x + c}\right ) + n \log \left ({\left (F^{d x + c}\right )}^{\frac{f}{d}}\right )\right )}}{{\left (d m + f n\right )}{\left (F^{\frac{c f}{d}}\right )}^{n} \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((F^(d*x + c)*a)^m*(F^(f*x + e)*b)^n,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.266936, size = 62, normalized size = 1.72 \[ \frac{e^{\left ({\left (d m x + c m\right )} \log \left (F\right ) +{\left (f n x + e n\right )} \log \left (F\right ) + m \log \left (a\right ) + n \log \left (b\right )\right )}}{{\left (d m + f n\right )} \log \left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((F^(d*x + c)*a)^m*(F^(f*x + e)*b)^n,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 104.7, size = 100, normalized size = 2.78 \[ \begin{cases} a^{m} b^{n} x & \text{for}\: F = 1 \wedge \left (F = 1 \vee d = - \frac{f n}{m}\right ) \\a^{m} b^{n} x \left (F^{c}\right )^{m} \left (F^{e}\right )^{n} \left (F^{f x}\right )^{n} \left (F^{- \frac{f n x}{m}}\right )^{m} & \text{for}\: d = - \frac{f n}{m} \\\frac{a^{m} b^{n} \left (F^{c}\right )^{m} \left (F^{e}\right )^{n} \left (F^{d x}\right )^{m} \left (F^{f x}\right )^{n}}{d m \log{\left (F \right )} + f n \log{\left (F \right )}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*F**(d*x+c))**m*(b*F**(f*x+e))**n,x)

[Out]

Piecewise((a**m*b**n*x, Eq(F, 1) & (Eq(F, 1) | Eq(d, -f*n/m))), (a**m*b**n*x*(F*
*c)**m*(F**e)**n*(F**(f*x))**n*(F**(-f*n*x/m))**m, Eq(d, -f*n/m)), (a**m*b**n*(F
**c)**m*(F**e)**n*(F**(d*x))**m*(F**(f*x))**n/(d*m*log(F) + f*n*log(F)), True))

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GIAC/XCAS [A]  time = 1.74042, size = 63, normalized size = 1.75 \[ \frac{e^{\left (d m x{\rm ln}\left (F\right ) + f n x{\rm ln}\left (F\right ) + c m{\rm ln}\left (F\right ) + n e{\rm ln}\left (F\right ) + m{\rm ln}\left (a\right ) + n{\rm ln}\left (b\right )\right )}}{d m{\rm ln}\left (F\right ) + f n{\rm ln}\left (F\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((F^(d*x + c)*a)^m*(F^(f*x + e)*b)^n,x, algorithm="giac")

[Out]

e^(d*m*x*ln(F) + f*n*x*ln(F) + c*m*ln(F) + n*e*ln(F) + m*ln(a) + n*ln(b))/(d*m*l
n(F) + f*n*ln(F))