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