Optimal. Leaf size=14 \[ \frac{a^x b^x}{\log (a)+\log (b)} \]
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Rubi [A] time = 0.0207525, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{a^x b^x}{\log (a)+\log (b)} \]
Antiderivative was successfully verified.
[In] Int[a^x*b^x,x]
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Rubi in Sympy [A] time = 3.52051, size = 15, normalized size = 1.07 \[ \frac{e^{x \left (\log{\left (a \right )} + \log{\left (b \right )}\right )}}{\log{\left (a \right )} + \log{\left (b \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(a**x*b**x,x)
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Mathematica [A] time = 0.00411626, size = 14, normalized size = 1. \[ \frac{a^x b^x}{\log (a)+\log (b)} \]
Antiderivative was successfully verified.
[In] Integrate[a^x*b^x,x]
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Maple [A] time = 0.003, size = 15, normalized size = 1.1 \[{\frac{{a}^{x}{b}^{x}}{\ln \left ( a \right ) +\ln \left ( b \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(a^x*b^x,x)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(a^x*b^x,x, algorithm="maxima")
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Fricas [A] time = 0.243938, size = 19, normalized size = 1.36 \[ \frac{a^{x} b^{x}}{\log \left (a\right ) + \log \left (b\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(a^x*b^x,x, algorithm="fricas")
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Sympy [A] time = 0.954187, size = 24, normalized size = 1.71 \[ \begin{cases} \frac{a^{x} b^{x}}{\log{\left (a \right )} + \log{\left (b \right )}} & \text{for}\: a \neq \frac{1}{b} \\\tilde{\infty } b^{x} \left (\frac{1}{b}\right )^{x} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(a**x*b**x,x)
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GIAC/XCAS [A] time = 0.243168, size = 327, normalized size = 23.36 \[ 2 \,{\left (\frac{2 \,{\left ({\rm ln}\left ({\left | a \right |}\right ) +{\rm ln}\left ({\left | b \right |}\right )\right )} \cos \left (-\frac{1}{2} \, \pi x{\rm sign}\left (a\right ) - \frac{1}{2} \, \pi x{\rm sign}\left (b\right ) + \pi x\right )}{{\left (2 \, \pi - \pi{\rm sign}\left (a\right ) - \pi{\rm sign}\left (b\right )\right )}^{2} + 4 \,{\left ({\rm ln}\left ({\left | a \right |}\right ) +{\rm ln}\left ({\left | b \right |}\right )\right )}^{2}} + \frac{{\left (2 \, \pi - \pi{\rm sign}\left (a\right ) - \pi{\rm sign}\left (b\right )\right )} \sin \left (-\frac{1}{2} \, \pi x{\rm sign}\left (a\right ) - \frac{1}{2} \, \pi x{\rm sign}\left (b\right ) + \pi x\right )}{{\left (2 \, \pi - \pi{\rm sign}\left (a\right ) - \pi{\rm sign}\left (b\right )\right )}^{2} + 4 \,{\left ({\rm ln}\left ({\left | a \right |}\right ) +{\rm ln}\left ({\left | b \right |}\right )\right )}^{2}}\right )} e^{\left (x{\left ({\rm ln}\left ({\left | a \right |}\right ) +{\rm ln}\left ({\left | b \right |}\right )\right )}\right )} - \frac{{\left (\frac{i e^{\left (\frac{1}{2} \,{\left (\pi{\left ({\rm sign}\left (a\right ) - 1\right )} + \pi{\left ({\rm sign}\left (b\right ) - 1\right )}\right )} i x\right )}}{\pi i{\rm sign}\left (a\right ) + \pi i{\rm sign}\left (b\right ) - 2 \, \pi i + 2 \,{\rm ln}\left ({\left | a \right |}\right ) + 2 \,{\rm ln}\left ({\left | b \right |}\right )} + \frac{i e^{\left (-\frac{1}{2} \,{\left (\pi{\left ({\rm sign}\left (a\right ) - 1\right )} + \pi{\left ({\rm sign}\left (b\right ) - 1\right )}\right )} i x\right )}}{\pi i{\rm sign}\left (a\right ) + \pi i{\rm sign}\left (b\right ) - 2 \, \pi i - 2 \,{\rm ln}\left ({\left | a \right |}\right ) - 2 \,{\rm ln}\left ({\left | b \right |}\right )}\right )} e^{\left (x{\left ({\rm ln}\left ({\left | a \right |}\right ) +{\rm ln}\left ({\left | b \right |}\right )\right )}\right )}}{i} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(a^x*b^x,x, algorithm="giac")
[Out]