Optimal. Leaf size=8 \[ \text{Ei}(x (\log (a)+\log (b))) \]
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Rubi [A] time = 0.0416269, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2287, 2178} \[ \text{Ei}(x (\log (a)+\log (b))) \]
Antiderivative was successfully verified.
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Rule 2287
Rule 2178
Rubi steps
\begin{align*} \int \frac{a^x b^x}{x} \, dx &=\int \frac{e^{x (\log (a)+\log (b))}}{x} \, dx\\ &=\text{Ei}(x (\log (a)+\log (b)))\\ \end{align*}
Mathematica [A] time = 0.0109685, size = 10, normalized size = 1.25 \[ \text{Ei}(x \log (a)+x \log (b)) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.034, size = 56, normalized size = 7. \begin{align*} \ln \left ( x \right ) +i\pi +\ln \left ( \ln \left ( b \right ) \right ) +\ln \left ( 1+{\frac{\ln \left ( a \right ) }{\ln \left ( b \right ) }} \right ) -\ln \left ( -x\ln \left ( b \right ) \left ( 1+{\frac{\ln \left ( a \right ) }{\ln \left ( b \right ) }} \right ) \right ) -{\it Ei} \left ( 1,-x\ln \left ( b \right ) \left ( 1+{\frac{\ln \left ( a \right ) }{\ln \left ( b \right ) }} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09311, size = 11, normalized size = 1.38 \begin{align*}{\rm Ei}\left (x{\left (\log \left (a\right ) + \log \left (b\right )\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.22634, size = 34, normalized size = 4.25 \begin{align*}{\rm Ei}\left (x \log \left (a\right ) + x \log \left (b\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a^{x} b^{x}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a^{x} b^{x}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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