Optimal. Leaf size=8 \[ \text {Ei}(x (\log (a)+\log (b))) \]
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Rubi [A]
time = 0.03, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2325, 2209}
\begin {gather*} \text {Ei}(x (\log (a)+\log (b))) \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2325
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*}
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Mathematica [A]
time = 0.02, size = 10, normalized size = 1.25 \begin {gather*} \text {Ei}(x \log (a)+x \log (b)) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.02, size = 56, normalized size = 7.00
method | result | size |
meijerg | \(-\ln \left (-x \ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )\right )-\expIntegral \left (1, -x \ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )\right )+\ln \left (x \right )+i \pi +\ln \left (\ln \left (b \right )\right )+\ln \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 8, normalized size = 1.00 \begin {gather*} {\rm Ei}\left (x {\left (\log \left (a\right ) + \log \left (b\right )\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 10, normalized size = 1.25 \begin {gather*} {\rm Ei}\left (x \log \left (a\right ) + x \log \left (b\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{x} b^{x}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 8, normalized size = 1.00 \begin {gather*} \mathrm {ei}\left (x\,\left (\ln \left (a\right )+\ln \left (b\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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