Optimal. Leaf size=14 \[ \frac {a^x b^x}{\log (a)+\log (b)} \]
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Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2287, 2194} \[ \frac {a^x b^x}{\log (a)+\log (b)} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2287
Rubi steps
\begin {align*} \int a^x b^x \, dx &=\int e^{x (\log (a)+\log (b))} \, dx\\ &=\frac {a^x b^x}{\log (a)+\log (b)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 1.00 \[ \frac {a^x b^x}{\log (a)+\log (b)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 14, normalized size = 1.00 \[ \frac {a^{x} b^{x}}{\log \relax (a) + \log \relax (b)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 242, normalized size = 17.29 \[ 2 \, {\left (\frac {2 \, {\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right )\right )} \cos \left (-\frac {1}{2} \, \pi x \mathrm {sgn}\relax (a) - \frac {1}{2} \, \pi x \mathrm {sgn}\relax (b) + \pi x\right )}{{\left (2 \, \pi - \pi \mathrm {sgn}\relax (a) - \pi \mathrm {sgn}\relax (b)\right )}^{2} + 4 \, {\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right )\right )}^{2}} + \frac {{\left (2 \, \pi - \pi \mathrm {sgn}\relax (a) - \pi \mathrm {sgn}\relax (b)\right )} \sin \left (-\frac {1}{2} \, \pi x \mathrm {sgn}\relax (a) - \frac {1}{2} \, \pi x \mathrm {sgn}\relax (b) + \pi x\right )}{{\left (2 \, \pi - \pi \mathrm {sgn}\relax (a) - \pi \mathrm {sgn}\relax (b)\right )}^{2} + 4 \, {\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right )\right )}^{2}}\right )} e^{\left (x {\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right )\right )}\right )} - \frac {{\left (\frac {i e^{\left (\frac {1}{2} \, {\left (\pi {\left (\mathrm {sgn}\relax (a) - 1\right )} + \pi {\left (\mathrm {sgn}\relax (b) - 1\right )}\right )} i x\right )}}{\pi i \mathrm {sgn}\relax (a) + \pi i \mathrm {sgn}\relax (b) - 2 \, \pi i + 2 \, \log \left ({\left | a \right |}\right ) + 2 \, \log \left ({\left | b \right |}\right )} + \frac {i e^{\left (-\frac {1}{2} \, {\left (\pi {\left (\mathrm {sgn}\relax (a) - 1\right )} + \pi {\left (\mathrm {sgn}\relax (b) - 1\right )}\right )} i x\right )}}{\pi i \mathrm {sgn}\relax (a) + \pi i \mathrm {sgn}\relax (b) - 2 \, \pi i - 2 \, \log \left ({\left | a \right |}\right ) - 2 \, \log \left ({\left | b \right |}\right )}\right )} e^{\left (x {\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right )\right )}\right )}}{i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 15, normalized size = 1.07 \[ \frac {a^{x} b^{x}}{\ln \relax (a )+\ln \relax (b )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.60, size = 14, normalized size = 1.00 \[ \frac {a^x\,b^x}{\ln \relax (a)+\ln \relax (b)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.57, size = 24, normalized size = 1.71 \[ \begin {cases} \frac {a^{x} b^{x}}{\log {\relax (a )} + \log {\relax (b )}} & \text {for}\: a \neq \frac {1}{b} \\\tilde {\infty } b^{x} \left (\frac {1}{b}\right )^{x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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