Integrand size = 8, antiderivative size = 31 \[ \int a^x b^x x \, dx=-\frac {a^x b^x}{(\log (a)+\log (b))^2}+\frac {a^x b^x x}{\log (a)+\log (b)} \]
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Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2325, 2207, 2225} \[ \int a^x b^x x \, dx=\frac {x a^x b^x}{\log (a)+\log (b)}-\frac {a^x b^x}{(\log (a)+\log (b))^2} \]
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Rule 2207
Rule 2225
Rule 2325
Rubi steps \begin{align*} \text {integral}& = \int e^{x (\log (a)+\log (b))} x \, dx \\ & = \frac {a^x b^x x}{\log (a)+\log (b)}-\frac {\int e^{x (\log (a)+\log (b))} \, dx}{\log (a)+\log (b)} \\ & = -\frac {a^x b^x}{(\log (a)+\log (b))^2}+\frac {a^x b^x x}{\log (a)+\log (b)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int a^x b^x x \, dx=a^x b^x \left (-\frac {1}{(\log (a)+\log (b))^2}+\frac {x}{\log (a)+\log (b)}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.81
method | result | size |
gosper | \(\frac {\left (\ln \left (a \right ) x +\ln \left (b \right ) x -1\right ) a^{x} b^{x}}{\left (\ln \left (a \right )+\ln \left (b \right )\right )^{2}}\) | \(25\) |
risch | \(\frac {\left (\ln \left (a \right ) x +\ln \left (b \right ) x -1\right ) a^{x} b^{x}}{\left (\ln \left (a \right )+\ln \left (b \right )\right )^{2}}\) | \(25\) |
parallelrisch | \(\frac {b^{x} x \,a^{x} \ln \left (a \right )+b^{x} x \,a^{x} \ln \left (b \right )-a^{x} b^{x}}{\left (\ln \left (a \right )+\ln \left (b \right )\right )^{2}}\) | \(38\) |
norman | \(\frac {x \,{\mathrm e}^{\ln \left (a \right ) x} {\mathrm e}^{\ln \left (b \right ) x}}{\ln \left (a \right )+\ln \left (b \right )}-\frac {{\mathrm e}^{\ln \left (a \right ) x} {\mathrm e}^{\ln \left (b \right ) x}}{\left (\ln \left (a \right )+\ln \left (b \right )\right )^{2}}\) | \(40\) |
meijerg | \(\frac {1-\frac {\left (2-2 x \ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )\right ) {\mathrm e}^{x \ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )}}{2}}{\ln \left (b \right )^{2} \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )^{2}}\) | \(51\) |
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Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int a^x b^x x \, dx=\frac {{\left (x \log \left (a\right ) + x \log \left (b\right ) - 1\right )} a^{x} b^{x}}{\log \left (a\right )^{2} + 2 \, \log \left (a\right ) \log \left (b\right ) + \log \left (b\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (29) = 58\).
Time = 0.37 (sec) , antiderivative size = 190, normalized size of antiderivative = 6.13 \[ \int a^x b^x x \, dx=\begin {cases} \frac {a^{x} b^{x} x \log {\left (a \right )}}{\log {\left (a \right )}^{2} + 2 \log {\left (a \right )} \log {\left (b \right )} + \log {\left (b \right )}^{2}} + \frac {a^{x} b^{x} x \log {\left (b \right )}}{\log {\left (a \right )}^{2} + 2 \log {\left (a \right )} \log {\left (b \right )} + \log {\left (b \right )}^{2}} - \frac {a^{x} b^{x}}{\log {\left (a \right )}^{2} + 2 \log {\left (a \right )} \log {\left (b \right )} + \log {\left (b \right )}^{2}} & \text {for}\: a \neq \frac {1}{b} \\\frac {b^{x} x \left (\frac {1}{b}\right )^{x} \log {\left (\frac {1}{b} \right )}}{\log {\left (\frac {1}{b} \right )}^{2} + 2 \log {\left (\frac {1}{b} \right )} \log {\left (b \right )} + \log {\left (b \right )}^{2}} + \frac {b^{x} x \left (\frac {1}{b}\right )^{x} \log {\left (b \right )}}{\log {\left (\frac {1}{b} \right )}^{2} + 2 \log {\left (\frac {1}{b} \right )} \log {\left (b \right )} + \log {\left (b \right )}^{2}} - \frac {b^{x} \left (\frac {1}{b}\right )^{x}}{\log {\left (\frac {1}{b} \right )}^{2} + 2 \log {\left (\frac {1}{b} \right )} \log {\left (b \right )} + \log {\left (b \right )}^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int a^x b^x x \, dx=\frac {{\left (x {\left (\log \left (a\right ) + \log \left (b\right )\right )} - 1\right )} e^{\left (x \log \left (a\right ) + x \log \left (b\right )\right )}}{\log \left (a\right )^{2} + 2 \, \log \left (a\right ) \log \left (b\right ) + \log \left (b\right )^{2}} \]
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Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 994, normalized size of antiderivative = 32.06 \[ \int a^x b^x x \, dx=\text {Too large to display} \]
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Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int a^x b^x x \, dx=\frac {a^x\,b^x\,\left (x\,\left (\ln \left (a\right )+\ln \left (b\right )\right )-1\right )}{{\left (\ln \left (a\right )+\ln \left (b\right )\right )}^2} \]
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