Integrand size = 10, antiderivative size = 19 \[ \int a^x b^x c^x \, dx=\frac {a^x b^x c^x}{\log (a)+\log (b)+\log (c)} \]
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Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2325, 2225} \[ \int a^x b^x c^x \, dx=\frac {a^x b^x c^x}{\log (a)+\log (b)+\log (c)} \]
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Rule 2225
Rule 2325
Rubi steps \begin{align*} \text {integral}& = \int c^x e^{x (\log (a)+\log (b))} \, dx \\ & = \int e^{x (\log (a)+\log (b)+\log (c))} \, dx \\ & = \frac {a^x b^x c^x}{\log (a)+\log (b)+\log (c)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int a^x b^x c^x \, dx=\frac {e^{x (\log (a)+\log (b)+\log (c))}}{\log (a)+\log (b)+\log (c)} \]
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Time = 0.51 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05
method | result | size |
gosper | \(\frac {a^{x} b^{x} c^{x}}{\ln \left (a \right )+\ln \left (b \right )+\ln \left (c \right )}\) | \(20\) |
risch | \(\frac {a^{x} b^{x} c^{x}}{\ln \left (a \right )+\ln \left (b \right )+\ln \left (c \right )}\) | \(20\) |
parallelrisch | \(\frac {a^{x} b^{x} c^{x}}{\ln \left (a \right )+\ln \left (b \right )+\ln \left (c \right )}\) | \(20\) |
norman | \(\frac {{\mathrm e}^{x \ln \left (c \right )} {\mathrm e}^{\ln \left (a \right ) x} {\mathrm e}^{\ln \left (b \right ) x}}{\ln \left (a \right )+\ln \left (b \right )+\ln \left (c \right )}\) | \(26\) |
meijerg | \(-\frac {1-{\mathrm e}^{x \ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right ) \left (1+\frac {\ln \left (c \right )}{\ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )}\right )}}{\ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right ) \left (1+\frac {\ln \left (c \right )}{\ln \left (b \right ) \left (1+\frac {\ln \left (a \right )}{\ln \left (b \right )}\right )}\right )}\) | \(78\) |
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none
Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int a^x b^x c^x \, dx=\frac {a^{x} b^{x} c^{x}}{\log \left (a\right ) + \log \left (b\right ) + \log \left (c\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (19) = 38\).
Time = 0.58 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.58 \[ \int a^x b^x c^x \, dx=\begin {cases} \frac {a^{x} b^{x} c^{x}}{\log {\left (a \right )} + \log {\left (b \right )} + \log {\left (c \right )}} & \text {for}\: a \neq \frac {1}{b c} \\\frac {b^{x} c^{x} \left (\frac {1}{b c}\right )^{x}}{\log {\left (b \right )} + \log {\left (c \right )} + \log {\left (\frac {1}{b c} \right )}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int a^x b^x c^x \, dx=\text {Exception raised: ValueError} \]
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Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 313, normalized size of antiderivative = 16.47 \[ \int a^x b^x c^x \, dx=2 \, {\left (\frac {2 \, {\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right ) + \log \left ({\left | c \right |}\right )\right )} \cos \left (-\frac {1}{2} \, \pi x \mathrm {sgn}\left (a\right ) - \frac {1}{2} \, \pi x \mathrm {sgn}\left (b\right ) - \frac {1}{2} \, \pi x \mathrm {sgn}\left (c\right ) + \frac {3}{2} \, \pi x\right )}{{\left (3 \, \pi - \pi \mathrm {sgn}\left (a\right ) - \pi \mathrm {sgn}\left (b\right ) - \pi \mathrm {sgn}\left (c\right )\right )}^{2} + 4 \, {\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right ) + \log \left ({\left | c \right |}\right )\right )}^{2}} + \frac {{\left (3 \, \pi - \pi \mathrm {sgn}\left (a\right ) - \pi \mathrm {sgn}\left (b\right ) - \pi \mathrm {sgn}\left (c\right )\right )} \sin \left (-\frac {1}{2} \, \pi x \mathrm {sgn}\left (a\right ) - \frac {1}{2} \, \pi x \mathrm {sgn}\left (b\right ) - \frac {1}{2} \, \pi x \mathrm {sgn}\left (c\right ) + \frac {3}{2} \, \pi x\right )}{{\left (3 \, \pi - \pi \mathrm {sgn}\left (a\right ) - \pi \mathrm {sgn}\left (b\right ) - \pi \mathrm {sgn}\left (c\right )\right )}^{2} + 4 \, {\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right ) + \log \left ({\left | c \right |}\right )\right )}^{2}}\right )} e^{\left (x {\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right ) + \log \left ({\left | c \right |}\right )\right )}\right )} + i \, {\left (\frac {i \, e^{\left (\frac {1}{2} i \, \pi x \mathrm {sgn}\left (a\right ) + \frac {1}{2} i \, \pi x \mathrm {sgn}\left (b\right ) + \frac {1}{2} i \, \pi x \mathrm {sgn}\left (c\right ) - \frac {3}{2} i \, \pi x\right )}}{-3 i \, \pi + i \, \pi \mathrm {sgn}\left (a\right ) + i \, \pi \mathrm {sgn}\left (b\right ) + i \, \pi \mathrm {sgn}\left (c\right ) + 2 \, \log \left ({\left | a \right |}\right ) + 2 \, \log \left ({\left | b \right |}\right ) + 2 \, \log \left ({\left | c \right |}\right )} - \frac {i \, e^{\left (-\frac {1}{2} i \, \pi x \mathrm {sgn}\left (a\right ) - \frac {1}{2} i \, \pi x \mathrm {sgn}\left (b\right ) - \frac {1}{2} i \, \pi x \mathrm {sgn}\left (c\right ) + \frac {3}{2} i \, \pi x\right )}}{3 i \, \pi - i \, \pi \mathrm {sgn}\left (a\right ) - i \, \pi \mathrm {sgn}\left (b\right ) - i \, \pi \mathrm {sgn}\left (c\right ) + 2 \, \log \left ({\left | a \right |}\right ) + 2 \, \log \left ({\left | b \right |}\right ) + 2 \, \log \left ({\left | c \right |}\right )}\right )} e^{\left (x {\left (\log \left ({\left | a \right |}\right ) + \log \left ({\left | b \right |}\right ) + \log \left ({\left | c \right |}\right )\right )}\right )} \]
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Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int a^x b^x c^x \, dx=\frac {a^x\,b^x\,c^x}{\ln \left (a\right )+\ln \left (b\right )+\ln \left (c\right )} \]
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