Optimal. Leaf size=22 \[ \frac {a^{m x} b^{n x}}{m \log (a)+n \log (b)} \]
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Rubi [A] time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2287, 2194} \[ \frac {a^{m x} b^{n x}}{m \log (a)+n \log (b)} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2287
Rubi steps
\begin {align*} \int a^{m x} b^{n x} \, dx &=\int e^{x (m \log (a)+n \log (b))} \, dx\\ &=\frac {a^{m x} b^{n x}}{m \log (a)+n \log (b)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \[ \frac {a^{m x} b^{n x}}{m \log (a)+n \log (b)} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int a^{m x} b^{n x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.08, size = 22, normalized size = 1.00 \[ \frac {a^{m x} b^{n x}}{m \log \relax (a) + n \log \relax (b)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.68, size = 325, normalized size = 14.77 \[ 2 \, {\left (\frac {2 \, {\left (m \log \left ({\left | a \right |}\right ) + n \log \left ({\left | b \right |}\right )\right )} \cos \left (-\frac {1}{2} \, \pi m x \mathrm {sgn}\relax (a) - \frac {1}{2} \, \pi n x \mathrm {sgn}\relax (b) + \frac {1}{2} \, \pi m x + \frac {1}{2} \, \pi n x\right )}{{\left (\pi m \mathrm {sgn}\relax (a) + \pi n \mathrm {sgn}\relax (b) - \pi m - \pi n\right )}^{2} + 4 \, {\left (m \log \left ({\left | a \right |}\right ) + n \log \left ({\left | b \right |}\right )\right )}^{2}} - \frac {{\left (\pi m \mathrm {sgn}\relax (a) + \pi n \mathrm {sgn}\relax (b) - \pi m - \pi n\right )} \sin \left (-\frac {1}{2} \, \pi m x \mathrm {sgn}\relax (a) - \frac {1}{2} \, \pi n x \mathrm {sgn}\relax (b) + \frac {1}{2} \, \pi m x + \frac {1}{2} \, \pi n x\right )}{{\left (\pi m \mathrm {sgn}\relax (a) + \pi n \mathrm {sgn}\relax (b) - \pi m - \pi n\right )}^{2} + 4 \, {\left (m \log \left ({\left | a \right |}\right ) + n \log \left ({\left | b \right |}\right )\right )}^{2}}\right )} e^{\left ({\left (m \log \left ({\left | a \right |}\right ) + n \log \left ({\left | b \right |}\right )\right )} x\right )} - \frac {1}{2} i \, {\left (-\frac {2 i \, e^{\left (\frac {1}{2} i \, \pi m x \mathrm {sgn}\relax (a) + \frac {1}{2} i \, \pi n x \mathrm {sgn}\relax (b) - \frac {1}{2} i \, \pi m x - \frac {1}{2} i \, \pi n x\right )}}{i \, \pi m \mathrm {sgn}\relax (a) + i \, \pi n \mathrm {sgn}\relax (b) - i \, \pi m - i \, \pi n + 2 \, m \log \left ({\left | a \right |}\right ) + 2 \, n \log \left ({\left | b \right |}\right )} + \frac {2 i \, e^{\left (-\frac {1}{2} i \, \pi m x \mathrm {sgn}\relax (a) - \frac {1}{2} i \, \pi n x \mathrm {sgn}\relax (b) + \frac {1}{2} i \, \pi m x + \frac {1}{2} i \, \pi n x\right )}}{-i \, \pi m \mathrm {sgn}\relax (a) - i \, \pi n \mathrm {sgn}\relax (b) + i \, \pi m + i \, \pi n + 2 \, m \log \left ({\left | a \right |}\right ) + 2 \, n \log \left ({\left | b \right |}\right )}\right )} e^{\left ({\left (m \log \left ({\left | a \right |}\right ) + n \log \left ({\left | b \right |}\right )\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 23, normalized size = 1.05
method | result | size |
gosper | \(\frac {a^{m x} b^{n x}}{m \ln \relax (a )+n \ln \relax (b )}\) | \(23\) |
risch | \(\frac {a^{m x} b^{n x}}{m \ln \relax (a )+n \ln \relax (b )}\) | \(23\) |
norman | \(\frac {{\mathrm e}^{m x \ln \relax (a )} {\mathrm e}^{n x \ln \relax (b )}}{m \ln \relax (a )+n \ln \relax (b )}\) | \(25\) |
meijerg | \(-\frac {1-{\mathrm e}^{x n \ln \relax (b ) \left (1+\frac {m \ln \relax (a )}{n \ln \relax (b )}\right )}}{n \ln \relax (b ) \left (1+\frac {m \ln \relax (a )}{n \ln \relax (b )}\right )}\) | \(48\) |
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 = 0.33, size = 22, normalized size = 1.00 \[ \frac {a^{m\,x}\,b^{n\,x}}{m\,\ln \relax (a)+n\,\ln \relax (b)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.70, size = 42, normalized size = 1.91 \[ \begin {cases} \frac {a^{m x} b^{n x}}{m \log {\relax (a )} + n \log {\relax (b )}} & \text {for}\: m \neq - \frac {n \log {\relax (b )}}{\log {\relax (a )}} \\b^{n x} x e^{- n x \log {\relax (b )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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