Optimal. Leaf size=27 \[ \frac {a^{k x}}{k \log (a)}+\frac {a^{l x}}{l \log (a)} \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2194} \[ \frac {a^{k x}}{k \log (a)}+\frac {a^{l x}}{l \log (a)} \]
Antiderivative was successfully verified.
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Rule 2194
Rubi steps
\begin {align*} \int \left (a^{k x}+a^{l x}\right ) \, dx &=\int a^{k x} \, dx+\int a^{l x} \, dx\\ &=\frac {a^{k x}}{k \log (a)}+\frac {a^{l x}}{l \log (a)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.00 \[ \frac {a^{k x}}{k \log (a)}+\frac {a^{l x}}{l \log (a)} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a^{k x}+a^{l x}\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.92, size = 26, normalized size = 0.96 \[ \frac {a^{l x} k + a^{k x} l}{k l \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 27, normalized size = 1.00 \[ \frac {a^{k x}}{k \log \relax (a)} + \frac {a^{l x}}{l \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 28, normalized size = 1.04
method | result | size |
default | \(\frac {a^{k x}}{k \ln \relax (a )}+\frac {a^{l x}}{l \ln \relax (a )}\) | \(28\) |
risch | \(\frac {a^{k x}}{k \ln \relax (a )}+\frac {a^{l x}}{l \ln \relax (a )}\) | \(28\) |
norman | \(\frac {{\mathrm e}^{k x \ln \relax (a )}}{k \ln \relax (a )}+\frac {{\mathrm e}^{l x \ln \relax (a )}}{l \ln \relax (a )}\) | \(30\) |
meijerg | \(-\frac {1-{\mathrm e}^{k x \ln \relax (a )}}{k \ln \relax (a )}-\frac {1-{\mathrm e}^{l x \ln \relax (a )}}{l \ln \relax (a )}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 27, normalized size = 1.00 \[ \frac {a^{k x}}{k \log \relax (a)} + \frac {a^{l x}}{l \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 26, normalized size = 0.96 \[ \frac {a^{k\,x}\,l+a^{l\,x}\,k}{k\,l\,\ln \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 29, normalized size = 1.07 \[ \begin {cases} \frac {a^{k x}}{k \log {\relax (a )}} & \text {for}\: k \log {\relax (a )} \neq 0 \\x & \text {otherwise} \end {cases} + \begin {cases} \frac {a^{l x}}{l \log {\relax (a )}} & \text {for}\: l \log {\relax (a )} \neq 0 \\x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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