Optimal. Leaf size=13 \[ \frac {1}{9} (3+x)^2 \log (2 x) \]
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Rubi [B] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 3.00, number of steps used = 7, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {12, 14, 2313, 9} \begin {gather*} \frac {x^2}{18}+\frac {1}{9} \left (x^2+6 x\right ) \log (2 x)+\frac {2 x}{3}-\frac {1}{18} (x+6)^2+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 9
Rule 12
Rule 14
Rule 2313
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \frac {9+6 x+x^2+\left (6 x+2 x^2\right ) \log (2 x)}{x} \, dx\\ &=\frac {1}{9} \int \left (\frac {9+6 x+x^2}{x}+2 (3+x) \log (2 x)\right ) \, dx\\ &=\frac {1}{9} \int \frac {9+6 x+x^2}{x} \, dx+\frac {2}{9} \int (3+x) \log (2 x) \, dx\\ &=\frac {1}{9} \left (6 x+x^2\right ) \log (2 x)+\frac {1}{9} \int \left (6+\frac {9}{x}+x\right ) \, dx-\frac {2}{9} \int \frac {6+x}{2} \, dx\\ &=\frac {2 x}{3}+\frac {x^2}{18}-\frac {1}{18} (6+x)^2+\log (x)+\frac {1}{9} \left (6 x+x^2\right ) \log (2 x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.00, size = 23, normalized size = 1.77 \begin {gather*} \log (x)+\frac {2}{3} x \log (2 x)+\frac {1}{9} x^2 \log (2 x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 14, normalized size = 1.08 \begin {gather*} \frac {1}{9} \, {\left (x^{2} + 6 \, x + 9\right )} \log \left (2 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 16, normalized size = 1.23 \begin {gather*} \frac {1}{9} \, {\left (x^{2} + 6 \, x\right )} \log \left (2 \, x\right ) + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 17, normalized size = 1.31
method | result | size |
risch | \(\frac {\left (x^{2}+6 x \right ) \ln \left (2 x \right )}{9}+\ln \relax (x )\) | \(17\) |
derivativedivides | \(\frac {x^{2} \ln \left (2 x \right )}{9}+\frac {2 x \ln \left (2 x \right )}{3}+\ln \left (2 x \right )\) | \(22\) |
default | \(\frac {x^{2} \ln \left (2 x \right )}{9}+\frac {2 x \ln \left (2 x \right )}{3}+\ln \left (2 x \right )\) | \(22\) |
norman | \(\frac {x^{2} \ln \left (2 x \right )}{9}+\frac {2 x \ln \left (2 x \right )}{3}+\ln \left (2 x \right )\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 19, normalized size = 1.46 \begin {gather*} \frac {1}{9} \, x^{2} \log \left (2 \, x\right ) + \frac {2}{3} \, x \log \left (2 \, x\right ) + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.44, size = 11, normalized size = 0.85 \begin {gather*} \frac {\ln \left (2\,x\right )\,{\left (x+3\right )}^2}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 17, normalized size = 1.31 \begin {gather*} \left (\frac {x^{2}}{9} + \frac {2 x}{3}\right ) \log {\left (2 x \right )} + \log {\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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