Optimal. Leaf size=20 \[ \frac {1}{2} x \left (3 e (-2+x)+\log \left (\frac {5}{4}\right )\right ) \log \left (x^3\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 28, normalized size of antiderivative = 1.40, number of steps used = 5, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {12, 2332, 2313} \begin {gather*} \frac {1}{2} \left (3 e x^2-x \left (6 e-\log \left (\frac {5}{4}\right )\right )\right ) \log \left (x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2313
Rule 2332
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \left (e (-18+9 x)+3 \log \left (\frac {5}{4}\right )+\left (e (-6+6 x)+\log \left (\frac {5}{4}\right )\right ) \log \left (x^3\right )\right ) \, dx\\ &=\frac {9}{4} e (2-x)^2+\frac {3}{2} x \log \left (\frac {5}{4}\right )+\frac {1}{2} \int \left (e (-6+6 x)+\log \left (\frac {5}{4}\right )\right ) \log \left (x^3\right ) \, dx\\ &=\frac {9}{4} e (2-x)^2+\frac {3}{2} x \log \left (\frac {5}{4}\right )+\frac {1}{2} \int \left (-6 e+6 e x+\log \left (\frac {5}{4}\right )\right ) \log \left (x^3\right ) \, dx\\ &=\frac {9}{4} e (2-x)^2+\frac {3}{2} x \log \left (\frac {5}{4}\right )+\frac {1}{2} \left (3 e x^2-x \left (6 e-\log \left (\frac {5}{4}\right )\right )\right ) \log \left (x^3\right )-\frac {3}{2} \int \left (3 e (-2+x)+\log \left (\frac {5}{4}\right )\right ) \, dx\\ &=\frac {1}{2} \left (3 e x^2-x \left (6 e-\log \left (\frac {5}{4}\right )\right )\right ) \log \left (x^3\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 1.70 \begin {gather*} -3 e x \log \left (x^3\right )+\frac {3}{2} e x^2 \log \left (x^3\right )+\frac {1}{2} x \log \left (\frac {5}{4}\right ) \log \left (x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 22, normalized size = 1.10 \begin {gather*} \frac {1}{2} \, {\left (3 \, {\left (x^{2} - 2 \, x\right )} e + x \log \left (\frac {5}{4}\right )\right )} \log \left (x^{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 46, normalized size = 2.30 \begin {gather*} -\frac {9}{4} \, x^{2} e + \frac {9}{4} \, {\left (x^{2} - 4 \, x\right )} e + 9 \, x e + \frac {1}{2} \, {\left (3 \, {\left (x^{2} - 2 \, x\right )} e + x \log \left (\frac {5}{4}\right )\right )} \log \left (x^{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 32, normalized size = 1.60
method | result | size |
norman | \(\left (-3 \,{\mathrm e}+\frac {\ln \relax (5)}{2}-\ln \relax (2)\right ) x \ln \left (x^{3}\right )+\frac {3 \,{\mathrm e} \ln \left (x^{3}\right ) x^{2}}{2}\) | \(32\) |
risch | \(\frac {3 \,{\mathrm e} \ln \left (x^{3}\right ) x^{2}}{2}+\frac {\ln \relax (5) \ln \left (x^{3}\right ) x}{2}-3 \,{\mathrm e} \ln \left (x^{3}\right ) x -\ln \relax (2) \ln \left (x^{3}\right ) x\) | \(40\) |
default | \(\frac {{\mathrm e} \left (\frac {9}{2} x^{2}-18 x \right )}{2}+\frac {\ln \relax (5) \ln \left (x^{3}\right ) x}{2}-\frac {3 x \ln \relax (5)}{2}-3 \,{\mathrm e} \ln \left (x^{3}\right ) x +9 x \,{\mathrm e}-\ln \relax (2) \ln \left (x^{3}\right ) x +3 x \ln \relax (2)+\frac {3 \,{\mathrm e} \ln \left (x^{3}\right ) x^{2}}{2}-\frac {9 x^{2} {\mathrm e}}{4}+\frac {3 \ln \left (\frac {5}{4}\right ) x}{2}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 58, normalized size = 2.90 \begin {gather*} -\frac {9}{4} \, x^{2} e + \frac {3}{2} \, x {\left (6 \, e - \log \left (\frac {5}{4}\right )\right )} + \frac {9}{4} \, {\left (x^{2} - 4 \, x\right )} e + \frac {3}{2} \, x \log \left (\frac {5}{4}\right ) + \frac {1}{2} \, {\left (3 \, {\left (x^{2} - 2 \, x\right )} e + x \log \left (\frac {5}{4}\right )\right )} \log \left (x^{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.59, size = 19, normalized size = 0.95 \begin {gather*} \frac {x\,\ln \left (x^3\right )\,\left (\ln \left (\frac {5}{4}\right )-6\,\mathrm {e}+3\,x\,\mathrm {e}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 32, normalized size = 1.60 \begin {gather*} \left (\frac {3 e x^{2}}{2} - 3 e x - x \log {\relax (2 )} + \frac {x \log {\relax (5 )}}{2}\right ) \log {\left (x^{3} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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