Integrand size = 65, antiderivative size = 19 \[ \int \frac {-6 x+4 x^2+(-3+2 x) \log \left (\frac {1}{2} \left (3 x-2 x^2\right )\right )+\log (x) \left (-3+4 x+(-3+2 x) \log \left (\frac {1}{2} \left (3 x-2 x^2\right )\right )\right )}{-3+2 x} \, dx=x \left (x+\log (x) \log \left (\frac {3 x}{2}-x^2\right )\right ) \]
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Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.169, Rules used = {6820, 2579, 29, 8, 6874, 45, 2404, 2332, 2353, 2352, 2636} \[ \int \frac {-6 x+4 x^2+(-3+2 x) \log \left (\frac {1}{2} \left (3 x-2 x^2\right )\right )+\log (x) \left (-3+4 x+(-3+2 x) \log \left (\frac {1}{2} \left (3 x-2 x^2\right )\right )\right )}{-3+2 x} \, dx=x^2+x \log (x) \log \left (\frac {1}{2} (3-2 x) x\right ) \]
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Rule 8
Rule 29
Rule 45
Rule 2332
Rule 2352
Rule 2353
Rule 2404
Rule 2579
Rule 2636
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (2 x+\log \left (\frac {1}{2} (3-2 x) x\right )+\frac {\log (x) \left (-3+4 x+(-3+2 x) \log \left (\frac {1}{2} (3-2 x) x\right )\right )}{-3+2 x}\right ) \, dx \\ & = x^2+\int \log \left (\frac {1}{2} (3-2 x) x\right ) \, dx+\int \frac {\log (x) \left (-3+4 x+(-3+2 x) \log \left (\frac {1}{2} (3-2 x) x\right )\right )}{-3+2 x} \, dx \\ & = x^2-\frac {1}{2} (3-2 x) \log \left (\frac {1}{2} (3-2 x) x\right )+\frac {3}{2} \int \frac {1}{x} \, dx-2 \int 1 \, dx+\int \left (\frac {(-3+4 x) \log (x)}{-3+2 x}+\log (x) \log \left (\frac {1}{2} (3-2 x) x\right )\right ) \, dx \\ & = -2 x+x^2+\frac {3 \log (x)}{2}-\frac {1}{2} (3-2 x) \log \left (\frac {1}{2} (3-2 x) x\right )+\int \frac {(-3+4 x) \log (x)}{-3+2 x} \, dx+\int \log (x) \log \left (\frac {1}{2} (3-2 x) x\right ) \, dx \\ & = -2 x+x^2+\frac {3 \log (x)}{2}-\frac {1}{2} (3-2 x) \log \left (\frac {1}{2} (3-2 x) x\right )+x \log (x) \log \left (\frac {1}{2} (3-2 x) x\right )-\int \frac {(3-4 x) \log (x)}{3-2 x} \, dx+\int \left (2 \log (x)+\frac {3 \log (x)}{-3+2 x}\right ) \, dx-\int \log \left (\frac {1}{2} (3-2 x) x\right ) \, dx \\ & = -2 x+x^2+\frac {3 \log (x)}{2}+x \log (x) \log \left (\frac {1}{2} (3-2 x) x\right )-\frac {3}{2} \int \frac {1}{x} \, dx+2 \int 1 \, dx+2 \int \log (x) \, dx+3 \int \frac {\log (x)}{-3+2 x} \, dx-\int \left (2 \log (x)+\frac {3 \log (x)}{-3+2 x}\right ) \, dx \\ & = -2 x+x^2+2 x \log (x)+x \log (x) \log \left (\frac {1}{2} (3-2 x) x\right )+\frac {3}{2} \log \left (\frac {3}{2}\right ) \log (-3+2 x)-2 \int \log (x) \, dx+3 \int \frac {\log \left (\frac {2 x}{3}\right )}{-3+2 x} \, dx-3 \int \frac {\log (x)}{-3+2 x} \, dx \\ & = x^2+x \log (x) \log \left (\frac {1}{2} (3-2 x) x\right )-\frac {3}{2} \operatorname {PolyLog}\left (2,1-\frac {2 x}{3}\right )-3 \int \frac {\log \left (\frac {2 x}{3}\right )}{-3+2 x} \, dx \\ & = x^2+x \log (x) \log \left (\frac {1}{2} (3-2 x) x\right ) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {-6 x+4 x^2+(-3+2 x) \log \left (\frac {1}{2} \left (3 x-2 x^2\right )\right )+\log (x) \left (-3+4 x+(-3+2 x) \log \left (\frac {1}{2} \left (3 x-2 x^2\right )\right )\right )}{-3+2 x} \, dx=x \left (x+\log (x) \log \left (\frac {1}{2} (3-2 x) x\right )\right ) \]
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Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00
method | result | size |
norman | \(x^{2}+\ln \left (x \right ) \ln \left (-x^{2}+\frac {3}{2} x \right ) x\) | \(19\) |
parallelrisch | \(\ln \left (x \right ) \ln \left (-x^{2}+\frac {3}{2} x \right ) x -\frac {9}{4}+x^{2}\) | \(20\) |
default | \(-x \ln \left (2\right ) \ln \left (x \right )-\frac {3 \ln \left (x \right )}{2}+\frac {3 \ln \left (-2 x^{2}+3 x \right )}{2}+x \ln \left (x \right ) \ln \left (-2 x^{2}+3 x \right )+x^{2}-\frac {3 \ln \left (-3+2 x \right )}{2}\) | \(50\) |
parts | \(-\frac {3 \ln \left (x \right )}{2}+\frac {3 \ln \left (-2 x^{2}+3 x \right )}{2}+x \ln \left (x \right ) \ln \left (-2 x^{2}+3 x \right )-\ln \left (2\right ) \left (x \ln \left (x \right )-x \right )+x^{2}-x \ln \left (2\right )-\frac {3 \ln \left (-3+2 x \right )}{2}\) | \(60\) |
risch | \(\ln \left (x -\frac {3}{2}\right ) x \ln \left (x \right )+x \ln \left (x \right )^{2}-\frac {i \ln \left (x \right ) x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left (x -\frac {3}{2}\right )\right ) \operatorname {csgn}\left (i x \left (x -\frac {3}{2}\right )\right )}{2}+\frac {i \ln \left (x \right ) x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \left (x -\frac {3}{2}\right )\right )^{2}}{2}-i \ln \left (x \right ) x \pi \operatorname {csgn}\left (i x \left (x -\frac {3}{2}\right )\right )^{2}+\frac {i \ln \left (x \right ) x \pi \,\operatorname {csgn}\left (i \left (x -\frac {3}{2}\right )\right ) \operatorname {csgn}\left (i x \left (x -\frac {3}{2}\right )\right )^{2}}{2}+\frac {i \ln \left (x \right ) x \pi \operatorname {csgn}\left (i x \left (x -\frac {3}{2}\right )\right )^{3}}{2}+i \ln \left (x \right ) x \pi -x \ln \left (2\right ) \ln \left (x \right )+x^{2}\) | \(140\) |
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Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {-6 x+4 x^2+(-3+2 x) \log \left (\frac {1}{2} \left (3 x-2 x^2\right )\right )+\log (x) \left (-3+4 x+(-3+2 x) \log \left (\frac {1}{2} \left (3 x-2 x^2\right )\right )\right )}{-3+2 x} \, dx=x \log \left (-x^{2} + \frac {3}{2} \, x\right ) \log \left (x\right ) + x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (15) = 30\).
Time = 0.50 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.68 \[ \int \frac {-6 x+4 x^2+(-3+2 x) \log \left (\frac {1}{2} \left (3 x-2 x^2\right )\right )+\log (x) \left (-3+4 x+(-3+2 x) \log \left (\frac {1}{2} \left (3 x-2 x^2\right )\right )\right )}{-3+2 x} \, dx=x^{2} + \left (x \log {\left (x \right )} - \frac {1}{24}\right ) \log {\left (- x^{2} + \frac {3 x}{2} \right )} + \frac {\log {\left (2 x^{2} - 3 x \right )}}{24} \]
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Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {-6 x+4 x^2+(-3+2 x) \log \left (\frac {1}{2} \left (3 x-2 x^2\right )\right )+\log (x) \left (-3+4 x+(-3+2 x) \log \left (\frac {1}{2} \left (3 x-2 x^2\right )\right )\right )}{-3+2 x} \, dx=-x \log \left (2\right ) \log \left (x\right ) + x \log \left (x\right )^{2} + x \log \left (x\right ) \log \left (-2 \, x + 3\right ) + x^{2} \]
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Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {-6 x+4 x^2+(-3+2 x) \log \left (\frac {1}{2} \left (3 x-2 x^2\right )\right )+\log (x) \left (-3+4 x+(-3+2 x) \log \left (\frac {1}{2} \left (3 x-2 x^2\right )\right )\right )}{-3+2 x} \, dx=-x \log \left (2\right ) \log \left (x\right ) + x \log \left (x\right )^{2} + x \log \left (x\right ) \log \left (-2 \, x + 3\right ) + x^{2} \]
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Time = 8.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {-6 x+4 x^2+(-3+2 x) \log \left (\frac {1}{2} \left (3 x-2 x^2\right )\right )+\log (x) \left (-3+4 x+(-3+2 x) \log \left (\frac {1}{2} \left (3 x-2 x^2\right )\right )\right )}{-3+2 x} \, dx=x^2+x\,\ln \left (\frac {3\,x}{2}-x^2\right )\,\ln \left (x\right ) \]
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