Integrand size = 57, antiderivative size = 28 \[ \int \frac {54+90 x+45 x^2+7 x^3+\left (9 x^2+2 x^3\right ) \log (x)+\left (27+18 x+3 x^2\right ) \log \left (x^2\right )}{27+18 x+3 x^2} \, dx=x^2-x \left (-\frac {x^2 \log (x)}{3 (3+x)}-\log \left (x^2\right )\right ) \]
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Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {27, 12, 6874, 45, 2404, 2332, 2341, 2351, 31} \[ \int \frac {54+90 x+45 x^2+7 x^3+\left (9 x^2+2 x^3\right ) \log (x)+\left (27+18 x+3 x^2\right ) \log \left (x^2\right )}{27+18 x+3 x^2} \, dx=x^2+\frac {1}{3} x^2 \log (x)+x \log \left (x^2\right )+\frac {3 x \log (x)}{x+3}-x \log (x) \]
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Rule 12
Rule 27
Rule 31
Rule 45
Rule 2332
Rule 2341
Rule 2351
Rule 2404
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {54+90 x+45 x^2+7 x^3+\left (9 x^2+2 x^3\right ) \log (x)+\left (27+18 x+3 x^2\right ) \log \left (x^2\right )}{3 (3+x)^2} \, dx \\ & = \frac {1}{3} \int \frac {54+90 x+45 x^2+7 x^3+\left (9 x^2+2 x^3\right ) \log (x)+\left (27+18 x+3 x^2\right ) \log \left (x^2\right )}{(3+x)^2} \, dx \\ & = \frac {1}{3} \int \left (\frac {54}{(3+x)^2}+\frac {90 x}{(3+x)^2}+\frac {45 x^2}{(3+x)^2}+\frac {7 x^3}{(3+x)^2}+\frac {x^2 (9+2 x) \log (x)}{(3+x)^2}+3 \log \left (x^2\right )\right ) \, dx \\ & = -\frac {18}{3+x}+\frac {1}{3} \int \frac {x^2 (9+2 x) \log (x)}{(3+x)^2} \, dx+\frac {7}{3} \int \frac {x^3}{(3+x)^2} \, dx+15 \int \frac {x^2}{(3+x)^2} \, dx+30 \int \frac {x}{(3+x)^2} \, dx+\int \log \left (x^2\right ) \, dx \\ & = -2 x-\frac {18}{3+x}+x \log \left (x^2\right )+\frac {1}{3} \int \left (-3 \log (x)+2 x \log (x)+\frac {27 \log (x)}{(3+x)^2}\right ) \, dx+\frac {7}{3} \int \left (-6+x-\frac {27}{(3+x)^2}+\frac {27}{3+x}\right ) \, dx+15 \int \left (1+\frac {9}{(3+x)^2}-\frac {6}{3+x}\right ) \, dx+30 \int \left (-\frac {3}{(3+x)^2}+\frac {1}{3+x}\right ) \, dx \\ & = -x+\frac {7 x^2}{6}+x \log \left (x^2\right )+3 \log (3+x)+\frac {2}{3} \int x \log (x) \, dx+9 \int \frac {\log (x)}{(3+x)^2} \, dx-\int \log (x) \, dx \\ & = x^2-x \log (x)+\frac {1}{3} x^2 \log (x)+\frac {3 x \log (x)}{3+x}+x \log \left (x^2\right )+3 \log (3+x)-3 \int \frac {1}{3+x} \, dx \\ & = x^2-x \log (x)+\frac {1}{3} x^2 \log (x)+\frac {3 x \log (x)}{3+x}+x \log \left (x^2\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {54+90 x+45 x^2+7 x^3+\left (9 x^2+2 x^3\right ) \log (x)+\left (27+18 x+3 x^2\right ) \log \left (x^2\right )}{27+18 x+3 x^2} \, dx=\frac {1}{3} \left (\frac {x^3 \log (x)}{3+x}+3 x \left (x+\log \left (x^2\right )\right )\right ) \]
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Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18
method | result | size |
default | \(x^{2}+\frac {x^{2} \ln \left (x \right )}{3}-x \ln \left (x \right )+\frac {3 \ln \left (x \right ) x}{3+x}+x \ln \left (x^{2}\right )\) | \(33\) |
parts | \(x^{2}+\frac {x^{2} \ln \left (x \right )}{3}-x \ln \left (x \right )+\frac {3 \ln \left (x \right ) x}{3+x}+x \ln \left (x^{2}\right )\) | \(33\) |
parallelrisch | \(\frac {18 x \ln \left (x \right )+3 x^{2} \ln \left (x^{2}\right )-27 \ln \left (x^{2}\right )+54 \ln \left (x \right )+3 x^{3}+9 x^{2}+x^{3} \ln \left (x \right )}{3 x +9}\) | \(49\) |
risch | \(\frac {\left (x^{3}+6 x^{2}+9 x -27\right ) \ln \left (x \right )}{3 x +9}-\frac {i \pi x \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )}{2}+i x \operatorname {csgn}\left (i x^{2}\right )^{2} \pi \,\operatorname {csgn}\left (i x \right )-\frac {i \pi x \operatorname {csgn}\left (i x^{2}\right )^{3}}{2}+x^{2}+3 \ln \left (x \right )\) | \(83\) |
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Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {54+90 x+45 x^2+7 x^3+\left (9 x^2+2 x^3\right ) \log (x)+\left (27+18 x+3 x^2\right ) \log \left (x^2\right )}{27+18 x+3 x^2} \, dx=\frac {3 \, x^{3} + 9 \, x^{2} + {\left (x^{3} + 6 \, x^{2} + 18 \, x\right )} \log \left (x\right )}{3 \, {\left (x + 3\right )}} \]
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Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {54+90 x+45 x^2+7 x^3+\left (9 x^2+2 x^3\right ) \log (x)+\left (27+18 x+3 x^2\right ) \log \left (x^2\right )}{27+18 x+3 x^2} \, dx=x^{2} + 3 \log {\left (x \right )} + \frac {\left (x^{3} + 6 x^{2} + 9 x - 27\right ) \log {\left (x \right )}}{3 x + 9} \]
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Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {54+90 x+45 x^2+7 x^3+\left (9 x^2+2 x^3\right ) \log (x)+\left (27+18 x+3 x^2\right ) \log \left (x^2\right )}{27+18 x+3 x^2} \, dx=\frac {7}{6} \, x^{2} + x - \frac {x^{3} + 9 \, x^{2} - 2 \, {\left (x^{3} + 6 \, x^{2} + 9 \, x - 27\right )} \log \left (x\right ) + 18 \, x}{6 \, {\left (x + 3\right )}} + 3 \, \log \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {54+90 x+45 x^2+7 x^3+\left (9 x^2+2 x^3\right ) \log (x)+\left (27+18 x+3 x^2\right ) \log \left (x^2\right )}{27+18 x+3 x^2} \, dx=x^{2} + \frac {1}{3} \, {\left (x^{2} + 3 \, x - \frac {27}{x + 3}\right )} \log \left (x\right ) + 3 \, \log \left (x\right ) \]
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Time = 13.61 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {54+90 x+45 x^2+7 x^3+\left (9 x^2+2 x^3\right ) \log (x)+\left (27+18 x+3 x^2\right ) \log \left (x^2\right )}{27+18 x+3 x^2} \, dx=\frac {x\,\left (9\,x+9\,\ln \left (x^2\right )+3\,x\,\ln \left (x^2\right )+x^2\,\ln \left (x\right )+3\,x^2\right )}{3\,\left (x+3\right )} \]
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