Integrand size = 50, antiderivative size = 26 \[ \int \frac {29 x-3 x^2+8 x^3+\left (3-55 x+17 x^2+18 x^3\right ) \log \left (-3+x+x^2\right )}{-15+5 x+5 x^2} \, dx=-x^2+\frac {1}{5} \left (-x+9 x^2\right ) \log \left (-3+x+x^2\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(124\) vs. \(2(26)=52\).
Time = 0.14 (sec) , antiderivative size = 124, normalized size of antiderivative = 4.77, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6860, 1642, 646, 31, 2605, 814} \[ \int \frac {29 x-3 x^2+8 x^3+\left (3-55 x+17 x^2+18 x^3\right ) \log \left (-3+x+x^2\right )}{-15+5 x+5 x^2} \, dx=-x^2+\frac {1}{180} (1-18 x)^2 \log \left (x^2+x-3\right )+\frac {1}{5} \left (32-5 \sqrt {13}\right ) \log \left (2 x-\sqrt {13}+1\right )-\frac {1}{180} \left (1153-180 \sqrt {13}\right ) \log \left (2 x-\sqrt {13}+1\right )-\frac {1}{180} \left (1153+180 \sqrt {13}\right ) \log \left (2 x+\sqrt {13}+1\right )+\frac {1}{5} \left (32+5 \sqrt {13}\right ) \log \left (2 x+\sqrt {13}+1\right ) \]
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Rule 31
Rule 646
Rule 814
Rule 1642
Rule 2605
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x \left (29-3 x+8 x^2\right )}{5 \left (-3+x+x^2\right )}+\frac {1}{5} (-1+18 x) \log \left (-3+x+x^2\right )\right ) \, dx \\ & = \frac {1}{5} \int \frac {x \left (29-3 x+8 x^2\right )}{-3+x+x^2} \, dx+\frac {1}{5} \int (-1+18 x) \log \left (-3+x+x^2\right ) \, dx \\ & = \frac {1}{180} (1-18 x)^2 \log \left (-3+x+x^2\right )-\frac {1}{180} \int \frac {(1+2 x) (-1+18 x)^2}{-3+x+x^2} \, dx+\frac {1}{5} \int \left (-11+8 x-\frac {33-64 x}{-3+x+x^2}\right ) \, dx \\ & = -\frac {11 x}{5}+\frac {4 x^2}{5}+\frac {1}{180} (1-18 x)^2 \log \left (-3+x+x^2\right )-\frac {1}{180} \int \left (-396+648 x-\frac {1187-2306 x}{-3+x+x^2}\right ) \, dx-\frac {1}{5} \int \frac {33-64 x}{-3+x+x^2} \, dx \\ & = -x^2+\frac {1}{180} (1-18 x)^2 \log \left (-3+x+x^2\right )+\frac {1}{180} \int \frac {1187-2306 x}{-3+x+x^2} \, dx-\frac {1}{5} \left (-32-5 \sqrt {13}\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {13}}{2}+x} \, dx-\frac {1}{5} \left (-32+5 \sqrt {13}\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {13}}{2}+x} \, dx \\ & = -x^2+\frac {1}{5} \left (32-5 \sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )+\frac {1}{5} \left (32+5 \sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )+\frac {1}{180} (1-18 x)^2 \log \left (-3+x+x^2\right )+\frac {1}{180} \left (-1153-180 \sqrt {13}\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {13}}{2}+x} \, dx+\frac {1}{180} \left (-1153+180 \sqrt {13}\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {13}}{2}+x} \, dx \\ & = -x^2-\frac {1}{180} \left (1153-180 \sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )+\frac {1}{5} \left (32-5 \sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )+\frac {1}{5} \left (32+5 \sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )-\frac {1}{180} \left (1153+180 \sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )+\frac {1}{180} (1-18 x)^2 \log \left (-3+x+x^2\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {29 x-3 x^2+8 x^3+\left (3-55 x+17 x^2+18 x^3\right ) \log \left (-3+x+x^2\right )}{-15+5 x+5 x^2} \, dx=\frac {1}{5} \left (-5 x^2-x \log \left (-3+x+x^2\right )+9 x^2 \log \left (-3+x+x^2\right )\right ) \]
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Time = 2.76 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92
method | result | size |
risch | \(\left (\frac {9}{5} x^{2}-\frac {1}{5} x \right ) \ln \left (x^{2}+x -3\right )-x^{2}\) | \(24\) |
default | \(\frac {9 \ln \left (x^{2}+x -3\right ) x^{2}}{5}-x^{2}-\frac {\ln \left (x^{2}+x -3\right ) x}{5}\) | \(29\) |
norman | \(\frac {9 \ln \left (x^{2}+x -3\right ) x^{2}}{5}-x^{2}-\frac {\ln \left (x^{2}+x -3\right ) x}{5}\) | \(29\) |
parts | \(\frac {9 \ln \left (x^{2}+x -3\right ) x^{2}}{5}-x^{2}-\frac {\ln \left (x^{2}+x -3\right ) x}{5}\) | \(29\) |
parallelrisch | \(\frac {9 \ln \left (x^{2}+x -3\right ) x^{2}}{5}-x^{2}-\frac {\ln \left (x^{2}+x -3\right ) x}{5}-5\) | \(30\) |
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Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {29 x-3 x^2+8 x^3+\left (3-55 x+17 x^2+18 x^3\right ) \log \left (-3+x+x^2\right )}{-15+5 x+5 x^2} \, dx=-x^{2} + \frac {1}{5} \, {\left (9 \, x^{2} - x\right )} \log \left (x^{2} + x - 3\right ) \]
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Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {29 x-3 x^2+8 x^3+\left (3-55 x+17 x^2+18 x^3\right ) \log \left (-3+x+x^2\right )}{-15+5 x+5 x^2} \, dx=- x^{2} + \left (\frac {9 x^{2}}{5} - \frac {x}{5}\right ) \log {\left (x^{2} + x - 3 \right )} \]
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Time = 0.33 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {29 x-3 x^2+8 x^3+\left (3-55 x+17 x^2+18 x^3\right ) \log \left (-3+x+x^2\right )}{-15+5 x+5 x^2} \, dx=-x^{2} + \frac {1}{5} \, {\left (9 \, x^{2} - x - 32\right )} \log \left (x^{2} + x - 3\right ) + \frac {32}{5} \, \log \left (x^{2} + x - 3\right ) \]
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Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {29 x-3 x^2+8 x^3+\left (3-55 x+17 x^2+18 x^3\right ) \log \left (-3+x+x^2\right )}{-15+5 x+5 x^2} \, dx=-x^{2} + \frac {1}{5} \, {\left (9 \, x^{2} - x\right )} \log \left (x^{2} + x - 3\right ) \]
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Time = 8.84 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {29 x-3 x^2+8 x^3+\left (3-55 x+17 x^2+18 x^3\right ) \log \left (-3+x+x^2\right )}{-15+5 x+5 x^2} \, dx=x^2\,\left (\frac {9\,\ln \left (x^2+x-3\right )}{5}-1\right )-\frac {x\,\ln \left (x^2+x-3\right )}{5} \]
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