Integrand size = 60, antiderivative size = 28 \[ \int \frac {-39-83 x+24 x^2+43 x^3-4 x^5+\left (18 x+3 x^2-9 x^3-3 x^4\right ) \log \left (-3+x+x^2\right )}{-15+5 x+5 x^2} \, dx=\frac {1}{5} (3+x) \left (10+x+x^2 \left (5-x-\log \left (-3+x+x^2\right )\right )\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(201\) vs. \(2(28)=56\).
Time = 0.21 (sec) , antiderivative size = 201, normalized size of antiderivative = 7.18, number of steps used = 21, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.117, Rules used = {6860, 1671, 646, 31, 2608, 2605, 814} \[ \int \frac {-39-83 x+24 x^2+43 x^3-4 x^5+\left (18 x+3 x^2-9 x^3-3 x^4\right ) \log \left (-3+x+x^2\right )}{-15+5 x+5 x^2} \, dx=-\frac {x^4}{5}+\frac {2 x^3}{5}+\frac {16 x^2}{5}-\frac {3}{5} x^2 \log \left (x^2+x-3\right )-\frac {1}{5} x^3 \log \left (x^2+x-3\right )+\frac {13 x}{5}-\frac {1}{10} \left (11+\sqrt {13}\right ) \log \left (2 x-\sqrt {13}+1\right )+\frac {3}{10} \left (7-\sqrt {13}\right ) \log \left (2 x-\sqrt {13}+1\right )-\frac {1}{5} \left (5-2 \sqrt {13}\right ) \log \left (2 x-\sqrt {13}+1\right )-\frac {1}{5} \left (5+2 \sqrt {13}\right ) \log \left (2 x+\sqrt {13}+1\right )+\frac {3}{10} \left (7+\sqrt {13}\right ) \log \left (2 x+\sqrt {13}+1\right )-\frac {1}{10} \left (11-\sqrt {13}\right ) \log \left (2 x+\sqrt {13}+1\right ) \]
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Rule 31
Rule 646
Rule 814
Rule 1671
Rule 2605
Rule 2608
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-39-83 x+24 x^2+43 x^3-4 x^5}{5 \left (-3+x+x^2\right )}-\frac {3}{5} x (2+x) \log \left (-3+x+x^2\right )\right ) \, dx \\ & = \frac {1}{5} \int \frac {-39-83 x+24 x^2+43 x^3-4 x^5}{-3+x+x^2} \, dx-\frac {3}{5} \int x (2+x) \log \left (-3+x+x^2\right ) \, dx \\ & = \frac {1}{5} \int \left (9+27 x+4 x^2-4 x^3-\frac {12+11 x}{-3+x+x^2}\right ) \, dx-\frac {3}{5} \int \left (2 x \log \left (-3+x+x^2\right )+x^2 \log \left (-3+x+x^2\right )\right ) \, dx \\ & = \frac {9 x}{5}+\frac {27 x^2}{10}+\frac {4 x^3}{15}-\frac {x^4}{5}-\frac {1}{5} \int \frac {12+11 x}{-3+x+x^2} \, dx-\frac {3}{5} \int x^2 \log \left (-3+x+x^2\right ) \, dx-\frac {6}{5} \int x \log \left (-3+x+x^2\right ) \, dx \\ & = \frac {9 x}{5}+\frac {27 x^2}{10}+\frac {4 x^3}{15}-\frac {x^4}{5}-\frac {3}{5} x^2 \log \left (-3+x+x^2\right )-\frac {1}{5} x^3 \log \left (-3+x+x^2\right )+\frac {1}{5} \int \frac {x^3 (1+2 x)}{-3+x+x^2} \, dx+\frac {3}{5} \int \frac {x^2 (1+2 x)}{-3+x+x^2} \, dx-\frac {1}{10} \left (11-\sqrt {13}\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {13}}{2}+x} \, dx-\frac {1}{10} \left (11+\sqrt {13}\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {13}}{2}+x} \, dx \\ & = \frac {9 x}{5}+\frac {27 x^2}{10}+\frac {4 x^3}{15}-\frac {x^4}{5}-\frac {1}{10} \left (11+\sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )-\frac {1}{10} \left (11-\sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )-\frac {3}{5} x^2 \log \left (-3+x+x^2\right )-\frac {1}{5} x^3 \log \left (-3+x+x^2\right )+\frac {1}{5} \int \left (7-x+2 x^2+\frac {21-10 x}{-3+x+x^2}\right ) \, dx+\frac {3}{5} \int \left (-1+2 x-\frac {3-7 x}{-3+x+x^2}\right ) \, dx \\ & = \frac {13 x}{5}+\frac {16 x^2}{5}+\frac {2 x^3}{5}-\frac {x^4}{5}-\frac {1}{10} \left (11+\sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )-\frac {1}{10} \left (11-\sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )-\frac {3}{5} x^2 \log \left (-3+x+x^2\right )-\frac {1}{5} x^3 \log \left (-3+x+x^2\right )+\frac {1}{5} \int \frac {21-10 x}{-3+x+x^2} \, dx-\frac {3}{5} \int \frac {3-7 x}{-3+x+x^2} \, dx \\ & = \frac {13 x}{5}+\frac {16 x^2}{5}+\frac {2 x^3}{5}-\frac {x^4}{5}-\frac {1}{10} \left (11+\sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )-\frac {1}{10} \left (11-\sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )-\frac {3}{5} x^2 \log \left (-3+x+x^2\right )-\frac {1}{5} x^3 \log \left (-3+x+x^2\right )+\frac {1}{5} \left (-5-2 \sqrt {13}\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {13}}{2}+x} \, dx+\frac {1}{10} \left (3 \left (7-\sqrt {13}\right )\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {13}}{2}+x} \, dx+\frac {1}{10} \left (3 \left (7+\sqrt {13}\right )\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {13}}{2}+x} \, dx+\frac {1}{5} \left (-5+2 \sqrt {13}\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {13}}{2}+x} \, dx \\ & = \frac {13 x}{5}+\frac {16 x^2}{5}+\frac {2 x^3}{5}-\frac {x^4}{5}-\frac {1}{5} \left (5-2 \sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )+\frac {3}{10} \left (7-\sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )-\frac {1}{10} \left (11+\sqrt {13}\right ) \log \left (1-\sqrt {13}+2 x\right )-\frac {1}{10} \left (11-\sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )+\frac {3}{10} \left (7+\sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )-\frac {1}{5} \left (5+2 \sqrt {13}\right ) \log \left (1+\sqrt {13}+2 x\right )-\frac {3}{5} x^2 \log \left (-3+x+x^2\right )-\frac {1}{5} x^3 \log \left (-3+x+x^2\right ) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {-39-83 x+24 x^2+43 x^3-4 x^5+\left (18 x+3 x^2-9 x^3-3 x^4\right ) \log \left (-3+x+x^2\right )}{-15+5 x+5 x^2} \, dx=\frac {1}{5} \left (13 x+16 x^2+2 x^3-x^4-3 x^2 \log \left (-3+x+x^2\right )-x^3 \log \left (-3+x+x^2\right )\right ) \]
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Time = 0.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39
method | result | size |
risch | \(\left (-\frac {1}{5} x^{3}-\frac {3}{5} x^{2}\right ) \ln \left (x^{2}+x -3\right )-\frac {x^{4}}{5}+\frac {2 x^{3}}{5}+\frac {16 x^{2}}{5}+\frac {13 x}{5}\) | \(39\) |
default | \(-\frac {x^{4}}{5}+\frac {2 x^{3}}{5}+\frac {16 x^{2}}{5}+\frac {13 x}{5}-\frac {\ln \left (x^{2}+x -3\right ) x^{3}}{5}-\frac {3 \ln \left (x^{2}+x -3\right ) x^{2}}{5}\) | \(44\) |
norman | \(-\frac {x^{4}}{5}+\frac {2 x^{3}}{5}+\frac {16 x^{2}}{5}+\frac {13 x}{5}-\frac {\ln \left (x^{2}+x -3\right ) x^{3}}{5}-\frac {3 \ln \left (x^{2}+x -3\right ) x^{2}}{5}\) | \(44\) |
parts | \(-\frac {x^{4}}{5}+\frac {2 x^{3}}{5}+\frac {16 x^{2}}{5}+\frac {13 x}{5}-\frac {\ln \left (x^{2}+x -3\right ) x^{3}}{5}-\frac {3 \ln \left (x^{2}+x -3\right ) x^{2}}{5}\) | \(44\) |
parallelrisch | \(-\frac {x^{4}}{5}-\frac {\ln \left (x^{2}+x -3\right ) x^{3}}{5}+\frac {2 x^{3}}{5}-\frac {3 \ln \left (x^{2}+x -3\right ) x^{2}}{5}+\frac {16 x^{2}}{5}+\frac {13 x}{5}+15\) | \(45\) |
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Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {-39-83 x+24 x^2+43 x^3-4 x^5+\left (18 x+3 x^2-9 x^3-3 x^4\right ) \log \left (-3+x+x^2\right )}{-15+5 x+5 x^2} \, dx=-\frac {1}{5} \, x^{4} + \frac {2}{5} \, x^{3} + \frac {16}{5} \, x^{2} - \frac {1}{5} \, {\left (x^{3} + 3 \, x^{2}\right )} \log \left (x^{2} + x - 3\right ) + \frac {13}{5} \, x \]
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Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {-39-83 x+24 x^2+43 x^3-4 x^5+\left (18 x+3 x^2-9 x^3-3 x^4\right ) \log \left (-3+x+x^2\right )}{-15+5 x+5 x^2} \, dx=- \frac {x^{4}}{5} + \frac {2 x^{3}}{5} + \frac {16 x^{2}}{5} + \frac {13 x}{5} + \left (- \frac {x^{3}}{5} - \frac {3 x^{2}}{5}\right ) \log {\left (x^{2} + x - 3 \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
Time = 0.31 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {-39-83 x+24 x^2+43 x^3-4 x^5+\left (18 x+3 x^2-9 x^3-3 x^4\right ) \log \left (-3+x+x^2\right )}{-15+5 x+5 x^2} \, dx=-\frac {1}{5} \, x^{4} + \frac {2}{5} \, x^{3} + \frac {16}{5} \, x^{2} - \frac {1}{10} \, {\left (2 \, x^{3} + 6 \, x^{2} - 11\right )} \log \left (x^{2} + x - 3\right ) + \frac {13}{5} \, x - \frac {11}{10} \, \log \left (x^{2} + x - 3\right ) \]
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Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {-39-83 x+24 x^2+43 x^3-4 x^5+\left (18 x+3 x^2-9 x^3-3 x^4\right ) \log \left (-3+x+x^2\right )}{-15+5 x+5 x^2} \, dx=-\frac {1}{5} \, x^{4} + \frac {2}{5} \, x^{3} + \frac {16}{5} \, x^{2} - \frac {1}{5} \, {\left (x^{3} + 3 \, x^{2}\right )} \log \left (x^{2} + x - 3\right ) + \frac {13}{5} \, x \]
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Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {-39-83 x+24 x^2+43 x^3-4 x^5+\left (18 x+3 x^2-9 x^3-3 x^4\right ) \log \left (-3+x+x^2\right )}{-15+5 x+5 x^2} \, dx=\frac {13\,x}{5}-x^3\,\left (\frac {\ln \left (x^2+x-3\right )}{5}-\frac {2}{5}\right )-x^2\,\left (\frac {3\,\ln \left (x^2+x-3\right )}{5}-\frac {16}{5}\right )-\frac {x^4}{5} \]
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