Integrand size = 16, antiderivative size = 47 \[ \int \frac {7-3 x}{-5+2 x+x^2} \, dx=-\frac {1}{6} \left (9-5 \sqrt {6}\right ) \log \left (1-\sqrt {6}+x\right )-\frac {1}{6} \left (9+5 \sqrt {6}\right ) \log \left (1+\sqrt {6}+x\right ) \]
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Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {646, 31} \[ \int \frac {7-3 x}{-5+2 x+x^2} \, dx=-\frac {1}{6} \left (9-5 \sqrt {6}\right ) \log \left (x-\sqrt {6}+1\right )-\frac {1}{6} \left (9+5 \sqrt {6}\right ) \log \left (x+\sqrt {6}+1\right ) \]
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Rule 31
Rule 646
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \left (-9+5 \sqrt {6}\right ) \int \frac {1}{1-\sqrt {6}+x} \, dx-\frac {1}{6} \left (9+5 \sqrt {6}\right ) \int \frac {1}{1+\sqrt {6}+x} \, dx \\ & = -\frac {1}{6} \left (9-5 \sqrt {6}\right ) \log \left (1-\sqrt {6}+x\right )-\frac {1}{6} \left (9+5 \sqrt {6}\right ) \log \left (1+\sqrt {6}+x\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00 \[ \int \frac {7-3 x}{-5+2 x+x^2} \, dx=\frac {1}{6} \left (-9+5 \sqrt {6}\right ) \log \left (-1+\sqrt {6}-x\right )+\frac {1}{6} \left (-9-5 \sqrt {6}\right ) \log \left (1+\sqrt {6}+x\right ) \]
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Time = 21.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.62
method | result | size |
default | \(-\frac {3 \ln \left (x^{2}+2 x -5\right )}{2}-\frac {5 \sqrt {6}\, \operatorname {arctanh}\left (\frac {\left (2+2 x \right ) \sqrt {6}}{12}\right )}{3}\) | \(29\) |
risch | \(-\frac {3 \ln \left (1+x -\sqrt {6}\right )}{2}+\frac {5 \ln \left (1+x -\sqrt {6}\right ) \sqrt {6}}{6}-\frac {3 \ln \left (1+x +\sqrt {6}\right )}{2}-\frac {5 \ln \left (1+x +\sqrt {6}\right ) \sqrt {6}}{6}\) | \(48\) |
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Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.15 \[ \int \frac {7-3 x}{-5+2 x+x^2} \, dx=\frac {5}{6} \, \sqrt {3} \sqrt {2} \log \left (-\frac {2 \, \sqrt {3} \sqrt {2} {\left (x + 1\right )} - x^{2} - 2 \, x - 7}{x^{2} + 2 \, x - 5}\right ) - \frac {3}{2} \, \log \left (x^{2} + 2 \, x - 5\right ) \]
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Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.94 \[ \int \frac {7-3 x}{-5+2 x+x^2} \, dx=- \left (\frac {3}{2} + \frac {5 \sqrt {6}}{6}\right ) \log {\left (x + 1 + \sqrt {6} \right )} - \left (\frac {3}{2} - \frac {5 \sqrt {6}}{6}\right ) \log {\left (x - \sqrt {6} + 1 \right )} \]
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Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \frac {7-3 x}{-5+2 x+x^2} \, dx=\frac {5}{6} \, \sqrt {6} \log \left (\frac {x - \sqrt {6} + 1}{x + \sqrt {6} + 1}\right ) - \frac {3}{2} \, \log \left (x^{2} + 2 \, x - 5\right ) \]
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Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.94 \[ \int \frac {7-3 x}{-5+2 x+x^2} \, dx=\frac {5}{6} \, \sqrt {6} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {6} + 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt {6} + 2 \right |}}\right ) - \frac {3}{2} \, \log \left ({\left | x^{2} + 2 \, x - 5 \right |}\right ) \]
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Time = 0.14 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.72 \[ \int \frac {7-3 x}{-5+2 x+x^2} \, dx=\ln \left (x-\sqrt {6}+1\right )\,\left (\frac {5\,\sqrt {6}}{6}-\frac {3}{2}\right )-\ln \left (x+\sqrt {6}+1\right )\,\left (\frac {5\,\sqrt {6}}{6}+\frac {3}{2}\right ) \]
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