Integrand size = 23, antiderivative size = 23 \[ \int \frac {-1+5 x+2 x^2}{-2 x+x^2+x^3} \, dx=2 \log (1-x)+\frac {\log (x)}{2}-\frac {1}{2} \log (2+x) \]
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Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1608, 1642} \[ \int \frac {-1+5 x+2 x^2}{-2 x+x^2+x^3} \, dx=2 \log (1-x)+\frac {\log (x)}{2}-\frac {1}{2} \log (x+2) \]
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Rule 1608
Rule 1642
Rubi steps \begin{align*} \text {integral}& = \int \frac {-1+5 x+2 x^2}{x \left (-2+x+x^2\right )} \, dx \\ & = \int \left (\frac {2}{-1+x}+\frac {1}{2 x}-\frac {1}{2 (2+x)}\right ) \, dx \\ & = 2 \log (1-x)+\frac {\log (x)}{2}-\frac {1}{2} \log (2+x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-1+5 x+2 x^2}{-2 x+x^2+x^3} \, dx=2 \log (1-x)+\frac {\log (x)}{2}-\frac {1}{2} \log (2+x) \]
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Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78
method | result | size |
default | \(2 \ln \left (-1+x \right )-\frac {\ln \left (2+x \right )}{2}+\frac {\ln \left (x \right )}{2}\) | \(18\) |
norman | \(2 \ln \left (-1+x \right )-\frac {\ln \left (2+x \right )}{2}+\frac {\ln \left (x \right )}{2}\) | \(18\) |
risch | \(2 \ln \left (-1+x \right )-\frac {\ln \left (2+x \right )}{2}+\frac {\ln \left (x \right )}{2}\) | \(18\) |
parallelrisch | \(2 \ln \left (-1+x \right )-\frac {\ln \left (2+x \right )}{2}+\frac {\ln \left (x \right )}{2}\) | \(18\) |
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none
Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {-1+5 x+2 x^2}{-2 x+x^2+x^3} \, dx=-\frac {1}{2} \, \log \left (x + 2\right ) + 2 \, \log \left (x - 1\right ) + \frac {1}{2} \, \log \left (x\right ) \]
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Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {-1+5 x+2 x^2}{-2 x+x^2+x^3} \, dx=\frac {\log {\left (x \right )}}{2} + 2 \log {\left (x - 1 \right )} - \frac {\log {\left (x + 2 \right )}}{2} \]
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none
Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {-1+5 x+2 x^2}{-2 x+x^2+x^3} \, dx=-\frac {1}{2} \, \log \left (x + 2\right ) + 2 \, \log \left (x - 1\right ) + \frac {1}{2} \, \log \left (x\right ) \]
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none
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {-1+5 x+2 x^2}{-2 x+x^2+x^3} \, dx=-\frac {1}{2} \, \log \left ({\left | x + 2 \right |}\right ) + 2 \, \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x \right |}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {-1+5 x+2 x^2}{-2 x+x^2+x^3} \, dx=2\,\ln \left (x-1\right )+\mathrm {atanh}\left (\frac {135}{11\,\left (11\,x-5\right )}+\frac {16}{11}\right ) \]
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