Integrand size = 22, antiderivative size = 15 \[ \int \frac {-1+3 x+2 x^2}{-x+2 x^2} \, dx=-5+x+\log \left (\frac {6 x (-1+2 x)}{e^5}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1607, 907} \[ \int \frac {-1+3 x+2 x^2}{-x+2 x^2} \, dx=x+\log (1-2 x)+\log (x) \]
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Rule 907
Rule 1607
Rubi steps \begin{align*} \text {integral}& = \int \frac {-1+3 x+2 x^2}{x (-1+2 x)} \, dx \\ & = \int \left (1+\frac {1}{x}+\frac {2}{-1+2 x}\right ) \, dx \\ & = x+\log (1-2 x)+\log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {-1+3 x+2 x^2}{-x+2 x^2} \, dx=x+\log (1-2 x)+\log (x) \]
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Time = 1.33 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60
method | result | size |
parallelrisch | \(x +\ln \left (x \right )+\ln \left (x -\frac {1}{2}\right )\) | \(9\) |
default | \(x +\ln \left (-1+2 x \right )+\ln \left (x \right )\) | \(11\) |
norman | \(x +\ln \left (-1+2 x \right )+\ln \left (x \right )\) | \(11\) |
risch | \(x +\ln \left (2 x^{2}-x \right )\) | \(13\) |
meijerg | \(\ln \left (x \right )+\ln \left (2\right )+i \pi +\ln \left (1-2 x \right )+x\) | \(17\) |
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none
Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {-1+3 x+2 x^2}{-x+2 x^2} \, dx=x + \log \left (2 \, x^{2} - x\right ) \]
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Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.53 \[ \int \frac {-1+3 x+2 x^2}{-x+2 x^2} \, dx=x + \log {\left (2 x^{2} - x \right )} \]
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none
Time = 0.19 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {-1+3 x+2 x^2}{-x+2 x^2} \, dx=x + \log \left (2 \, x - 1\right ) + \log \left (x\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {-1+3 x+2 x^2}{-x+2 x^2} \, dx=x + \log \left ({\left | 2 \, x - 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \]
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Time = 9.72 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {-1+3 x+2 x^2}{-x+2 x^2} \, dx=x+\ln \left (x\,\left (2\,x-1\right )\right ) \]
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