Integrand size = 12, antiderivative size = 20 \[ \int \frac {x^2}{-6+x+x^2} \, dx=x+\frac {4}{5} \log (2-x)-\frac {9}{5} \log (3+x) \]
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Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {717, 646, 31} \[ \int \frac {x^2}{-6+x+x^2} \, dx=x+\frac {4}{5} \log (2-x)-\frac {9}{5} \log (x+3) \]
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Rule 31
Rule 646
Rule 717
Rubi steps \begin{align*} \text {integral}& = x+\int \frac {6-x}{-6+x+x^2} \, dx \\ & = x+\frac {4}{5} \int \frac {1}{-2+x} \, dx-\frac {9}{5} \int \frac {1}{3+x} \, dx \\ & = x+\frac {4}{5} \log (2-x)-\frac {9}{5} \log (3+x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{-6+x+x^2} \, dx=x+\frac {4}{5} \log (2-x)-\frac {9}{5} \log (3+x) \]
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Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75
method | result | size |
default | \(x -\frac {9 \ln \left (3+x \right )}{5}+\frac {4 \ln \left (-2+x \right )}{5}\) | \(15\) |
norman | \(x -\frac {9 \ln \left (3+x \right )}{5}+\frac {4 \ln \left (-2+x \right )}{5}\) | \(15\) |
risch | \(x -\frac {9 \ln \left (3+x \right )}{5}+\frac {4 \ln \left (-2+x \right )}{5}\) | \(15\) |
parallelrisch | \(x -\frac {9 \ln \left (3+x \right )}{5}+\frac {4 \ln \left (-2+x \right )}{5}\) | \(15\) |
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none
Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {x^2}{-6+x+x^2} \, dx=x - \frac {9}{5} \, \log \left (x + 3\right ) + \frac {4}{5} \, \log \left (x - 2\right ) \]
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Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {x^2}{-6+x+x^2} \, dx=x + \frac {4 \log {\left (x - 2 \right )}}{5} - \frac {9 \log {\left (x + 3 \right )}}{5} \]
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none
Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {x^2}{-6+x+x^2} \, dx=x - \frac {9}{5} \, \log \left (x + 3\right ) + \frac {4}{5} \, \log \left (x - 2\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {x^2}{-6+x+x^2} \, dx=x - \frac {9}{5} \, \log \left ({\left | x + 3 \right |}\right ) + \frac {4}{5} \, \log \left ({\left | x - 2 \right |}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {x^2}{-6+x+x^2} \, dx=x+\frac {4\,\ln \left (x-2\right )}{5}-\frac {9\,\ln \left (x+3\right )}{5} \]
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