Integrand size = 18, antiderivative size = 23 \[ \int \frac {3 x+x^3}{-3-2 x+x^2} \, dx=2 x+\frac {x^2}{2}+9 \log (3-x)+\log (1+x) \]
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Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1607, 1642, 646, 31} \[ \int \frac {3 x+x^3}{-3-2 x+x^2} \, dx=\frac {x^2}{2}+2 x+9 \log (3-x)+\log (x+1) \]
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Rule 31
Rule 646
Rule 1607
Rule 1642
Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (3+x^2\right )}{-3-2 x+x^2} \, dx \\ & = \int \left (2+x+\frac {2 (3+5 x)}{-3-2 x+x^2}\right ) \, dx \\ & = 2 x+\frac {x^2}{2}+2 \int \frac {3+5 x}{-3-2 x+x^2} \, dx \\ & = 2 x+\frac {x^2}{2}+9 \int \frac {1}{-3+x} \, dx+\int \frac {1}{1+x} \, dx \\ & = 2 x+\frac {x^2}{2}+9 \log (3-x)+\log (1+x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {3 x+x^3}{-3-2 x+x^2} \, dx=2 x+\frac {x^2}{2}+9 \log (3-x)+\log (1+x) \]
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Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87
method | result | size |
default | \(2 x +\frac {x^{2}}{2}+\ln \left (1+x \right )+9 \ln \left (-3+x \right )\) | \(20\) |
norman | \(2 x +\frac {x^{2}}{2}+\ln \left (1+x \right )+9 \ln \left (-3+x \right )\) | \(20\) |
risch | \(2 x +\frac {x^{2}}{2}+\ln \left (1+x \right )+9 \ln \left (-3+x \right )\) | \(20\) |
parallelrisch | \(2 x +\frac {x^{2}}{2}+\ln \left (1+x \right )+9 \ln \left (-3+x \right )\) | \(20\) |
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none
Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {3 x+x^3}{-3-2 x+x^2} \, dx=\frac {1}{2} \, x^{2} + 2 \, x + \log \left (x + 1\right ) + 9 \, \log \left (x - 3\right ) \]
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Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {3 x+x^3}{-3-2 x+x^2} \, dx=\frac {x^{2}}{2} + 2 x + 9 \log {\left (x - 3 \right )} + \log {\left (x + 1 \right )} \]
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none
Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {3 x+x^3}{-3-2 x+x^2} \, dx=\frac {1}{2} \, x^{2} + 2 \, x + \log \left (x + 1\right ) + 9 \, \log \left (x - 3\right ) \]
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Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {3 x+x^3}{-3-2 x+x^2} \, dx=\frac {1}{2} \, x^{2} + 2 \, x + \log \left ({\left | x + 1 \right |}\right ) + 9 \, \log \left ({\left | x - 3 \right |}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {3 x+x^3}{-3-2 x+x^2} \, dx=2\,x+\ln \left (x+1\right )+9\,\ln \left (x-3\right )+\frac {x^2}{2} \]
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