Integrand size = 14, antiderivative size = 14 \[ \int \frac {2+2 x+x^2}{2+x} \, dx=\frac {x^2}{2}+2 \log (2+x) \]
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Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {712} \[ \int \frac {2+2 x+x^2}{2+x} \, dx=\frac {x^2}{2}+2 \log (x+2) \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (x+\frac {2}{2+x}\right ) \, dx \\ & = \frac {x^2}{2}+2 \log (2+x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {2+2 x+x^2}{2+x} \, dx=\frac {1}{2} \left (-4+x^2+4 \log (2+x)\right ) \]
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Time = 16.17 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {x^{2}}{2}+2 \ln \left (2+x \right )\) | \(13\) |
norman | \(\frac {x^{2}}{2}+2 \ln \left (2+x \right )\) | \(13\) |
risch | \(\frac {x^{2}}{2}+2 \ln \left (2+x \right )\) | \(13\) |
parallelrisch | \(\frac {x^{2}}{2}+2 \ln \left (2+x \right )\) | \(13\) |
meijerg | \(2 \ln \left (\frac {x}{2}+1\right )-\frac {x \left (-\frac {3 x}{2}+6\right )}{3}+2 x\) | \(21\) |
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none
Time = 0.38 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {2+2 x+x^2}{2+x} \, dx=\frac {1}{2} \, x^{2} + 2 \, \log \left (x + 2\right ) \]
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Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {2+2 x+x^2}{2+x} \, dx=\frac {x^{2}}{2} + 2 \log {\left (x + 2 \right )} \]
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none
Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {2+2 x+x^2}{2+x} \, dx=\frac {1}{2} \, x^{2} + 2 \, \log \left (x + 2\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {2+2 x+x^2}{2+x} \, dx=\frac {1}{2} \, x^{2} + 2 \, \log \left ({\left | x + 2 \right |}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {2+2 x+x^2}{2+x} \, dx=2\,\ln \left (x+2\right )+\frac {x^2}{2} \]
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