Integrand size = 15, antiderivative size = 33 \[ \int \frac {x (1+x)^2}{\left (1+x+x^2\right )^3} \, dx=-\frac {(1+x) (1+2 x)}{6 \left (1+x+x^2\right )^2}-\frac {1}{6 \left (1+x+x^2\right )} \]
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Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {832, 643} \[ \int \frac {x (1+x)^2}{\left (1+x+x^2\right )^3} \, dx=-\frac {(x+1) (2 x+1)}{6 \left (x^2+x+1\right )^2}-\frac {1}{6 \left (x^2+x+1\right )} \]
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Rule 643
Rule 832
Rubi steps \begin{align*} \text {integral}& = -\frac {(1+x) (1+2 x)}{6 \left (1+x+x^2\right )^2}-\frac {1}{6} \int \frac {-1-2 x}{\left (1+x+x^2\right )^2} \, dx \\ & = -\frac {(1+x) (1+2 x)}{6 \left (1+x+x^2\right )^2}-\frac {1}{6 \left (1+x+x^2\right )} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.67 \[ \int \frac {x (1+x)^2}{\left (1+x+x^2\right )^3} \, dx=-\frac {2+4 x+3 x^2}{6 \left (1+x+x^2\right )^2} \]
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Time = 0.38 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.61
method | result | size |
default | \(\frac {-\frac {1}{2} x^{2}-\frac {2}{3} x -\frac {1}{3}}{\left (x^{2}+x +1\right )^{2}}\) | \(20\) |
norman | \(\frac {-\frac {1}{2} x^{2}-\frac {2}{3} x -\frac {1}{3}}{\left (x^{2}+x +1\right )^{2}}\) | \(20\) |
risch | \(\frac {-\frac {1}{2} x^{2}-\frac {2}{3} x -\frac {1}{3}}{\left (x^{2}+x +1\right )^{2}}\) | \(20\) |
gosper | \(-\frac {3 x^{2}+4 x +2}{6 \left (x^{2}+x +1\right )^{2}}\) | \(21\) |
parallelrisch | \(\frac {-3 x^{2}-4 x -2}{6 \left (x^{2}+x +1\right )^{2}}\) | \(21\) |
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none
Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {x (1+x)^2}{\left (1+x+x^2\right )^3} \, dx=-\frac {3 \, x^{2} + 4 \, x + 2}{6 \, {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} \]
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Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {x (1+x)^2}{\left (1+x+x^2\right )^3} \, dx=\frac {- 3 x^{2} - 4 x - 2}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} \]
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none
Time = 0.18 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {x (1+x)^2}{\left (1+x+x^2\right )^3} \, dx=-\frac {3 \, x^{2} + 4 \, x + 2}{6 \, {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} \]
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none
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.61 \[ \int \frac {x (1+x)^2}{\left (1+x+x^2\right )^3} \, dx=-\frac {3 \, x^{2} + 4 \, x + 2}{6 \, {\left (x^{2} + x + 1\right )}^{2}} \]
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Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.61 \[ \int \frac {x (1+x)^2}{\left (1+x+x^2\right )^3} \, dx=-\frac {3\,x^2+4\,x+2}{6\,{\left (x^2+x+1\right )}^2} \]
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