Integrand size = 23, antiderivative size = 23 \[ \int \frac {x^2}{(1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx=-\frac {2}{9 (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {927} \[ \int \frac {x^2}{(1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx=-\frac {2}{9 (x+1)^{3/2} \left (x^2-x+1\right )^{3/2}} \]
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Rule 927
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{9 (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \\ \end{align*}
Time = 10.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{(1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx=-\frac {2}{9 (1+x)^{3/2} \left (1-x+x^2\right )^{3/2}} \]
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Time = 0.66 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78
method | result | size |
gosper | \(-\frac {2}{9 \left (1+x \right )^{\frac {3}{2}} \left (x^{2}-x +1\right )^{\frac {3}{2}}}\) | \(18\) |
default | \(-\frac {2}{9 \left (x^{3}+1\right ) \sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(25\) |
elliptic | \(-\frac {2 \sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}}{9 \sqrt {1+x}\, \sqrt {x^{2}-x +1}\, \left (x^{3}+1\right )^{\frac {3}{2}}}\) | \(39\) |
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none
Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {x^2}{(1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx=-\frac {2 \, \sqrt {x^{2} - x + 1} \sqrt {x + 1}}{9 \, {\left (x^{6} + 2 \, x^{3} + 1\right )}} \]
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\[ \int \frac {x^2}{(1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx=\int \frac {x^{2}}{\left (x + 1\right )^{\frac {5}{2}} \left (x^{2} - x + 1\right )^{\frac {5}{2}}}\, dx \]
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none
Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {x^2}{(1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx=-\frac {2}{9 \, {\left (x^{3} + 1\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1}} \]
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\[ \int \frac {x^2}{(1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx=\int { \frac {x^{2}}{{\left (x^{2} - x + 1\right )}^{\frac {5}{2}} {\left (x + 1\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 12.44 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.57 \[ \int \frac {x^2}{(1+x)^{5/2} \left (1-x+x^2\right )^{5/2}} \, dx=\frac {18\,\sqrt {x+1}\,{\left (x^2-x+1\right )}^{5/2}-18\,x\,\sqrt {x+1}\,{\left (x^2-x+1\right )}^{5/2}}{\left (x+1\right )\,\left (81\,x\,{\left (x^2-x+1\right )}^4-162\,{\left (x^2-x+1\right )}^4+81\,{\left (x^2-x+1\right )}^5\right )} \]
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