Integrand size = 18, antiderivative size = 94 \[ \int (1-x)^n x^2 (1+x)^{-n} \, dx=\frac {1}{3} n (1-x)^{1+n} (1+x)^{1-n}-\frac {1}{3} (1-x)^{1+n} x (1+x)^{1-n}-\frac {2^{-n} \left (1+2 n^2\right ) (1-x)^{1+n} \operatorname {Hypergeometric2F1}\left (n,1+n,2+n,\frac {1-x}{2}\right )}{3 (1+n)} \]
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Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {92, 81, 71} \[ \int (1-x)^n x^2 (1+x)^{-n} \, dx=-\frac {2^{-n} \left (2 n^2+1\right ) (1-x)^{n+1} \operatorname {Hypergeometric2F1}\left (n,n+1,n+2,\frac {1-x}{2}\right )}{3 (n+1)}+\frac {1}{3} n (1-x)^{n+1} (x+1)^{1-n}-\frac {1}{3} x (1-x)^{n+1} (x+1)^{1-n} \]
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Rule 71
Rule 81
Rule 92
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{3} (1-x)^{1+n} x (1+x)^{1-n}-\frac {1}{3} \int (1-x)^n (1+x)^{-n} (-1+2 n x) \, dx \\ & = \frac {1}{3} n (1-x)^{1+n} (1+x)^{1-n}-\frac {1}{3} (1-x)^{1+n} x (1+x)^{1-n}-\frac {1}{3} \left (-1-2 n^2\right ) \int (1-x)^n (1+x)^{-n} \, dx \\ & = \frac {1}{3} n (1-x)^{1+n} (1+x)^{1-n}-\frac {1}{3} (1-x)^{1+n} x (1+x)^{1-n}-\frac {2^{-n} \left (1+2 n^2\right ) (1-x)^{1+n} \, _2F_1\left (n,1+n;2+n;\frac {1-x}{2}\right )}{3 (1+n)} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.81 \[ \int (1-x)^n x^2 (1+x)^{-n} \, dx=-\frac {2^{-n} (1-x)^{1+n} (1+x)^{-n} \left (2^n (1+n) (1+x) (-n+x)+\left (1+2 n^2\right ) (1+x)^n \operatorname {Hypergeometric2F1}\left (n,1+n,2+n,\frac {1-x}{2}\right )\right )}{3 (1+n)} \]
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\[\int \left (1-x \right )^{n} x^{2} \left (1+x \right )^{-n}d x\]
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\[ \int (1-x)^n x^2 (1+x)^{-n} \, dx=\int { \frac {x^{2} {\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}} \,d x } \]
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\[ \int (1-x)^n x^2 (1+x)^{-n} \, dx=\int x^{2} \left (1 - x\right )^{n} \left (x + 1\right )^{- n}\, dx \]
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\[ \int (1-x)^n x^2 (1+x)^{-n} \, dx=\int { \frac {x^{2} {\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}} \,d x } \]
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\[ \int (1-x)^n x^2 (1+x)^{-n} \, dx=\int { \frac {x^{2} {\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}} \,d x } \]
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Timed out. \[ \int (1-x)^n x^2 (1+x)^{-n} \, dx=\int \frac {x^2\,{\left (1-x\right )}^n}{{\left (x+1\right )}^n} \,d x \]
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