Optimal. Leaf size=94 \[ \frac {1}{3} n (1-x)^{1+n} (1+x)^{1-n}-\frac {1}{3} (1-x)^{1+n} x (1+x)^{1-n}-\frac {2^{-n} \left (1+2 n^2\right ) (1-x)^{1+n} \, _2F_1\left (n,1+n;2+n;\frac {1-x}{2}\right )}{3 (1+n)} \]
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Rubi [A]
time = 0.02, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {92, 81, 71}
\begin {gather*} -\frac {2^{-n} \left (2 n^2+1\right ) (1-x)^{n+1} \, _2F_1\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 81
Rule 92
Rubi steps
\begin {align*} \int (1-x)^n x^2 (1+x)^{-n} \, dx &=-\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*}
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Mathematica [A]
time = 0.05, size = 76, normalized size = 0.81 \begin {gather*} -\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 \, _2F_1\left (n,1+n;2+n;\frac {1-x}{2}\right )\right )}{3 (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (1-x \right )^{n} x^{2} \left (1+x \right )^{-n}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (1 - x\right )^{n} \left (x + 1\right )^{- n}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,{\left (1-x\right )}^n}{{\left (x+1\right )}^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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