Optimal. Leaf size=105 \[ \frac {2^{-n} n \left (n^2+2\right ) (1-x)^{n+1} \, _2F_1\left (n,n+1;n+2;\frac {1-x}{2}\right )}{3 (n+1)}-\frac {1}{12} (1-x)^{n+1} \left (2 n^2-2 n x+3\right ) (x+1)^{1-n}-\frac {1}{4} x^2 (1-x)^{n+1} (x+1)^{1-n} \]
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Rubi [A] time = 0.08, antiderivative size = 105, 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 = {100, 147, 69} \[ \frac {2^{-n} n \left (n^2+2\right ) (1-x)^{n+1} \, _2F_1\left (n,n+1;n+2;\frac {1-x}{2}\right )}{3 (n+1)}-\frac {1}{12} (1-x)^{n+1} \left (2 n^2-2 n x+3\right ) (x+1)^{1-n}-\frac {1}{4} x^2 (1-x)^{n+1} (x+1)^{1-n} \]
Antiderivative was successfully verified.
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Rule 69
Rule 100
Rule 147
Rubi steps
\begin {align*} \int (1-x)^n x^3 (1+x)^{-n} \, dx &=-\frac {1}{4} (1-x)^{1+n} x^2 (1+x)^{1-n}-\frac {1}{4} \int (1-x)^n x (1+x)^{-n} (-2+2 n x) \, dx\\ &=-\frac {1}{4} (1-x)^{1+n} x^2 (1+x)^{1-n}-\frac {1}{12} (1-x)^{1+n} (1+x)^{1-n} \left (3+2 n^2-2 n x\right )-\frac {1}{3} \left (n \left (2+n^2\right )\right ) \int (1-x)^n (1+x)^{-n} \, dx\\ &=-\frac {1}{4} (1-x)^{1+n} x^2 (1+x)^{1-n}-\frac {1}{12} (1-x)^{1+n} (1+x)^{1-n} \left (3+2 n^2-2 n x\right )+\frac {2^{-n} n \left (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 = 0.13, size = 128, normalized size = 1.22 \[ \frac {2^{-n-2} (x-1) (1-x)^n (x+1)^{-n} \left ((n+1) \left (2^n x^2 (x+1)-2 (x+1)^n \, _2F_1\left (n,n+1;n+2;\frac {1-x}{2}\right )\right )-8 n (x+1)^n \, _2F_1\left (n-2,n+1;n+2;\frac {1-x}{2}\right )+4 (2 n+1) (x+1)^n \, _2F_1\left (n-1,n+1;n+2;\frac {1-x}{2}\right )\right )}{n+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3} {\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}}, x\right ) \]
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 \[ \int \frac {x^{3} {\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int x^{3} \left (-x +1\right )^{n} \left (x +1\right )^{-n}\, dx \]
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 \[ \int \frac {x^{3} {\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}}\,{d x} \]
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 \[ \int \frac {x^3\,{\left (1-x\right )}^n}{{\left (x+1\right )}^n} \,d x \]
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 \[ \int x^{3} \left (1 - x\right )^{n} \left (x + 1\right )^{- n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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