3.979 \(\int (1-x)^n x^2 (1+x)^{-n} \, dx\)

Optimal. Leaf size=94 \[ -\frac{2^{-n} \left (2 n^2+1\right ) (1-x)^{n+1} \, _2F_1\left (n,n+1;n+2;\frac{1-x}{2}\right )}{3 (n+1)}+\frac{1}{3} n (1-x)^{n+1} (x+1)^{1-n}-\frac{1}{3} x (1-x)^{n+1} (x+1)^{1-n} \]

[Out]

(n*(1 - x)^(1 + n)*(1 + x)^(1 - n))/3 - ((1 - x)^(1 + n)*x*(1 + x)^(1 - n))/3 - ((1 + 2*n^2)*(1 - x)^(1 + n)*H
ypergeometric2F1[n, 1 + n, 2 + n, (1 - x)/2])/(3*2^n*(1 + n))

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Rubi [A]  time = 0.032461, antiderivative size = 94, normalized size of antiderivative = 1., 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 = {90, 80, 69} \[ -\frac{2^{-n} \left (2 n^2+1\right ) (1-x)^{n+1} \, _2F_1\left (n,n+1;n+2;\frac{1-x}{2}\right )}{3 (n+1)}+\frac{1}{3} n (1-x)^{n+1} (x+1)^{1-n}-\frac{1}{3} x (1-x)^{n+1} (x+1)^{1-n} \]

Antiderivative was successfully verified.

[In]

Int[((1 - x)^n*x^2)/(1 + x)^n,x]

[Out]

(n*(1 - x)^(1 + n)*(1 + x)^(1 - n))/3 - ((1 - x)^(1 + n)*x*(1 + x)^(1 - n))/3 - ((1 + 2*n^2)*(1 - x)^(1 + n)*H
ypergeometric2F1[n, 1 + n, 2 + n, (1 - x)/2])/(3*2^n*(1 + n))

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

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*}

Mathematica [A]  time = 0.0302527, size = 76, normalized size = 0.81 \[ -\frac{2^{-n} (1-x)^{n+1} (x+1)^{-n} \left (\left (2 n^2+1\right ) (x+1)^n \, _2F_1\left (n,n+1;n+2;\frac{1-x}{2}\right )+2^n (n+1) (x+1) (x-n)\right )}{3 (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - x)^n*x^2)/(1 + x)^n,x]

[Out]

-((1 - x)^(1 + n)*(2^n*(1 + n)*(1 + x)*(-n + x) + (1 + 2*n^2)*(1 + x)^n*Hypergeometric2F1[n, 1 + n, 2 + n, (1
- x)/2]))/(3*2^n*(1 + n)*(1 + x)^n)

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2} \left ( 1-x \right ) ^{n}}{ \left ( 1+x \right ) ^{n}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x)^n*x^2/((1+x)^n),x)

[Out]

int((1-x)^n*x^2/((1+x)^n),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^n*x^2/((1+x)^n),x, algorithm="maxima")

[Out]

integrate(x^2*(-x + 1)^n/(x + 1)^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2}{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^n*x^2/((1+x)^n),x, algorithm="fricas")

[Out]

integral(x^2*(-x + 1)^n/(x + 1)^n, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**n*x**2/((1+x)**n),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}{\left (-x + 1\right )}^{n}}{{\left (x + 1\right )}^{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)^n*x^2/((1+x)^n),x, algorithm="giac")

[Out]

integrate(x^2*(-x + 1)^n/(x + 1)^n, x)