3.998 \(\int \frac{(1-\frac{x}{a})^{-n/2} (1+\frac{x}{a})^{n/2}}{x^2} \, dx\)

Optimal. Leaf size=70 \[ -\frac{4 \left (1-\frac{x}{a}\right )^{1-\frac{n}{2}} \left (\frac{x}{a}+1\right )^{\frac{n-2}{2}} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{a-x}{a+x}\right )}{a (2-n)} \]

[Out]

(-4*(1 - x/a)^(1 - n/2)*(1 + x/a)^((-2 + n)/2)*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (a - x)/(a + x)])/(a*(2
- n))

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Rubi [A]  time = 0.0169073, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032, Rules used = {131} \[ -\frac{4 \left (1-\frac{x}{a}\right )^{1-\frac{n}{2}} \left (\frac{x}{a}+1\right )^{\frac{n-2}{2}} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{a-x}{a+x}\right )}{a (2-n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(1 - x/a)^(1 - n/2)*(1 + x/a)^((-2 + n)/2)*Hypergeometric2F1[2, 1 - n/2, 2 - n/2, (a - x)/(a + x)])/(a*(2
- n))

Rule 131

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((b*c -
a*d)^n*(a + b*x)^(m + 1)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rubi steps

\begin{align*} \int \frac{\left (1-\frac{x}{a}\right )^{-n/2} \left (1+\frac{x}{a}\right )^{n/2}}{x^2} \, dx &=-\frac{4 \left (1-\frac{x}{a}\right )^{1-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{\frac{1}{2} (-2+n)} \, _2F_1\left (2,1-\frac{n}{2};2-\frac{n}{2};\frac{a-x}{a+x}\right )}{a (2-n)}\\ \end{align*}

Mathematica [A]  time = 0.0331712, size = 70, normalized size = 1. \[ -\frac{4 \left (\frac{a+x}{a}\right )^{\frac{n+2}{2}} \left (1-\frac{x}{a}\right )^{-n/2} \, _2F_1\left (2,\frac{n}{2}+1;\frac{n}{2}+2;\frac{a+x}{a-x}\right )}{(n+2) (x-a)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*((a + x)/a)^((2 + n)/2)*Hypergeometric2F1[2, 1 + n/2, 2 + n/2, (a + x)/(a - x)])/((2 + n)*(-a + x)*(1 - x/
a)^(n/2))

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Maple [F]  time = 0.104, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( 1+{\frac{x}{a}} \right ) ^{{\frac{n}{2}}} \left ( \left ( 1-{\frac{x}{a}} \right ) ^{{\frac{n}{2}}} \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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