\(\int \frac {(1-\frac {x}{a})^{-n/2} (1+\frac {x}{a})^{n/2}}{x^2} \, dx\) [335]

Optimal result

Integrand size = 31, antiderivative size = 70 \[ \int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx=-\frac {4 \left (1-\frac {x}{a}\right )^{1-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {a-x}{a+x}\right )}{a (2-n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx=-\frac {4 \left (\frac {a+x}{a}\right )^{\frac {2+n}{2}} \left (1-\frac {x}{a}\right )^{-n/2} \operatorname {Hypergeometric2F1}\left (2,1+\frac {n}{2},2+\frac {n}{2},\frac {a+x}{a-x}\right )}{(2+n) (-a+x)} \] Input:

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

Output:

(-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))
 

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {141}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 141

\(\displaystyle -\frac {4 \left (1-\frac {x}{a}\right )^{1-\frac {n}{2}} \left (\frac {x}{a}+1\right )^{\frac {n-2}{2}} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {a-x}{a+x}\right )}{a (2-n)}\)

Input:

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

Output:

(-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))
 

Defintions of rubi rules used

rule 141

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

Maple [F]

\[\int \frac {\left (1+\frac {x}{a}\right )^{\frac {n}{2}} \left (1-\frac {x}{a}\right )^{-\frac {n}{2}}}{x^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx=\int { \frac {{\left (\frac {x}{a} + 1\right )}^{\frac {1}{2} \, n}}{x^{2} {\left (-\frac {x}{a} + 1\right )}^{\frac {1}{2} \, n}} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx=\int \frac {\left (1 - \frac {x}{a}\right )^{- \frac {n}{2}} \left (1 + \frac {x}{a}\right )^{\frac {n}{2}}}{x^{2}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx=\int { \frac {{\left (\frac {x}{a} + 1\right )}^{\frac {1}{2} \, n}}{x^{2} {\left (-\frac {x}{a} + 1\right )}^{\frac {1}{2} \, n}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx=\int { \frac {{\left (\frac {x}{a} + 1\right )}^{\frac {1}{2} \, n}}{x^{2} {\left (-\frac {x}{a} + 1\right )}^{\frac {1}{2} \, n}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx=\int \frac {{\left (\frac {x}{a}+1\right )}^{n/2}}{x^2\,{\left (1-\frac {x}{a}\right )}^{n/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (1-\frac {x}{a}\right )^{-n/2} \left (1+\frac {x}{a}\right )^{n/2}}{x^2} \, dx=\int \frac {\left (a +x \right )^{\frac {n}{2}}}{\left (a -x \right )^{\frac {n}{2}} x^{2}}d x \] Input:

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

Output:

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