\[ x (n-v-1) (n+v) y(x)-\left (2 (n-1) x^2+2 n-1\right ) y'(x)+x \left (x^2+1\right ) y''(x)=0 \] ✓ Mathematica : cpu = 0.137336 (sec), leaf count = 87
DSolve[(-1 + n - v)*(n + v)*x*y[x] - (-1 + 2*n + 2*(-1 + n)*x^2)*Derivative[1][y][x] + x*(1 + x^2)*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_1 \, _2F_1\left (-\frac {n}{2}-\frac {v}{2},-\frac {n}{2}+\frac {v}{2}+\frac {1}{2};1-n;-x^2\right )+c_2 x^{2 n} \, _2F_1\left (\frac {n}{2}-\frac {v}{2},\frac {n}{2}+\frac {v}{2}+\frac {1}{2};n+1;-x^2\right )\right \}\right \}\] ✓ Maple : cpu = 0.064 (sec), leaf count = 33
dsolve(x*(x^2+1)*diff(diff(y(x),x),x)-(2*(n-1)*x^2+2*n-1)*diff(y(x),x)+(v+n)*(-v+n-1)*x*y(x)=0,y(x))
\[y \left (x \right ) = x^{n} \left (\LegendreQ \left (v , n , \sqrt {x^{2}+1}\right ) c_{2}+\LegendreP \left (v , n , \sqrt {x^{2}+1}\right ) c_{1}\right )\]