\[ y''(x)=-\frac {y(x) \left (-n^2-v (v+1) x^2\right )}{x^2 \left (x^2+1\right )}-\frac {\left (2 x^2+1\right ) y'(x)}{x \left (x^2+1\right )} \] ✓ Mathematica : cpu = 0.162732 (sec), leaf count = 90
DSolve[Derivative[2][y][x] == -(((-n^2 - v*(1 + v)*x^2)*y[x])/(x^2*(1 + x^2))) - ((1 + 2*x^2)*Derivative[1][y][x])/(x*(1 + x^2)),y[x],x]
\[\left \{\left \{y(x)\to c_1 x^{-n} \operatorname {Hypergeometric2F1}\left (-\frac {n}{2}-\frac {v}{2},-\frac {n}{2}+\frac {v}{2}+\frac {1}{2},1-n,-x^2\right )+c_2 x^n \operatorname {Hypergeometric2F1}\left (\frac {n}{2}-\frac {v}{2},\frac {n}{2}+\frac {v}{2}+\frac {1}{2},n+1,-x^2\right )\right \}\right \}\] ✓ Maple : cpu = 0.128 (sec), leaf count = 29
dsolve(diff(diff(y(x),x),x) = -(2*x^2+1)/x/(x^2+1)*diff(y(x),x)-(-v*(v+1)*x^2-n^2)/x^2/(x^2+1)*y(x),y(x))
\[y \left (x \right ) = c_{1} \operatorname {LegendreP}\left (v , n , \sqrt {x^{2}+1}\right )+c_{2} \operatorname {LegendreQ}\left (v , n , \sqrt {x^{2}+1}\right )\]