\[ y''(x)=-\frac {y(x) \left (-n^2-v (v+1) x^2\right )}{x^2 \left (x^2+1\right )}-\frac {\left (2 x^2+1\right ) y'(x)}{x \left (x^2+1\right )} \] ✓ Mathematica : cpu = 0.228603 (sec), leaf count = 90
\[\left \{\left \{y(x)\to c_1 x^{-n} \, _2F_1\left (-\frac {n}{2}-\frac {v}{2},-\frac {n}{2}+\frac {v}{2}+\frac {1}{2};1-n;-x^2\right )+c_2 x^n \, _2F_1\left (\frac {n}{2}-\frac {v}{2},\frac {n}{2}+\frac {v}{2}+\frac {1}{2};n+1;-x^2\right )\right \}\right \}\] ✓ Maple : cpu = 0.456 (sec), leaf count = 29
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\it LegendreP} \left ( v,n,\sqrt {{x}^{2}+1} \right ) +{\it \_C2}\,{\it LegendreQ} \left ( v,n,\sqrt {{x}^{2}+1} \right ) \right \} \]