\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) =-{\frac { \left ( 2\,{x}^{2}+1 \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }{x \left ( {x}^{2}+1 \right ) }}-{\frac { \left ( -v \left ( v+1 \right ) {x}^{2}-{n}^{2} \right ) y \left ( x \right ) }{{x}^{2} \left ( {x}^{2}+1 \right ) }}=0} \]
Mathematica: cpu = 0.287536 (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 = 0.063 (sec), leaf count = 29 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\it LegendreP} \left ( v,n, \sqrt {{x}^{2}+1} \right ) +{\it \_C2}\,{\it LegendreQ} \left ( v,n, \sqrt {{x}^{2}+1} \right ) \right \} \]