\[ \boxed { x \left ( {x}^{2}+1 \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( 2\,{x}^{2}+1 \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -v \left ( v+1 \right ) xy \left ( x \right ) =0} \]
Mathematica: cpu = 0.134517 (sec), leaf count = 63 \[ \left \{\left \{y(x)\to c_2 G_{2,2}^{2,0}\left (-x^2| \begin {array}{c} \frac {1-v}{2},\frac {v+2}{2} \\ 0,0 \\ \end {array} \right )+c_1 \, _2F_1\left (\frac {v}{2}+\frac {1}{2},-\frac {v}{2};1;-x^2\right )\right \}\right \} \]
Maple: cpu = 0.109 (sec), leaf count = 52 \[ \left \{ y \left ( x \right ) ={\it \_C1}\, {\mbox {$_2$F$_1$}(-{\frac {v}{2}},{\frac {1}{2}}+{\frac {v}{2}};\,{\frac {1}{2}};\,{x}^{2}+1)} +{\it \_C2}\,\sqrt {{x}^{2}+1} {\mbox {$_2$F$_1$}(1+{\frac {v}{2}},{\frac {1}{2}}-{\frac {v}{2}};\,{\frac {3}{2}};\,{x}^{2}+1)} \right \} \]