\[ -v (v+1) x y(x)+x \left (x^2+1\right ) y''(x)+\left (2 x^2+1\right ) y'(x)=0 \] ✓ Mathematica : cpu = 0.0909222 (sec), leaf count = 63
DSolve[-(v*(1 + v)*x*y[x]) + (1 + 2*x^2)*Derivative[1][y][x] + x*(1 + x^2)*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_2 G_{2,2}^{2,0}\left (-x^2|\begin {array}{c} \frac {1-v}{2},\frac {v+2}{2} \\ 0,0 \\\end {array}\right )+c_1 \, _2F_1\left (\frac {v}{2}+\frac {1}{2},-\frac {v}{2};1;-x^2\right )\right \}\right \}\] ✓ Maple : cpu = 0.122 (sec), leaf count = 52
dsolve(x*(x^2+1)*diff(diff(y(x),x),x)+(2*x^2+1)*diff(y(x),x)-v*(v+1)*x*y(x)=0,y(x))
\[y \left (x \right ) = c_{1} \hypergeom \left (\left [-\frac {v}{2}, \frac {1}{2}+\frac {v}{2}\right ], \left [\frac {1}{2}\right ], x^{2}+1\right )+c_{2} \sqrt {x^{2}+1}\, \hypergeom \left (\left [1+\frac {v}{2}, \frac {1}{2}-\frac {v}{2}\right ], \left [\frac {3}{2}\right ], x^{2}+1\right )\]