\[ x (n-v) (n+v+1) y(x)+\left (2 (n+1) x^2+2 n+1\right ) y'(x)+x \left (x^2+1\right ) y''(x)=0 \] ✓ Mathematica : cpu = 0.149839 (sec), leaf count = 87
DSolve[(n - v)*(1 + n + v)*x*y[x] + (1 + 2*n + 2*(1 + n)*x^2)*Derivative[1][y][x] + x*(1 + x^2)*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_1 \, _2F_1\left (\frac {n}{2}-\frac {v}{2},\frac {n}{2}+\frac {v}{2}+\frac {1}{2};n+1;-x^2\right )+c_2 x^{-2 n} \, _2F_1\left (-\frac {n}{2}-\frac {v}{2},-\frac {n}{2}+\frac {v}{2}+\frac {1}{2};1-n;-x^2\right )\right \}\right \}\] ✓ Maple : cpu = 0.078 (sec), leaf count = 35
dsolve(x*(x^2+1)*diff(diff(y(x),x),x)+(2*(n+1)*x^2+2*n+1)*diff(y(x),x)-(v-n)*(v+n+1)*x*y(x)=0,y(x))
\[y \left (x \right ) = x^{-n} \left (\LegendreQ \left (v , n , \sqrt {x^{2}+1}\right ) c_{2}+\LegendreP \left (v , n , \sqrt {x^{2}+1}\right ) c_{1}\right )\]