\[ y''(x)=-\frac {\left (-(2 v+1)^2+x^2-1\right ) y(x)}{\left (x^2-1\right )^2}-\frac {\left (3 x^2-1\right ) y'(x)}{x \left (x^2-1\right )} \] ✓ Mathematica : cpu = 0.123707 (sec), leaf count = 72
DSolve[Derivative[2][y][x] == -(((-1 - (1 + 2*v)^2 + x^2)*y[x])/(-1 + x^2)^2) - ((-1 + 3*x^2)*Derivative[1][y][x])/(x*(-1 + x^2)),y[x],x]
\[\left \{\left \{y(x)\to c_1 \left (x^2-1\right )^{-v-\frac {1}{2}} \, _2F_1\left (-v,-v;-2 v;1-x^2\right )+c_2 \left (x^2-1\right )^{v+\frac {1}{2}} \, _2F_1\left (v+1,v+1;2 v+2;1-x^2\right )\right \}\right \}\] ✓ Maple : cpu = 0.155 (sec), leaf count = 69
dsolve(diff(diff(y(x),x),x) = -1/(x^2-1)*(3*x^2-1)/x*diff(y(x),x)-(x^2-1-(2*v+1)^2)/(x^2-1)^2*y(x),y(x))
\[y \left (x \right ) = c_{1} \left (x^{2}-1\right )^{-\frac {1}{2}-v} \hypergeom \left (\left [-v , -v \right ], \left [-2 v \right ], -x^{2}+1\right )+c_{2} \left (x^{2}-1\right )^{v +\frac {1}{2}} \hypergeom \left (\left [v +1, v +1\right ], \left [2 v +2\right ], -x^{2}+1\right )\]