\[ y''(x)=-\frac {y(x) \left (4 a (a+1) x^4-2 a \left (x^2-1\right ) x^2+\left (x^2-1\right )^2 \left (x^2-v^2\right )\right )}{x^2 \left (x^2-1\right )^2}-\frac {\left ((1-4 a) x^2-1\right ) y'(x)}{x \left (x^2-1\right )} \] ✓ Mathematica : cpu = 0.621271 (sec), leaf count = 86
DSolve[Derivative[2][y][x] == -(((4*a*(1 + a)*x^4 - 2*a*x^2*(-1 + x^2) + (-1 + x^2)^2*(-v^2 + x^2))*y[x])/(x^2*(-1 + x^2)^2)) - ((-1 + (1 - 4*a)*x^2)*Derivative[1][y][x])/(x*(-1 + x^2)),y[x],x]
\[\left \{\left \{y(x)\to c_2 \left (x^2-1\right )^{a+1} x^{-v} \text {HeunC}\left [-\frac {a}{2}+v-\frac {3}{4},\frac {1}{4},1-v,2,0,x^2\right ]+c_1 \left (x^2-1\right )^{a+1} x^v \text {HeunC}\left [\frac {1}{4} (-2 a-4 v-3),\frac {1}{4},v+1,2,0,x^2\right ]\right \}\right \}\] ✓ Maple : cpu = 0.531 (sec), leaf count = 59
dsolve(diff(diff(y(x),x),x) = -1/x/(x^2-1)*((1-4*a)*x^2-1)*diff(y(x),x)-((-v^2+x^2)*(x^2-1)^2+4*a*(a+1)*x^4-2*a*x^2*(x^2-1))/x^2/(x^2-1)^2*y(x),y(x))
\[y \left (x \right ) = -\left (x^{2}-1\right ) \left (x^{2}-1\right )^{a} \left (c_{1} x^{v} \operatorname {HeunC}\left (0, v , 1, \frac {1}{4}, \frac {1}{4}+\frac {a}{2}, x^{2}\right )+c_{2} x^{-v} \operatorname {HeunC}\left (0, -v , 1, \frac {1}{4}, \frac {1}{4}+\frac {a}{2}, x^{2}\right )\right )\]