\[ y''(x)=-\frac {y(x) \left (a^2 \left (x^2+1\right )^2+m^2-n (n+1) \left (x^2+1\right )\right )}{\left (x^2+1\right )^2}-\frac {2 x y'(x)}{x^2+1} \] ✓ Mathematica : cpu = 0.208025 (sec), leaf count = 229
DSolve[Derivative[2][y][x] == -(((m^2 - n*(1 + n)*(1 + x^2) + a^2*(1 + x^2)^2)*y[x])/(1 + x^2)^2) - (2*x*Derivative[1][y][x])/(1 + x^2),y[x],x]
\[\left \{\left \{y(x)\to c_1 \left (x^2+1\right )^{\frac {\sqrt {m^2}}{2}} \text {HeunC}\left [\frac {1}{4} \left (-a^2-\sqrt {m^2} \left (\sqrt {m^2}+1\right )+\frac {1}{4} \left (\sqrt {(2 n+1)^2}-1\right )^2+\frac {1}{2} \left (\sqrt {(2 n+1)^2}-1\right )\right ),-\frac {a^2}{4},\frac {1}{2},\sqrt {m^2}+1,0,-x^2\right ]+c_2 x \left (x^2+1\right )^{\frac {\sqrt {m^2}}{2}} \text {HeunC}\left [\frac {1}{4} \left (-a^2-\left (\sqrt {m^2}+\frac {1}{2} \left (1-\sqrt {(2 n+1)^2}\right )+1\right ) \left (\sqrt {m^2}+\frac {1}{2} \left (\sqrt {(2 n+1)^2}-1\right )+2\right )\right ),-\frac {a^2}{4},\frac {3}{2},\sqrt {m^2}+1,0,-x^2\right ]\right \}\right \}\] ✓ Maple : cpu = 0.198 (sec), leaf count = 88
dsolve(diff(diff(y(x),x),x) = -2/(x^2+1)*x*diff(y(x),x)-(a^2*(x^2+1)^2-n*(n+1)*(x^2+1)+m^2)/(x^2+1)^2*y(x),y(x))
\[y \left (x \right ) = \left (x^{2}+1\right )^{\frac {m}{2}} \left (\HeunC \left (0, \frac {1}{2}, m , -\frac {a^{2}}{4}, \frac {1}{4}+\frac {1}{4} a^{2}+\frac {1}{4} m^{2}-\frac {1}{4} n^{2}-\frac {1}{4} n , -x^{2}\right ) c_{2} x +\HeunC \left (0, -\frac {1}{2}, m , -\frac {a^{2}}{4}, \frac {1}{4}+\frac {1}{4} a^{2}+\frac {1}{4} m^{2}-\frac {1}{4} n^{2}-\frac {1}{4} n , -x^{2}\right ) c_{1}\right )\]