\[ y''(x)=-\frac {y(x) \left (a^2 \left (x^2+1\right )^2+m^2-n (n+1) \left (x^2+1\right )\right )}{\left (x^2+1\right )^2}-\frac {2 x y'(x)}{x^2+1} \] ✗ Mathematica : cpu = 2.24208 (sec), leaf count = 0 , DifferentialRoot result
\[\left \{\left \{y(x)\to \text {DifferentialRoot}\left (\{\unicode {f818},\unicode {f817}\}\unicode {f4a1}\left \{\unicode {f818}''(\unicode {f817}) \left (\unicode {f817}^2+1\right )^2+2 \unicode {f817} \unicode {f818}'(\unicode {f817}) \left (\unicode {f817}^2+1\right )+\left (a^2 \unicode {f817}^4+2 a^2 \unicode {f817}^2-n^2 \unicode {f817}^2-n \unicode {f817}^2+a^2+m^2-n^2-n\right ) \unicode {f818}(\unicode {f817})=0,\unicode {f818}(0)=c_1,\unicode {f818}'(0)=c_2\right \}\right )(x)\right \}\right \}\]
✓ Maple : cpu = 0.512 (sec), leaf count = 88
\[ \left \{ y \left ( x \right ) = \left ( {x}^{2}+1 \right ) ^{{\frac {m}{2}}} \left ( {\it HeunC} \left ( 0,{\frac {1}{2}},m,-{\frac {{a}^{2}}{4}},{\frac {1}{4}}+{\frac {{a}^{2}}{4}}+{\frac {{m}^{2}}{4}}-{\frac {{n}^{2}}{4}}-{\frac {n}{4}},-{x}^{2} \right ) {\it \_C2}\,x+{\it HeunC} \left ( 0,-{\frac {1}{2}},m,-{\frac {{a}^{2}}{4}},{\frac {1}{4}}+{\frac {{a}^{2}}{4}}+{\frac {{m}^{2}}{4}}-{\frac {{n}^{2}}{4}}-{\frac {n}{4}},-{x}^{2} \right ) {\it \_C1} \right ) \right \} \]