\[ y''(x)=-\frac {a x y'(x)}{x^2+1}-\frac {b y(x)}{\left (x^2+1\right )^2} \] ✓ Mathematica : cpu = 0.0219905 (sec), leaf count = 106
\[\left \{\left \{y(x)\to c_1 \left (x^2+1\right )^{\frac {2-a}{4}} P_{\frac {a-2}{2}}^{\frac {1}{2} \sqrt {a^2-4 a+4 b+4}}(i x)+c_2 \left (x^2+1\right )^{\frac {2-a}{4}} Q_{\frac {a-2}{2}}^{\frac {1}{2} \sqrt {a^2-4 a+4 b+4}}(i x)\right \}\right \}\] ✓ Maple : cpu = 0.334 (sec), leaf count = 71
\[ \left \{ y \left ( x \right ) = \left ( {x}^{2}+1 \right ) ^{{\frac {1}{2}}-{\frac {a}{4}}} \left ( {\it LegendreQ} \left ( {\frac {a}{2}}-1,{\frac {1}{2}\sqrt {{a}^{2}-4\,a+4\,b+4}},ix \right ) {\it \_C2}+{\it LegendreP} \left ( {\frac {a}{2}}-1,{\frac {1}{2}\sqrt {{a}^{2}-4\,a+4\,b+4}},ix \right ) {\it \_C1} \right ) \right \} \]