\[ y''(x)=-\frac {b y(x)}{x^2 \left (a+x^2\right )}-\frac {\left (a+2 x^2\right ) y'(x)}{x \left (a+x^2\right )} \] ✓ Mathematica : cpu = 0.0294589 (sec), leaf count = 69
DSolve[Derivative[2][y][x] == -((b*y[x])/(x^2*(a + x^2))) - ((a + 2*x^2)*Derivative[1][y][x])/(x*(a + x^2)),y[x],x]
\[\left \{\left \{y(x)\to c_1 \cos \left (\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {a+x^2}}{\sqrt {a}}\right )}{\sqrt {a}}\right )-c_2 \sin \left (\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {a+x^2}}{\sqrt {a}}\right )}{\sqrt {a}}\right )\right \}\right \}\] ✓ Maple : cpu = 0.05 (sec), leaf count = 73
dsolve(diff(diff(y(x),x),x) = -1/x*(2*x^2+a)/(x^2+a)*diff(y(x),x)-b/x^2/(x^2+a)*y(x),y(x))
\[y \left (x \right ) = \left (c_{2} \left (\frac {2 a +2 \sqrt {a}\, \sqrt {x^{2}+a}}{x}\right )^{\frac {2 i \sqrt {b}}{\sqrt {a}}}+c_{1}\right ) \left (\frac {2 a +2 \sqrt {a}\, \sqrt {x^{2}+a}}{x}\right )^{-\frac {i \sqrt {b}}{\sqrt {a}}}\]