\[ y''(x)=-\frac {\left (a x^2+a-3\right ) y(x)}{4 \left (x^2+1\right )^2} \] ✓ Mathematica : cpu = 0.0129023 (sec), leaf count = 78
DSolve[Derivative[2][y][x] == -1/4*((-3 + a + a*x^2)*y[x])/(1 + x^2)^2,y[x],x]
\[\left \{\left \{y(x)\to c_1 \sqrt {x^2+1} P_{\frac {1}{2} \left (\sqrt {1-a}-1\right )}^{\frac {1}{2}}(i x)+c_2 \sqrt {x^2+1} Q_{\frac {1}{2} \left (\sqrt {1-a}-1\right )}^{\frac {1}{2}}(i x)\right \}\right \}\] ✓ Maple : cpu = 0.058 (sec), leaf count = 55
dsolve(diff(diff(y(x),x),x) = -1/4*(a*x^2+a-3)/(x^2+1)^2*y(x),y(x))
\[y \left (x \right ) = \left (x^{2}+1\right )^{\frac {1}{4}} \left (\left (x +\sqrt {x^{2}+1}\right )^{-\frac {\sqrt {1-a}}{2}} c_{2}+\left (x +\sqrt {x^{2}+1}\right )^{\frac {\sqrt {1-a}}{2}} c_{1}\right )\]