\[ y''(x)=\frac {\left (7 a x^2+5\right ) y'(x)}{x \left (a x^2+1\right )}-\frac {\left (15 a x^2+5\right ) y(x)}{x^2 \left (a x^2+1\right )} \] ✓ Mathematica : cpu = 0.0382677 (sec), leaf count = 27
DSolve[Derivative[2][y][x] == -(((5 + 15*a*x^2)*y[x])/(x^2*(1 + a*x^2))) + ((5 + 7*a*x^2)*Derivative[1][y][x])/(x*(1 + a*x^2)),y[x],x]
\[\left \{\left \{y(x)\to c_1 x^5-\frac {1}{4} c_2 x \left (2 a x^2+1\right )\right \}\right \}\] ✓ Maple : cpu = 0.024 (sec), leaf count = 20
dsolve(diff(diff(y(x),x),x) = 1/x*(7*a*x^2+5)/(a*x^2+1)*diff(y(x),x)-(15*a*x^2+5)/x^2/(a*x^2+1)*y(x),y(x))
\[y \left (x \right ) = x^{5} c_{1}+2 a \,x^{3} c_{2}+x c_{2}\]