\[ y''(x)=-\frac {A y(x)}{\left (a x^2+b x+c\right )^2} \] ✓ Mathematica : cpu = 0.560307 (sec), leaf count = 211
DSolve[Derivative[2][y][x] == -((A*y[x])/(c + b*x + a*x^2)^2),y[x],x]
\[\left \{\left \{y(x)\to \frac {c_2 \sqrt {a x^2+b x+c} \exp \left (-\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}+c_1 \sqrt {x (a x+b)+c} \exp \left (\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \arctan \left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )\right \}\right \}\] ✓ Maple : cpu = 0.152 (sec), leaf count = 178
dsolve(diff(diff(y(x),x),x) = -A/(a*x^2+b*x+c)^2*y(x),y(x))
\[y \left (x \right ) = \sqrt {a \,x^{2}+b x +c}\, \left ({\left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )}^{\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} c_{1}+{\left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )}^{-\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} c_{2}\right )\]