\[ y''(x)=-\frac {A y(x)}{\left (a x^2+b x+c\right )^2} \] ✓ Mathematica : cpu = 0.864956 (sec), leaf count = 211
\[\left \{\left \{y(x)\to \frac {c_2 \sqrt {a x^2+b x+c} \exp \left (-\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}+c_1 \sqrt {x (a x+b)+c} \exp \left (\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )\right \}\right \}\] ✓ Maple : cpu = 0.183 (sec), leaf count = 178
\[y \relax (x ) = \sqrt {a \,x^{2}+b x +c}\, \left (\left (\frac {i \sqrt {4 c a -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 c a -b^{2}}}\right )^{\frac {a \sqrt {\frac {-4 c a +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 c a +b^{2}}}} c_{1}+\left (\frac {i \sqrt {4 c a -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 c a -b^{2}}}\right )^{-\frac {a \sqrt {\frac {-4 c a +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 c a +b^{2}}}} c_{2}\right )\]