\[ \left (y'(x)+y(x)^2\right ) \left (a x^2+b x+c\right )^2+A=0 \] ✓ Mathematica : cpu = 1.13879 (sec), leaf count = 704
DSolve[A + (c + b*x + a*x^2)^2*(y[x]^2 + Derivative[1][y][x]) == 0,y[x],x]
\[\left \{\left \{y(x)\to -\frac {-\frac {2 a \sqrt {a x^2+b x+c} \exp \left (-\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \left (\frac {(2 a x+b)^2}{4 a c-b^2}+1\right )}+\frac {(2 a x+b) \exp \left (-\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}} \sqrt {a x^2+b x+c}}+c_1 \left (\frac {2 a \sqrt {1-\frac {4 A}{b^2-4 a c}} \sqrt {x (a x+b)+c} \exp \left (\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (\frac {(2 a x+b)^2}{4 a c-b^2}+1\right )}+\frac {(2 a x+b) \exp \left (\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {x (a x+b)+c}}\right )}{-\frac {\sqrt {a x^2+b x+c} \exp \left (-\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}+c_1 \sqrt {x (a x+b)+c} \left (-\exp \left (\frac {\sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \tan ^{-1}\left (\frac {2 a x+b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )\right )}\right \}\right \}\] ✓ Maple : cpu = 0.396 (sec), leaf count = 493
dsolve((a*x^2+b*x+c)^2*(diff(y(x),x)+y(x)^2)+A = 0,y(x))
\[y \left (x \right ) = \frac {2 \left (\left (i \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}\, a \sqrt {4 a c -b^{2}}-2 \left (a x +\frac {b}{2}\right ) \sqrt {-4 a c +b^{2}}\right ) c_{1} \left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )^{-\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}-\left (i \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}\, a \sqrt {4 a c -b^{2}}+2 \left (a x +\frac {b}{2}\right ) \sqrt {-4 a c +b^{2}}\right ) \left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )^{\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right ) a}{\sqrt {-4 a c +b^{2}}\, \left (2 a x +b +i \sqrt {4 a c -b^{2}}\right ) \left (i \sqrt {4 a c -b^{2}}-2 a x -b \right ) \left (c_{1} \left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )^{-\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}+\left (\frac {i \sqrt {4 a c -b^{2}}-2 a x -b}{2 a x +b +i \sqrt {4 a c -b^{2}}}\right )^{\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right )}\]