\[ y'(x)=F\left (\frac {a x^2}{4}+\frac {b x}{2}+y(x)\right )-\frac {a x}{2} \] ✓ Mathematica : cpu = 16.4478 (sec), leaf count = 510
\[\text {Solve}\left [\int _1^{y(x)} -\frac {2 F\left (K[2]+\frac {a x^2}{4}+\frac {b x}{2}\right ) \int _1^x \left (\frac {2 a K[1] F'\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )+b\right )^2}+\frac {2 F'\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )+b}-\frac {4 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right ) F'\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )+b\right )^2}\right ) \, dK[1]+b \int _1^x \left (\frac {2 a K[1] F'\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )+b\right )^2}+\frac {2 F'\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )+b}-\frac {4 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right ) F'\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )+b\right )^2}\right ) \, dK[1]+2}{2 F\left (K[2]+\frac {a x^2}{4}+\frac {b x}{2}\right )+b} \, dK[2]+\int _1^x \left (\frac {2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+y(x)\right )}{2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+y(x)\right )+b}-\frac {a K[1]}{2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+y(x)\right )+b}\right ) \, dK[1]=c_1,y(x)\right ]\] ✓ Maple : cpu = 0.057 (sec), leaf count = 35
\[ \left \{ y \left ( x \right ) =-{\frac {a{x}^{2}}{4}}-{\frac {bx}{2}}+{\it RootOf} \left ( -x+2\,\int ^{{\it \_Z}}\! \left ( 2\,F \left ( {\it \_a} \right ) +b \right ) ^{-1}{d{\it \_a}}+{\it \_C1} \right ) \right \} \]