\[ y'(x)=F\left (\frac {a x^2}{4}+\frac {b x}{2}+y(x)\right )-\frac {a x}{2} \] ✓ Mathematica : cpu = 0.284012 (sec), leaf count = 514
DSolve[Derivative[1][y][x] == -1/2*(a*x) + F[(b*x)/2 + (a*x^2)/4 + y[x]],y[x],x]
\[\text {Solve}\left [\int _1^{y(x)}-\frac {b \int _1^x\left (\frac {2 a K[1] F'\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )\right )^2}+\frac {2 F'\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}-\frac {4 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right ) F'\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )\right )^2}\right )dK[1]+2 F\left (\frac {a x^2}{4}+\frac {b x}{2}+K[2]\right ) \int _1^x\left (\frac {2 a K[1] F'\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )\right )^2}+\frac {2 F'\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}-\frac {4 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right ) F'\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )}{\left (b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+K[2]\right )\right )^2}\right )dK[1]+2}{b+2 F\left (\frac {a x^2}{4}+\frac {b x}{2}+K[2]\right )}dK[2]+\int _1^x\left (\frac {2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+y(x)\right )}{b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+y(x)\right )}-\frac {a K[1]}{b+2 F\left (\frac {1}{4} a K[1]^2+\frac {1}{2} b K[1]+y(x)\right )}\right )dK[1]=c_1,y(x)\right ]\] ✓ Maple : cpu = 0.058 (sec), leaf count = 35
dsolve(diff(y(x),x) = -1/2*a*x+F(y(x)+1/4*a*x^2+1/2*b*x),y(x))
\[y \left (x \right ) = -\frac {a \,x^{2}}{4}-\frac {b x}{2}+\RootOf \left (-x +2 \left (\int _{}^{\textit {\_Z}}\frac {1}{2 F \left (\textit {\_a} \right )+b}d \textit {\_a} \right )+c_{1}\right )\]