\[ y'(x)=\frac {a^3 x^3}{8}+\frac {3 a^2 x^4}{16}+\frac {3}{4} a^2 x^2 y(x)+\frac {a^2 x^2}{4}+\frac {3 a x^5}{32}+\frac {3}{4} a x^3 y(x)+\frac {a x^3}{4}+\frac {3}{2} a x y(x)^2+a x y(x)+\frac {x^6}{64}+\frac {3}{16} x^4 y(x)+\frac {x^4}{16}+\frac {3}{4} x^2 y(x)^2+\frac {1}{2} x^2 y(x)+y(x)^3+y(x)^2-\frac {x}{2}+1 \] ✓ Mathematica : cpu = 6.2811 (sec), leaf count = 906
DSolve[Derivative[1][y][x] == 1 - x/2 + (a^2*x^2)/4 + (a*x^3)/4 + (a^3*x^3)/8 + x^4/16 + (3*a^2*x^4)/16 + (3*a*x^5)/32 + x^6/64 + a*x*y[x] + (x^2*y[x])/2 + (3*a^2*x^2*y[x])/4 + (3*a*x^3*y[x])/4 + (3*x^4*y[x])/16 + y[x]^2 + (3*a*x*y[x]^2)/2 + (3*x^2*y[x]^2)/4 + y[x]^3,y[x],x]
\[\text {Solve}\left [\frac {1}{9} \text {RootSum}\left [729 a^2 \text {$\#$1}^9+3132 a \text {$\#$1}^9+3364 \text {$\#$1}^9+2187 a^2 \text {$\#$1}^6+9396 a \text {$\#$1}^6+10092 \text {$\#$1}^6+2187 a^2 \text {$\#$1}^3+9396 a \text {$\#$1}^3+9984 \text {$\#$1}^3+729 a^2+3132 a+3364\& ,\frac {729 a^2 \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}^6+3132 a \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}^6+3364 \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}^6+81\ 2^{2/3} a \sqrt [3]{27 a+58} \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}^4+174\ 2^{2/3} \sqrt [3]{27 a+58} \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}^4+1458 a^2 \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}^3+6264 a \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}^3+6728 \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}^3+18 \sqrt [3]{2} (27 a+58)^{2/3} \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}^2+81\ 2^{2/3} a \sqrt [3]{27 a+58} \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}+174\ 2^{2/3} \sqrt [3]{27 a+58} \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right ) \text {$\#$1}+729 a^2 \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right )+3132 a \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right )+3364 \log \left (\frac {\sqrt [3]{2} \left (\frac {1}{4} \left (3 x^2+6 a x+4\right )+3 y(x)\right )}{\sqrt [3]{27 a+58}}-\text {$\#$1}\right )}{729 a^2 \text {$\#$1}^8+3132 a \text {$\#$1}^8+3364 \text {$\#$1}^8+1458 a^2 \text {$\#$1}^5+6264 a \text {$\#$1}^5+6728 \text {$\#$1}^5+729 a^2 \text {$\#$1}^2+3132 a \text {$\#$1}^2+3328 \text {$\#$1}^2}\& \right ]=\frac {(27 a+58)^{2/3} x}{9\ 2^{2/3}}+c_1,y(x)\right ]\] ✓ Maple : cpu = 0.063 (sec), leaf count = 41
dsolve(diff(y(x),x) = -1/2*x+1+y(x)^2+1/2*x^2*y(x)+a*x*y(x)+1/16*x^4+1/4*a*x^3+1/4*a^2*x^2+y(x)^3+3/4*y(x)^2*x^2+3/2*a*x*y(x)^2+3/16*y(x)*x^4+3/4*y(x)*a*x^3+3/4*a^2*x^2*y(x)+1/64*x^6+3/32*x^5*a+3/16*a^2*x^4+1/8*a^3*x^3,y(x))
\[y \left (x \right ) = -\frac {x^{2}}{4}-\frac {a x}{2}+\operatorname {RootOf}\left (-x +2 \left (\int _{}^{\textit {\_Z}}\frac {1}{2 \textit {\_a}^{3}+2 \textit {\_a}^{2}+a +2}d \textit {\_a} \right )+c_{1}\right )\]