\[ \text {a0} x+y'(x) (\text {a1} x+\text {b1} y(x)+\text {c1})+y'(x)^2 (\text {a2} x+\text {b2} y(x)+\text {c2})+\text {b0} y(x)+\text {c0}=0 \] ✓ Mathematica : cpu = 1037.13 (sec), leaf count = 508
\[\text {Solve}\left [\left \{x=c_1 \left (\text {b0}+\text {b1} \text {K$\$$12542503}+\text {b2} \text {K$\$$12542503}^2\right ) \exp \left (-\text {RootSum}\left [\text {$\#$1}^3 \text {b2}+\text {$\#$1}^2 \text {a2}+\text {$\#$1}^2 \text {b1}+\text {$\#$1} \text {a1}+\text {$\#$1} \text {b0}+\text {a0}\& ,\frac {2 \text {$\#$1}^2 \text {b2} \log (\text {K$\$$12542503}-\text {$\#$1})+\text {a1} \log (\text {K$\$$12542503}-\text {$\#$1})+2 \text {$\#$1} \text {a2} \log (\text {K$\$$12542503}-\text {$\#$1})+\text {$\#$1} \text {b1} \log (\text {K$\$$12542503}-\text {$\#$1})}{3 \text {$\#$1}^2 \text {b2}+2 \text {$\#$1} \text {a2}+2 \text {$\#$1} \text {b1}+\text {a1}+\text {b0}}\& \right ]\right )+\left (\text {b0}+\text {b1} \text {K$\$$12542503}+\text {b2} \text {K$\$$12542503}^2\right ) \exp \left (-\text {RootSum}\left [\text {$\#$1}^3 \text {b2}+\text {$\#$1}^2 \text {a2}+\text {$\#$1}^2 \text {b1}+\text {$\#$1} \text {a1}+\text {$\#$1} \text {b0}+\text {a0}\& ,\frac {2 \text {$\#$1}^2 \text {b2} \log (\text {K$\$$12542503}-\text {$\#$1})+\text {a1} \log (\text {K$\$$12542503}-\text {$\#$1})+2 \text {$\#$1} \text {a2} \log (\text {K$\$$12542503}-\text {$\#$1})+\text {$\#$1} \text {b1} \log (\text {K$\$$12542503}-\text {$\#$1})}{3 \text {$\#$1}^2 \text {b2}+2 \text {$\#$1} \text {a2}+2 \text {$\#$1} \text {b1}+\text {a1}+\text {b0}}\& \right ]\right ) \int \frac {\left (\frac {-\text {c1}-2 \text {c2} \text {K$\$$12542503}}{\text {b0}+\text {b1} \text {K$\$$12542503}+\text {b2} \text {K$\$$12542503}^2}-\frac {(\text {b1}+2 \text {b2} \text {K$\$$12542503}) \left (-\text {c0}-\text {c1} \text {K$\$$12542503}-\text {c2} \text {K$\$$12542503}^2\right )}{\left (\text {b0}+\text {b1} \text {K$\$$12542503}+\text {b2} \text {K$\$$12542503}^2\right )^2}\right ) \exp \left (\text {RootSum}\left [\text {$\#$1}^3 \text {b2}+\text {$\#$1}^2 \text {a2}+\text {$\#$1}^2 \text {b1}+\text {$\#$1} \text {a1}+\text {$\#$1} \text {b0}+\text {a0}\& ,\frac {2 \text {$\#$1}^2 \text {b2} \log (\text {K$\$$12542503}-\text {$\#$1})+\text {a1} \log (\text {K$\$$12542503}-\text {$\#$1})+2 \text {$\#$1} \text {a2} \log (\text {K$\$$12542503}-\text {$\#$1})+\text {$\#$1} \text {b1} \log (\text {K$\$$12542503}-\text {$\#$1})}{3 \text {$\#$1}^2 \text {b2}+2 \text {$\#$1} \text {a2}+2 \text {$\#$1} \text {b1}+\text {a1}+\text {b0}}\& \right ]\right )}{\left (\text {b0}+\text {b1} \text {K$\$$12542503}+\text {b2} \text {K$\$$12542503}^2\right ) \left (\text {K$\$$12542503}-\frac {-\text {a0}-\text {a1} \text {K$\$$12542503}-\text {a2} \text {K$\$$12542503}^2}{\text {b0}+\text {b1} \text {K$\$$12542503}+\text {b2} \text {K$\$$12542503}^2}\right )} \, d\text {K$\$$12542503},y(x)=\frac {x \left (-\text {a0}-\text {a1} \text {K$\$$12542503}-\text {a2} \text {K$\$$12542503}^2\right )}{\text {b0}+\text {b1} \text {K$\$$12542503}+\text {b2} \text {K$\$$12542503}^2}+\frac {-\text {c0}-\text {c1} \text {K$\$$12542503}-\text {c2} \text {K$\$$12542503}^2}{\text {b0}+\text {b1} \text {K$\$$12542503}+\text {b2} \text {K$\$$12542503}^2}\right \},\{y(x),\text {K$\$$12542503}\}\right ]\] ✓ Maple : cpu = 0.623 (sec), leaf count = 929
\[ \left \{ x-{{\rm e}^{\int ^{{\frac {1}{2\,{\it b2}\,y \left ( x \right ) +2\,{\it a2}\,x+2\,{\it c2}} \left ( -{\it a1}\,x-{\it b1}\,y \left ( x \right ) -{\it c1}+\sqrt {-4\,{\it a0}\,{\it a2}\,{x}^{2}-4\,{\it a0}\,{\it b2}\,xy \left ( x \right ) +{{\it a1}}^{2}{x}^{2}+2\,{\it a1}\,{\it b1}\,xy \left ( x \right ) -4\,{\it a2}\,{\it b0}\,xy \left ( x \right ) -4\,{\it b0}\,{\it b2}\, \left ( y \left ( x \right ) \right ) ^{2}+{{\it b1}}^{2} \left ( y \left ( x \right ) \right ) ^{2}-4\,{\it a0}\,{\it c2}\,x+2\,{\it a1}\,{\it c1}\,x-4\,{\it a2}\,{\it c0}\,x-4\,{\it b0}\,{\it c2}\,y \left ( x \right ) +2\,{\it b1}\,{\it c1}\,y \left ( x \right ) -4\,{\it b2}\,{\it c0}\,y \left ( x \right ) -4\,{\it c0}\,{\it c2}+{{\it c1}}^{2}} \right ) }}\!{\frac { \left ( {\it a1}\,{\it b2}-{\it a2}\,{\it b1} \right ) {{\it \_a}}^{2}+ \left ( 2\,{\it a0}\,{\it b2}-2\,{\it a2}\,{\it b0} \right ) {\it \_a}+{\it a0}\,{\it b1}-{\it a1}\,{\it b0}}{ \left ( {{\it \_a}}^{2}{\it b2}+{\it \_a}\,{\it b1}+{\it b0} \right ) \left ( {{\it \_a}}^{3}{\it b2}+ \left ( {\it a2}+{\it b1} \right ) {{\it \_a}}^{2}+ \left ( {\it a1}+{\it b0} \right ) {\it \_a}+{\it a0} \right ) }}{d{\it \_a}}}} \left ( \int ^{{\frac {1}{2\,{\it b2}\,y \left ( x \right ) +2\,{\it a2}\,x+2\,{\it c2}} \left ( -{\it a1}\,x-{\it b1}\,y \left ( x \right ) -{\it c1}+\sqrt {-4\,{\it a0}\,{\it a2}\,{x}^{2}-4\,{\it a0}\,{\it b2}\,xy \left ( x \right ) +{{\it a1}}^{2}{x}^{2}+2\,{\it a1}\,{\it b1}\,xy \left ( x \right ) -4\,{\it a2}\,{\it b0}\,xy \left ( x \right ) -4\,{\it b0}\,{\it b2}\, \left ( y \left ( x \right ) \right ) ^{2}+{{\it b1}}^{2} \left ( y \left ( x \right ) \right ) ^{2}-4\,{\it a0}\,{\it c2}\,x+2\,{\it a1}\,{\it c1}\,x-4\,{\it a2}\,{\it c0}\,x-4\,{\it b0}\,{\it c2}\,y \left ( x \right ) +2\,{\it b1}\,{\it c1}\,y \left ( x \right ) -4\,{\it b2}\,{\it c0}\,y \left ( x \right ) -4\,{\it c0}\,{\it c2}+{{\it c1}}^{2}} \right ) }}\!-{\frac {{{\it \_b}}^{2}{\it b1}\,{\it c2}-{{\it \_b}}^{2}{\it b2}\,{\it c1}+2\,{\it \_b}\,{\it b0}\,{\it c2}-2\,{\it \_b}\,{\it b2}\,{\it c0}+{\it b0}\,{\it c1}-{\it b1}\,{\it c0}}{ \left ( {{\it \_b}}^{2}{\it b2}+{\it \_b}\,{\it b1}+{\it b0} \right ) \left ( {{\it \_b}}^{3}{\it b2}+ \left ( {\it a2}+{\it b1} \right ) {{\it \_b}}^{2}+ \left ( {\it a1}+{\it b0} \right ) {\it \_b}+{\it a0} \right ) }{{\rm e}^{-\int \!{\frac { \left ( {\it a1}\,{\it b2}-{\it a2}\,{\it b1} \right ) {{\it \_b}}^{2}+ \left ( 2\,{\it a0}\,{\it b2}-2\,{\it a2}\,{\it b0} \right ) {\it \_b}+{\it a0}\,{\it b1}-{\it a1}\,{\it b0}}{ \left ( {{\it \_b}}^{2}{\it b2}+{\it \_b}\,{\it b1}+{\it b0} \right ) \left ( {{\it \_b}}^{3}{\it b2}+ \left ( {\it a2}+{\it b1} \right ) {{\it \_b}}^{2}+ \left ( {\it a1}+{\it b0} \right ) {\it \_b}+{\it a0} \right ) }}\,{\rm d}{\it \_b}}}}{d{\it \_b}}+{\it \_C1} \right ) =0,x-{{\rm e}^{\int ^{{\frac {1}{2\,{\it b2}\,y \left ( x \right ) +2\,{\it a2}\,x+2\,{\it c2}} \left ( -{\it a1}\,x-{\it b1}\,y \left ( x \right ) -\sqrt {-4\,{\it a0}\,{\it a2}\,{x}^{2}-4\,{\it a0}\,{\it b2}\,xy \left ( x \right ) +{{\it a1}}^{2}{x}^{2}+2\,{\it a1}\,{\it b1}\,xy \left ( x \right ) -4\,{\it a2}\,{\it b0}\,xy \left ( x \right ) -4\,{\it b0}\,{\it b2}\, \left ( y \left ( x \right ) \right ) ^{2}+{{\it b1}}^{2} \left ( y \left ( x \right ) \right ) ^{2}-4\,{\it a0}\,{\it c2}\,x+2\,{\it a1}\,{\it c1}\,x-4\,{\it a2}\,{\it c0}\,x-4\,{\it b0}\,{\it c2}\,y \left ( x \right ) +2\,{\it b1}\,{\it c1}\,y \left ( x \right ) -4\,{\it b2}\,{\it c0}\,y \left ( x \right ) -4\,{\it c0}\,{\it c2}+{{\it c1}}^{2}}-{\it c1} \right ) }}\!{\frac { \left ( {\it a1}\,{\it b2}-{\it a2}\,{\it b1} \right ) {{\it \_a}}^{2}+ \left ( 2\,{\it a0}\,{\it b2}-2\,{\it a2}\,{\it b0} \right ) {\it \_a}+{\it a0}\,{\it b1}-{\it a1}\,{\it b0}}{ \left ( {{\it \_a}}^{2}{\it b2}+{\it \_a}\,{\it b1}+{\it b0} \right ) \left ( {{\it \_a}}^{3}{\it b2}+ \left ( {\it a2}+{\it b1} \right ) {{\it \_a}}^{2}+ \left ( {\it a1}+{\it b0} \right ) {\it \_a}+{\it a0} \right ) }}{d{\it \_a}}}} \left ( \int ^{{\frac {1}{2\,{\it b2}\,y \left ( x \right ) +2\,{\it a2}\,x+2\,{\it c2}} \left ( -{\it a1}\,x-{\it b1}\,y \left ( x \right ) -\sqrt {-4\,{\it a0}\,{\it a2}\,{x}^{2}-4\,{\it a0}\,{\it b2}\,xy \left ( x \right ) +{{\it a1}}^{2}{x}^{2}+2\,{\it a1}\,{\it b1}\,xy \left ( x \right ) -4\,{\it a2}\,{\it b0}\,xy \left ( x \right ) -4\,{\it b0}\,{\it b2}\, \left ( y \left ( x \right ) \right ) ^{2}+{{\it b1}}^{2} \left ( y \left ( x \right ) \right ) ^{2}-4\,{\it a0}\,{\it c2}\,x+2\,{\it a1}\,{\it c1}\,x-4\,{\it a2}\,{\it c0}\,x-4\,{\it b0}\,{\it c2}\,y \left ( x \right ) +2\,{\it b1}\,{\it c1}\,y \left ( x \right ) -4\,{\it b2}\,{\it c0}\,y \left ( x \right ) -4\,{\it c0}\,{\it c2}+{{\it c1}}^{2}}-{\it c1} \right ) }}\!-{\frac {{{\it \_b}}^{2}{\it b1}\,{\it c2}-{{\it \_b}}^{2}{\it b2}\,{\it c1}+2\,{\it \_b}\,{\it b0}\,{\it c2}-2\,{\it \_b}\,{\it b2}\,{\it c0}+{\it b0}\,{\it c1}-{\it b1}\,{\it c0}}{ \left ( {{\it \_b}}^{2}{\it b2}+{\it \_b}\,{\it b1}+{\it b0} \right ) \left ( {{\it \_b}}^{3}{\it b2}+ \left ( {\it a2}+{\it b1} \right ) {{\it \_b}}^{2}+ \left ( {\it a1}+{\it b0} \right ) {\it \_b}+{\it a0} \right ) }{{\rm e}^{-\int \!{\frac { \left ( {\it a1}\,{\it b2}-{\it a2}\,{\it b1} \right ) {{\it \_b}}^{2}+ \left ( 2\,{\it a0}\,{\it b2}-2\,{\it a2}\,{\it b0} \right ) {\it \_b}+{\it a0}\,{\it b1}-{\it a1}\,{\it b0}}{ \left ( {{\it \_b}}^{2}{\it b2}+{\it \_b}\,{\it b1}+{\it b0} \right ) \left ( {{\it \_b}}^{3}{\it b2}+ \left ( {\it a2}+{\it b1} \right ) {{\it \_b}}^{2}+ \left ( {\it a1}+{\it b0} \right ) {\it \_b}+{\it a0} \right ) }}\,{\rm d}{\it \_b}}}}{d{\it \_b}}+{\it \_C1} \right ) =0 \right \} \]