\[ \text {a0} x+y'(x) (\text {a1} x+\text {b1} y(x)+\text {c1})+y'(x)^2 (\text {a2} x+\text {b2} y(x)+\text {c2})+\text {b0} y(x)+\text {c0}=0 \] ✓ Mathematica : cpu = 8.24177 (sec), leaf count = 576
\[\text {Solve}\left [\left \{x=-\frac {-(K[2] (\text {b2} K[2]+\text {b1})+\text {b0}) \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 {\text {$\#$1}^2 \text {b2} \log (K[2]-\text {$\#$1})+\text {b0} \log (K[2]-\text {$\#$1})+\text {$\#$1} \text {b1} \log (K[2]-\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 (\int _1^{K[2]}\frac {\exp \left (-\text {RootSum}\left [\text {b2} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {b1} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {b0} \text {$\#$1}+\text {a0}\& ,\frac {\text {b2} \log (K[1]-\text {$\#$1}) \text {$\#$1}^2+\text {b1} \log (K[1]-\text {$\#$1}) \text {$\#$1}+\text {b0} \log (K[1]-\text {$\#$1})}{3 \text {b2} \text {$\#$1}^2+2 \text {a2} \text {$\#$1}+2 \text {b1} \text {$\#$1}+\text {a1}+\text {b0}}\& \right ]\right ) (-\text {c0}-K[1] (\text {c1}+\text {c2} K[1]))}{\text {a0}+K[1] (\text {a1}+\text {b0}+K[1] (\text {a2}+\text {b1}+\text {b2} K[1]))}dK[1]+c_1\right )+\text {c1} K[2]+\text {c2} K[2]^2+\text {c0}}{K[2] (K[2] (\text {b2} K[2]+\text {a2}+\text {b1})+\text {a1}+\text {b0})+\text {a0}},y(x)=-\frac {K[2] (K[2] (\text {c2} K[2]+\text {c1})+\text {c0})+(K[2] (\text {a2} K[2]+\text {a1})+\text {a0}) \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 {\text {$\#$1}^2 \text {b2} \log (K[2]-\text {$\#$1})+\text {b0} \log (K[2]-\text {$\#$1})+\text {$\#$1} \text {b1} \log (K[2]-\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 (\int _1^{K[2]}\frac {\exp \left (-\text {RootSum}\left [\text {b2} \text {$\#$1}^3+\text {a2} \text {$\#$1}^2+\text {b1} \text {$\#$1}^2+\text {a1} \text {$\#$1}+\text {b0} \text {$\#$1}+\text {a0}\& ,\frac {\text {b2} \log (K[1]-\text {$\#$1}) \text {$\#$1}^2+\text {b1} \log (K[1]-\text {$\#$1}) \text {$\#$1}+\text {b0} \log (K[1]-\text {$\#$1})}{3 \text {b2} \text {$\#$1}^2+2 \text {a2} \text {$\#$1}+2 \text {b1} \text {$\#$1}+\text {a1}+\text {b0}}\& \right ]\right ) (-\text {c0}-K[1] (\text {c1}+\text {c2} K[1]))}{\text {a0}+K[1] (\text {a1}+\text {b0}+K[1] (\text {a2}+\text {b1}+\text {b2} K[1]))}dK[1]+c_1\right )}{K[2] (K[2] (\text {b2} K[2]+\text {a2}+\text {b1})+\text {a1}+\text {b0})+\text {a0}}\right \},\{y(x),K[2]\}\right ]\] ✓ Maple : cpu = 0.375 (sec), leaf count = 929
\[\left \{x -\left (c_{1}+\int _{}^{\frac {-\mathit {a1} x -\mathit {b1} y \left (x \right )-\mathit {c1} +\sqrt {-4 \mathit {a0} \mathit {a2} \,x^{2}-4 \mathit {a0} \mathit {b2} x y \left (x \right )+\mathit {a1}^{2} x^{2}+2 \mathit {a1} \mathit {b1} x y \left (x \right )-4 \mathit {a2} \mathit {b0} x y \left (x \right )-4 \mathit {b0} \mathit {b2} y \left (x \right )^{2}+\mathit {b1}^{2} y \left (x \right )^{2}-4 \mathit {a0} \mathit {c2} x +2 \mathit {a1} \mathit {c1} x -4 \mathit {a2} \mathit {c0} x -4 \mathit {b0} \mathit {c2} y \left (x \right )+2 \mathit {b1} \mathit {c1} y \left (x \right )-4 \mathit {b2} \mathit {c0} y \left (x \right )-4 \mathit {c0} \mathit {c2} +\mathit {c1}^{2}}}{2 \mathit {a2} x +2 \mathit {b2} y \left (x \right )+2 \mathit {c2}}}-\frac {\left (\textit {\_b}^{2} \mathit {b1} \mathit {c2} -\textit {\_b}^{2} \mathit {b2} \mathit {c1} +2 \textit {\_b} \mathit {b0} \mathit {c2} -2 \textit {\_b} \mathit {b2} \mathit {c0} +\mathit {b0} \mathit {c1} -\mathit {b1} \mathit {c0} \right ) {\mathrm e}^{-\left (\int \frac {\left (\mathit {a1} \mathit {b2} -\mathit {a2} \mathit {b1} \right ) \textit {\_b}^{2}+\mathit {a0} \mathit {b1} -\mathit {a1} \mathit {b0} +\left (2 \mathit {a0} \mathit {b2} -2 \mathit {a2} \mathit {b0} \right ) \textit {\_b}}{\left (\textit {\_b}^{2} \mathit {b2} +\textit {\_b} \mathit {b1} +\mathit {b0} \right ) \left (\textit {\_b}^{3} \mathit {b2} +\left (\mathit {a2} +\mathit {b1} \right ) \textit {\_b}^{2}+\left (\mathit {a1} +\mathit {b0} \right ) \textit {\_b} +\mathit {a0} \right )}d \textit {\_b} \right )}}{\left (\textit {\_b}^{3} \mathit {b2} +\left (\mathit {a2} +\mathit {b1} \right ) \textit {\_b}^{2}+\left (\mathit {a1} +\mathit {b0} \right ) \textit {\_b} +\mathit {a0} \right ) \left (\textit {\_b}^{2} \mathit {b2} +\textit {\_b} \mathit {b1} +\mathit {b0} \right )}d \textit {\_b} \right ) {\mathrm e}^{\int _{}^{\frac {-\mathit {a1} x -\mathit {b1} y \left (x \right )-\mathit {c1} +\sqrt {-4 \mathit {a0} \mathit {a2} \,x^{2}-4 \mathit {a0} \mathit {b2} x y \left (x \right )+\mathit {a1}^{2} x^{2}+2 \mathit {a1} \mathit {b1} x y \left (x \right )-4 \mathit {a2} \mathit {b0} x y \left (x \right )-4 \mathit {b0} \mathit {b2} y \left (x \right )^{2}+\mathit {b1}^{2} y \left (x \right )^{2}-4 \mathit {a0} \mathit {c2} x +2 \mathit {a1} \mathit {c1} x -4 \mathit {a2} \mathit {c0} x -4 \mathit {b0} \mathit {c2} y \left (x \right )+2 \mathit {b1} \mathit {c1} y \left (x \right )-4 \mathit {b2} \mathit {c0} y \left (x \right )-4 \mathit {c0} \mathit {c2} +\mathit {c1}^{2}}}{2 \mathit {a2} x +2 \mathit {b2} y \left (x \right )+2 \mathit {c2}}}\frac {\left (\mathit {a1} \mathit {b2} -\mathit {a2} \mathit {b1} \right ) \textit {\_a}^{2}+\mathit {a0} \mathit {b1} -\mathit {a1} \mathit {b0} +\left (2 \mathit {a0} \mathit {b2} -2 \mathit {a2} \mathit {b0} \right ) \textit {\_a}}{\left (\textit {\_a}^{2} \mathit {b2} +\textit {\_a} \mathit {b1} +\mathit {b0} \right ) \left (\textit {\_a}^{3} \mathit {b2} +\left (\mathit {a2} +\mathit {b1} \right ) \textit {\_a}^{2}+\left (\mathit {a1} +\mathit {b0} \right ) \textit {\_a} +\mathit {a0} \right )}d \textit {\_a}} = 0, x -\left (c_{1}+\int _{}^{\frac {-\mathit {a1} x -\mathit {b1} y \left (x \right )-\mathit {c1} -\sqrt {-4 \mathit {a0} \mathit {a2} \,x^{2}-4 \mathit {a0} \mathit {b2} x y \left (x \right )+\mathit {a1}^{2} x^{2}+2 \mathit {a1} \mathit {b1} x y \left (x \right )-4 \mathit {a2} \mathit {b0} x y \left (x \right )-4 \mathit {b0} \mathit {b2} y \left (x \right )^{2}+\mathit {b1}^{2} y \left (x \right )^{2}-4 \mathit {a0} \mathit {c2} x +2 \mathit {a1} \mathit {c1} x -4 \mathit {a2} \mathit {c0} x -4 \mathit {b0} \mathit {c2} y \left (x \right )+2 \mathit {b1} \mathit {c1} y \left (x \right )-4 \mathit {b2} \mathit {c0} y \left (x \right )-4 \mathit {c0} \mathit {c2} +\mathit {c1}^{2}}}{2 \mathit {a2} x +2 \mathit {b2} y \left (x \right )+2 \mathit {c2}}}-\frac {\left (\textit {\_b}^{2} \mathit {b1} \mathit {c2} -\textit {\_b}^{2} \mathit {b2} \mathit {c1} +2 \textit {\_b} \mathit {b0} \mathit {c2} -2 \textit {\_b} \mathit {b2} \mathit {c0} +\mathit {b0} \mathit {c1} -\mathit {b1} \mathit {c0} \right ) {\mathrm e}^{-\left (\int \frac {\left (\mathit {a1} \mathit {b2} -\mathit {a2} \mathit {b1} \right ) \textit {\_b}^{2}+\mathit {a0} \mathit {b1} -\mathit {a1} \mathit {b0} +\left (2 \mathit {a0} \mathit {b2} -2 \mathit {a2} \mathit {b0} \right ) \textit {\_b}}{\left (\textit {\_b}^{2} \mathit {b2} +\textit {\_b} \mathit {b1} +\mathit {b0} \right ) \left (\textit {\_b}^{3} \mathit {b2} +\left (\mathit {a2} +\mathit {b1} \right ) \textit {\_b}^{2}+\left (\mathit {a1} +\mathit {b0} \right ) \textit {\_b} +\mathit {a0} \right )}d \textit {\_b} \right )}}{\left (\textit {\_b}^{3} \mathit {b2} +\left (\mathit {a2} +\mathit {b1} \right ) \textit {\_b}^{2}+\left (\mathit {a1} +\mathit {b0} \right ) \textit {\_b} +\mathit {a0} \right ) \left (\textit {\_b}^{2} \mathit {b2} +\textit {\_b} \mathit {b1} +\mathit {b0} \right )}d \textit {\_b} \right ) {\mathrm e}^{\int _{}^{\frac {-\mathit {a1} x -\mathit {b1} y \left (x \right )-\mathit {c1} -\sqrt {-4 \mathit {a0} \mathit {a2} \,x^{2}-4 \mathit {a0} \mathit {b2} x y \left (x \right )+\mathit {a1}^{2} x^{2}+2 \mathit {a1} \mathit {b1} x y \left (x \right )-4 \mathit {a2} \mathit {b0} x y \left (x \right )-4 \mathit {b0} \mathit {b2} y \left (x \right )^{2}+\mathit {b1}^{2} y \left (x \right )^{2}-4 \mathit {a0} \mathit {c2} x +2 \mathit {a1} \mathit {c1} x -4 \mathit {a2} \mathit {c0} x -4 \mathit {b0} \mathit {c2} y \left (x \right )+2 \mathit {b1} \mathit {c1} y \left (x \right )-4 \mathit {b2} \mathit {c0} y \left (x \right )-4 \mathit {c0} \mathit {c2} +\mathit {c1}^{2}}}{2 \mathit {a2} x +2 \mathit {b2} y \left (x \right )+2 \mathit {c2}}}\frac {\left (\mathit {a1} \mathit {b2} -\mathit {a2} \mathit {b1} \right ) \textit {\_a}^{2}+\mathit {a0} \mathit {b1} -\mathit {a1} \mathit {b0} +\left (2 \mathit {a0} \mathit {b2} -2 \mathit {a2} \mathit {b0} \right ) \textit {\_a}}{\left (\textit {\_a}^{2} \mathit {b2} +\textit {\_a} \mathit {b1} +\mathit {b0} \right ) \left (\textit {\_a}^{3} \mathit {b2} +\left (\mathit {a2} +\mathit {b1} \right ) \textit {\_a}^{2}+\left (\mathit {a1} +\mathit {b0} \right ) \textit {\_a} +\mathit {a0} \right )}d \textit {\_a}} = 0\right \}\]