\[ \text {a0} x+y'(x) (\text {a1} x+\text {b1} y(x)+\text {c1})+(\text {a2} x+\text {c2}) y'(x)^2+\text {b0} y(x)+\text {c0}=0 \] ✓ Mathematica : cpu = 235.303 (sec), leaf count = 507
DSolve[c0 + a0*x + b0*y[x] + (c1 + a1*x + b1*y[x])*Derivative[1][y][x] + (c2 + a2*x)*Derivative[1][y][x]^2 == 0,y[x],x]
\[\text {Solve}\left [\left \{x=(\text {b1} K[1]+\text {b0}) \exp \left (\frac {(\text {b1} (\text {b0}-\text {a1})+2 \text {a2} \text {b0}) \tan ^{-1}\left (\frac {2 (\text {a2}+\text {b1}) K[1]+\text {a1}+\text {b0}}{\sqrt {4 \text {a0} (\text {a2}+\text {b1})-\text {a1}^2-2 \text {a1} \text {b0}-\text {b0}^2}}\right )}{(\text {a2}+\text {b1}) \sqrt {4 \text {a0} (\text {a2}+\text {b1})-\text {a1}^2-2 \text {a1} \text {b0}-\text {b0}^2}}-\frac {(2 \text {a2}+\text {b1}) \log (K[1] ((\text {a2}+\text {b1}) K[1]+\text {a1}+\text {b0})+\text {a0})}{2 (\text {a2}+\text {b1})}\right ) \left (\int \frac {\left (\frac {-2 \text {c2} K[1]-\text {c1}}{\text {b1} K[1]+\text {b0}}-\frac {\text {b1} \left (-\text {c1} K[1]-\text {c2} K[1]^2-\text {c0}\right )}{(\text {b1} K[1]+\text {b0})^2}\right ) \exp \left (\frac {(2 \text {a2}+\text {b1}) \log (K[1] ((\text {a2}+\text {b1}) K[1]+\text {a1}+\text {b0})+\text {a0})}{2 (\text {a2}+\text {b1})}-\frac {(\text {b1} (\text {b0}-\text {a1})+2 \text {a2} \text {b0}) \tan ^{-1}\left (\frac {2 (\text {a2}+\text {b1}) K[1]+\text {a1}+\text {b0}}{\sqrt {4 \text {a0} (\text {a2}+\text {b1})-\text {a1}^2-2 \text {a1} \text {b0}-\text {b0}^2}}\right )}{(\text {a2}+\text {b1}) \sqrt {4 \text {a0} (\text {a2}+\text {b1})-\text {a1}^2-2 \text {a1} \text {b0}-\text {b0}^2}}\right )}{(\text {b1} K[1]+\text {b0}) \left (K[1]-\frac {-\text {a1} K[1]-\text {a2} K[1]^2-\text {a0}}{\text {b1} K[1]+\text {b0}}\right )} \, dK[1]\right )+c_1 (\text {b1} K[1]+\text {b0}) \exp \left (\frac {(\text {b1} (\text {b0}-\text {a1})+2 \text {a2} \text {b0}) \tan ^{-1}\left (\frac {2 (\text {a2}+\text {b1}) K[1]+\text {a1}+\text {b0}}{\sqrt {4 \text {a0} (\text {a2}+\text {b1})-\text {a1}^2-2 \text {a1} \text {b0}-\text {b0}^2}}\right )}{(\text {a2}+\text {b1}) \sqrt {4 \text {a0} (\text {a2}+\text {b1})-\text {a1}^2-2 \text {a1} \text {b0}-\text {b0}^2}}-\frac {(2 \text {a2}+\text {b1}) \log (K[1] ((\text {a2}+\text {b1}) K[1]+\text {a1}+\text {b0})+\text {a0})}{2 (\text {a2}+\text {b1})}\right ),y(x)=\frac {x \left (-\text {a1} K[1]-\text {a2} K[1]^2-\text {a0}\right )}{\text {b1} K[1]+\text {b0}}+\frac {-\text {c1} K[1]-\text {c2} K[1]^2-\text {c0}}{\text {b1} K[1]+\text {b0}}\right \},\{y(x),K[1]\}\right ]\] ✓ Maple : cpu = 6.387 (sec), leaf count = 1602
dsolve((a2*x+c2)*diff(y(x),x)^2+(a1*x+b1*y(x)+c1)*diff(y(x),x)+a0*x+b0*y(x)+c0 = 0,y(x))
\[\frac {2 \,{\mathrm e}^{\frac {\left (-2 \mathit {a2} \mathit {b0} +\mathit {b1} \left (\mathit {a1} -\mathit {b0} \right )\right ) \arctan \left (\frac {\left (\mathit {a2} +\mathit {b1} \right ) \sqrt {\mathit {b1}^{2} y \left (x \right )^{2}+\left (\left (2 \mathit {a1} \mathit {b1} -4 \mathit {a2} \mathit {b0} \right ) x -4 \mathit {b0} \mathit {c2} +2 \mathit {b1} \mathit {c1} \right ) y \left (x \right )+\left (-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}\right ) x^{2}+\left (-4 \mathit {a0} \mathit {c2} +2 \mathit {a1} \mathit {c1} -4 \mathit {c0} \mathit {a2} \right ) x -4 \mathit {c0} \mathit {c2} +\mathit {c1}^{2}}+\mathit {b1} \left (\mathit {a2} +\mathit {b1} \right ) y \left (x \right )+\left (-\mathit {b0} x +\mathit {c1} \right ) \mathit {a2} +\left (\mathit {a1} x +\mathit {c1} \right ) \mathit {b1} -\mathit {c2} \left (\mathit {a1} +\mathit {b0} \right )}{\sqrt {\left (4 \mathit {a2} +4 \mathit {b1} \right ) \mathit {a0} -\left (\mathit {a1} +\mathit {b0} \right )^{2}}\, \left (\mathit {a2} x +\mathit {c2} \right )}\right )}{\sqrt {\left (4 \mathit {a2} +4 \mathit {b1} \right ) \mathit {a0} -\left (\mathit {a1} +\mathit {b0} \right )^{2}}\, \left (\mathit {a2} +\mathit {b1} \right )}} \left (\mathit {b1} \sqrt {\mathit {b1}^{2} y \left (x \right )^{2}+\left (\left (2 \mathit {a1} \mathit {b1} -4 \mathit {a2} \mathit {b0} \right ) x -4 \mathit {b0} \mathit {c2} +2 \mathit {b1} \mathit {c1} \right ) y \left (x \right )+\left (-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}\right ) x^{2}+\left (-4 \mathit {a0} \mathit {c2} +2 \mathit {a1} \mathit {c1} -4 \mathit {c0} \mathit {a2} \right ) x -4 \mathit {c0} \mathit {c2} +\mathit {c1}^{2}}+\mathit {b1}^{2} y \left (x \right )+\left (\mathit {a1} x +\mathit {c1} \right ) \mathit {b1} -2 \mathit {b0} \left (\mathit {a2} x +\mathit {c2} \right )\right ) \left (c_{1}+\frac {\left (\int _{}^{\frac {-\mathit {a1} x -\mathit {b1} y \left (x \right )-\sqrt {\mathit {b1}^{2} y \left (x \right )^{2}+\left (\left (2 \mathit {a1} \mathit {b1} -4 \mathit {a2} \mathit {b0} \right ) x -4 \mathit {b0} \mathit {c2} +2 \mathit {b1} \mathit {c1} \right ) y \left (x \right )+\left (-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}\right ) x^{2}+\left (-4 \mathit {a0} \mathit {c2} +2 \mathit {a1} \mathit {c1} -4 \mathit {c0} \mathit {a2} \right ) x -4 \mathit {c0} \mathit {c2} +\mathit {c1}^{2}}-\mathit {c1}}{2 \mathit {a2} x +2 \mathit {c2}}}-\frac {\left (\left (\mathit {a2} +\mathit {b1} \right ) \textit {\_a}^{2}+\left (\mathit {a1} +\mathit {b0} \right ) \textit {\_a} +\mathit {a0} \right )^{-\frac {\mathit {b1}}{2 \mathit {a2} +2 \mathit {b1}}} {\mathrm e}^{\frac {\arctan \left (\frac {\left (2 \mathit {a2} +2 \mathit {b1} \right ) \textit {\_a} +\mathit {a1} +\mathit {b0}}{\sqrt {\left (4 \mathit {a2} +4 \mathit {b1} \right ) \mathit {a0} -\left (\mathit {a1} +\mathit {b0} \right )^{2}}}\right ) \left (-2 \mathit {a2} \mathit {b0} +\mathit {b1} \left (\mathit {a1} -\mathit {b0} \right )\right )}{\sqrt {\left (4 \mathit {a2} +4 \mathit {b1} \right ) \mathit {a0} -\left (\mathit {a1} +\mathit {b0} \right )^{2}}\, \left (\mathit {a2} +\mathit {b1} \right )}} \left (\left (2 \textit {\_a} \mathit {c2} +\mathit {c1} \right ) \mathit {b0} +\mathit {b1} \left (\textit {\_a}^{2} \mathit {c2} -\mathit {c0} \right )\right )}{\left (\textit {\_a} \mathit {b1} +\mathit {b0} \right )^{2}}d \textit {\_a} \right )}{2}\right ) \left (\frac {\left (\mathit {b1} \left (\mathit {a2} +\mathit {b1} \right ) y \left (x \right )+\left (-\mathit {b0} x +\mathit {c1} \right ) \mathit {a2} +\left (\mathit {a1} x +\mathit {c1} \right ) \mathit {b1} -\mathit {c2} \left (\mathit {a1} +\mathit {b0} \right )\right ) \sqrt {\mathit {b1}^{2} y \left (x \right )^{2}+\left (\left (2 \mathit {a1} \mathit {b1} -4 \mathit {a2} \mathit {b0} \right ) x -4 \mathit {b0} \mathit {c2} +2 \mathit {b1} \mathit {c1} \right ) y \left (x \right )+\left (-4 \mathit {a0} \mathit {a2} +\mathit {a1}^{2}\right ) x^{2}+\left (-4 \mathit {a0} \mathit {c2} +2 \mathit {a1} \mathit {c1} -4 \mathit {c0} \mathit {a2} \right ) x -4 \mathit {c0} \mathit {c2} +\mathit {c1}^{2}}+\mathit {b1}^{2} \left (\mathit {a2} +\mathit {b1} \right ) y \left (x \right )^{2}+\left (\left (2 \mathit {a1} x +2 \mathit {c1} \right ) \mathit {b1}^{2}+\left (\left (\left (\mathit {a1} -3 \mathit {b0} \right ) x +2 \mathit {c1} \right ) \mathit {a2} -\mathit {c2} \left (\mathit {a1} +3 \mathit {b0} \right )\right ) \mathit {b1} -2 \mathit {a2} \mathit {b0} \left (\mathit {a2} x +\mathit {c2} \right )\right ) y \left (x \right )+\left (\left (-2 \mathit {a0} \,x^{2}-2 \mathit {c0} x \right ) \mathit {a2} +\mathit {a1}^{2} x^{2}+\left (-2 \mathit {a0} \mathit {c2} +2 \mathit {a1} \mathit {c1} \right ) x -2 \mathit {c0} \mathit {c2} +\mathit {c1}^{2}\right ) \mathit {b1} -2 \mathit {a2}^{2} \mathit {c0} x +\left (-\mathit {a1} \mathit {b0} \,x^{2}+\left (2 \mathit {a0} \mathit {c2} +\mathit {c1} \left (\mathit {a1} -\mathit {b0} \right )\right ) x -2 \mathit {c0} \mathit {c2} +\mathit {c1}^{2}\right ) \mathit {a2} -\mathit {c2} \left (\mathit {a1} \left (\mathit {a1} +\mathit {b0} \right ) x -2 \mathit {a0} \mathit {c2} +\mathit {c1} \left (\mathit {a1} +\mathit {b0} \right )\right )}{2 \left (\mathit {a2} x +\mathit {c2} \right )^{2}}\right )^{\frac {-2 \mathit {a2} -\mathit {b1}}{2 \mathit {a2} +2 \mathit {b1}}}+2 \mathit {a2} \,x^{2}+2 x \mathit {c2}}{2 \mathit {a2} x +2 \mathit {c2}} = 0\]