\[ a x y'(x)-b x^2-c+y'(x)^2=0 \] ✓ Mathematica : cpu = 0.445394 (sec), leaf count = 207
DSolve[-c - b*x^2 + a*x*Derivative[1][y][x] + Derivative[1][y][x]^2 == 0,y[x],x]
\[\left \{\left \{y(x)\to \frac {1}{2} \left (-\frac {4 c \arctan \left (\frac {x \sqrt {-a^2-4 b}}{\sqrt {x^2 \left (a^2+4 b\right )+4 c}-2 \sqrt {c}}\right )}{\sqrt {-a^2-4 b}}-\frac {1}{2} x \left (\sqrt {x^2 \left (a^2+4 b\right )+4 c}+a x\right )\right )+c_1\right \},\left \{y(x)\to \frac {1}{2} \left (\frac {4 c \arctan \left (\frac {x \sqrt {-a^2-4 b}}{\sqrt {x^2 \left (a^2+4 b\right )+4 c}-2 \sqrt {c}}\right )}{\sqrt {-a^2-4 b}}+\frac {1}{2} x \sqrt {x^2 \left (a^2+4 b\right )+4 c}-\frac {a x^2}{2}\right )+c_1\right \}\right \}\] ✓ Maple : cpu = 0.02 (sec), leaf count = 146
dsolve(diff(y(x),x)^2+a*x*diff(y(x),x)-b*x^2-c = 0,y(x))
\[y \left (x \right ) = -\frac {x \sqrt {\left (a^{2}+4 b \right ) x^{2}+4 c}}{4}-\frac {c \ln \left (\sqrt {a^{2}+4 b}\, x +\sqrt {\left (a^{2}+4 b \right ) x^{2}+4 c}\right )}{\sqrt {a^{2}+4 b}}-\frac {a \,x^{2}}{4}+c_{1}\]