\[ \left (y'(x)^2+1\right ) (a y(x)+b)-c=0 \] ✓ Mathematica : cpu = 0.30966 (sec), leaf count = 223
DSolve[-c + (b + a*y[x])*(1 + Derivative[1][y][x]^2) == 0,y[x],x]
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {\frac {c \sqrt {-a c} \sqrt {\frac {\text {$\#$1} a+b}{c}} \sin ^{-1}\left (\frac {a \sqrt {-\text {$\#$1} a-b+c}}{\sqrt {-a} \sqrt {-a c}}\right )}{\sqrt {-a}}-(\text {$\#$1} a+b) \sqrt {-\text {$\#$1} a-b+c}}{a \sqrt {\text {$\#$1} a+b}}\& \right ][-x+c_1]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\frac {c \sqrt {-a c} \sqrt {\frac {\text {$\#$1} a+b}{c}} \sin ^{-1}\left (\frac {a \sqrt {-\text {$\#$1} a-b+c}}{\sqrt {-a} \sqrt {-a c}}\right )}{\sqrt {-a}}-(\text {$\#$1} a+b) \sqrt {-\text {$\#$1} a-b+c}}{a \sqrt {\text {$\#$1} a+b}}\& \right ][x+c_1]\right \}\right \}\] ✓ Maple : cpu = 0.108 (sec), leaf count = 88
dsolve((a*y(x)+b)*(diff(y(x),x)^2+1)-c = 0,y(x))
\[y \left (x \right ) = \frac {-b +c}{a}\]