\[ \boxed { \left ( ay \left ( x \right ) +b \right ) \left ( \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+1 \right ) -c=0} \]
Mathematica: cpu = 0.166521 (sec), leaf count = 141 \[ \left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {c \tan ^{-1}\left (\frac {\sqrt {\text {$\#$1} a+b}}{\sqrt {-\text {$\#$1} a-b+c}}\right )-\sqrt {\text {$\#$1} a+b} \sqrt {-\text {$\#$1} a-b+c}}{a}\& \right ]\left [c_1-x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {c \tan ^{-1}\left (\frac {\sqrt {\text {$\#$1} a+b}}{\sqrt {-\text {$\#$1} a-b+c}}\right )-\sqrt {\text {$\#$1} a+b} \sqrt {-\text {$\#$1} a-b+c}}{a}\& \right ]\left [c_1+x\right ]\right \}\right \} \]
Maple: cpu = 0.609 (sec), leaf count = 88 \[ \left \{ x-\int ^{y \left ( x \right ) }\!{({\it \_a}\,a+b){\frac {1}{ \sqrt {- \left ( {\it \_a}\,a+b \right ) \left ( {\it \_a}\,a+b-c \right ) }}}}{d{\it \_a}}-{\it \_C1}=0,x-\int ^{y \left ( x \right ) }\! -{({\it \_a}\,a+b){\frac {1}{\sqrt {- \left ( {\it \_a}\,a+b \right ) \left ( {\it \_a}\,a+b-c \right ) }}}}{d{\it \_a}}-{\it \_C1}=0,y \left ( x \right ) ={\frac {-b+c}{a}} \right \} \]