\[ \boxed { \left ( ay \left ( x \right ) +bx+c \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) +\alpha \,y \left ( x \right ) +\beta \,x+\gamma =0} \]
Mathematica: cpu = 2.446311 (sec), leaf count = 252 \[ \text {Solve}\left [\frac {(\alpha -b)^2 \left (-\log \left (\frac {(a y(x)+b x+c)^2 \left (-\frac {(\alpha (b x+c)-a (\beta x+\gamma )) \left (a (\alpha -b) y(x)+a (\beta x+\gamma )+b^2 (-x)-b c\right )}{(a y(x)+b x+c)^2}+a \beta -\alpha b\right )}{(\alpha (b x+c)-a (\beta x+\gamma ))^2}\right )-\frac {2 \tan ^{-1}\left (\frac {\frac {2 (a (\beta x+\gamma )-\alpha (b x+c))}{a y(x)+b x+c}+\alpha -b}{(\alpha -b) \sqrt {\frac {4 (a \beta -\alpha b)}{(\alpha -b)^2}-1}}\right )}{\sqrt {\frac {4 (a \beta -\alpha b)}{(\alpha -b)^2}-1}}\right )}{2 (a \beta -\alpha b)}=\frac {(\alpha -b)^2 \log (a (\beta x+\gamma )-\alpha (b x+c))}{a \beta -\alpha b}+c_1,y(x)\right ] \]
Maple: cpu = 0.156 (sec), leaf count = 206 \[ \left \{ y \left ( x \right ) ={\frac {1}{-a\beta +b\alpha } \left ( -b \gamma +\beta \,c+{\frac {x \left ( a\beta -b\alpha \right ) +a\gamma - \alpha \,c}{2\,a} \left ( \sqrt {4\,a\beta -{\alpha }^{2}-2\,b\alpha -{b}^{ 2}}\tan \left ( {\it RootOf} \left ( \sqrt {4\,a\beta -{\alpha }^{2}-2\,b \alpha -{b}^{2}}\ln \left ( {\frac { \left ( a\beta \,x-\alpha \,bx+a \gamma -\alpha \,c \right ) ^{2} \left ( 4\,a\beta \, \left ( \tan \left ( { \it \_Z} \right ) \right ) ^{2}-{\alpha }^{2} \left ( \tan \left ( {\it \_Z} \right ) \right ) ^{2}-2\,\alpha \,b \left ( \tan \left ( {\it \_Z} \right ) \right ) ^{2}-{b}^{2} \left ( \tan \left ( {\it \_Z} \right ) \right ) ^{2}+4\,a\beta -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{4\,a }} \right ) +2\,{\it \_C1}\,\sqrt {4\,a\beta -{\alpha }^{2}-2\,b\alpha -{b }^{2}}+2\,{\it \_Z}\,\alpha -2\,{\it \_Z}\,b \right ) \right ) +\alpha +b \right ) } \right ) } \right \} \]