\[ y'(x) (a y(x)+b x+c)+\alpha y(x)+\beta x+\gamma =0 \] ✓ Mathematica : cpu = 2.46037 (sec), leaf count = 252
DSolve[EulerGamma + beta*x + alpha*y[x] + (c + b*x + a*y[x])*Derivative[1][y][x] == 0,y[x],x]
\[\text {Solve}\left [\frac {(\alpha -b)^2 \left (-\log \left (\frac {(a y(x)+b x+c)^2 \left (-\frac {(\alpha (b x+c)-a (\beta x+\gamma )) \left (a (\alpha -b) y(x)+a (\beta x+\gamma )+b^2 (-x)-b c\right )}{(a y(x)+b x+c)^2}+a \beta -\alpha b\right )}{(\alpha (b x+c)-a (\beta x+\gamma ))^2}\right )-\frac {2 \tan ^{-1}\left (\frac {\frac {2 a (\beta x+\gamma )-2 \alpha (b x+c)}{a y(x)+b x+c}+\alpha -b}{(\alpha -b) \sqrt {\frac {4 (a \beta -\alpha b)}{(\alpha -b)^2}-1}}\right )}{\sqrt {\frac {4 (a \beta -\alpha b)}{(\alpha -b)^2}-1}}\right )}{2 (a \beta -\alpha b)}=\frac {(\alpha -b)^2 \log (a (\beta x+\gamma )-\alpha (b x+c))}{a \beta -\alpha b}+c_1,y(x)\right ]\] ✓ Maple : cpu = 0.237 (sec), leaf count = 178
dsolve((a*y(x)+b*x+c)*diff(y(x),x)+alpha*y(x)+beta*x+gamma = 0,y(x))
\[y \left (x \right ) = \frac {-b \gamma +\beta c +\frac {\left (x \left (a \beta -b \alpha \right )+a \gamma -\alpha c \right ) \left (\sqrt {4 a \beta -\alpha ^{2}-2 b \alpha -b^{2}}\, \tan \left (\RootOf \left (\sqrt {4 a \beta -\alpha ^{2}-2 b \alpha -b^{2}}\, \ln \left (\frac {\left (a \beta x -\alpha b x +a \gamma -\alpha c \right )^{2} \left (\tan ^{2}\left (\textit {\_Z} \right )+1\right ) \left (4 a \beta -\alpha ^{2}-2 b \alpha -b^{2}\right )}{4 a}\right )+2 c_{1} \sqrt {4 a \beta -\alpha ^{2}-2 b \alpha -b^{2}}+2 \textit {\_Z} \alpha -2 \textit {\_Z} b \right )\right )+\alpha +b \right )}{2 a}}{-a \beta +b \alpha }\]