#### 2.231   ODE No. 231

$y'(x) (a y(x)+b x+c)+\alpha y(x)+\beta x+\gamma =0$ Mathematica : cpu = 1.91958 (sec), leaf count = 252

$\text {Solve}\left [\frac {(\alpha -b)^2 \left (-\log \left (\frac {(a y(x)+b x+c)^2 \left (-\frac {(\alpha (b x+c)-a (\beta x+\gamma )) \left (a (\alpha -b) y(x)+a (\beta x+\gamma )+b^2 (-x)-b c\right )}{(a y(x)+b x+c)^2}+a \beta -\alpha b\right )}{(\alpha (b x+c)-a (\beta x+\gamma ))^2}\right )-\frac {2 \tan ^{-1}\left (\frac {\frac {2 a (\beta x+\gamma )-2 \alpha (b x+c)}{a y(x)+b x+c}+\alpha -b}{(\alpha -b) \sqrt {\frac {4 (a \beta -\alpha b)}{(\alpha -b)^2}-1}}\right )}{\sqrt {\frac {4 (a \beta -\alpha b)}{(\alpha -b)^2}-1}}\right )}{2 (a \beta -\alpha b)}=\frac {(\alpha -b)^2 \log (a (\beta x+\gamma )-\alpha (b x+c))}{a \beta -\alpha b}+c_1,y(x)\right ]$ Maple : cpu = 0.194 (sec), leaf count = 178

$\left \{ y \left ( x \right ) ={\frac {1}{-a\beta +b\alpha } \left ( -b\gamma +\beta \,c+{\frac {x \left ( a\beta -b\alpha \right ) +a\gamma -\alpha \,c}{2\,a} \left ( \sqrt {4\,a\beta -{\alpha }^{2}-2\,b\alpha -{b}^{2}}\tan \left ( {\it RootOf} \left ( \sqrt {4\,a\beta -{\alpha }^{2}-2\,b\alpha -{b}^{2}}\ln \left ( {\frac { \left ( a\beta \,x-\alpha \,bx+a\gamma -\alpha \,c \right ) ^{2} \left ( \left ( \tan \left ( {\it \_Z} \right ) \right ) ^{2}+1 \right ) \left ( 4\,a\beta -{\alpha }^{2}-2\,b\alpha -{b}^{2} \right ) }{4\,a}} \right ) +2\,{\it \_C1}\,\sqrt {4\,a\beta -{\alpha }^{2}-2\,b\alpha -{b}^{2}}+2\,{\it \_Z}\,\alpha -2\,{\it \_Z}\,b \right ) \right ) +\alpha +b \right ) } \right ) } \right \}$