##### 4.9.45 $$(y(x)+x+1) y'(x)+3 y(x)+4 x+1=0$$

ODE
$(y(x)+x+1) y'(x)+3 y(x)+4 x+1=0$ ODE Classiﬁcation

[[_homogeneous, class C], _rational, [_Abel, 2nd type, class A]]

Book solution method
Equation linear in the variables, $$y'(x)=f\left ( \frac {X_1}{X_2} \right )$$

Mathematica
cpu = 2.19809 (sec), leaf count = 159

$\text {Solve}\left [\frac {(-2)^{2/3} \left (-2 x \log \left (\frac {3 (-2)^{2/3} (y(x)+2 x-1)}{y(x)+x+1}\right )+(2 x-1) \log \left (-\frac {3 (-2)^{2/3} (x-2)}{y(x)+x+1}\right )+\log \left (\frac {3 (-2)^{2/3} (y(x)+2 x-1)}{y(x)+x+1}\right )+y(x) \left (\log \left (-\frac {3 (-2)^{2/3} (x-2)}{y(x)+x+1}\right )-\log \left (\frac {3 (-2)^{2/3} (y(x)+2 x-1)}{y(x)+x+1}\right )+1\right )+x+1\right )}{9 (y(x)+2 x-1)}=\frac {1}{9} (-2)^{2/3} \log (x-2)+c_1,y(x)\right ]$

Maple
cpu = 0.125 (sec), leaf count = 29

$\left [y \left (x \right ) = -3-\frac {\left (x -2\right ) \left (2 \LambertW \left (\textit {\_C1} \left (x -2\right )\right )+1\right )}{\LambertW \left (\textit {\_C1} \left (x -2\right )\right )}\right ]$ Mathematica raw input

DSolve[1 + 4*x + 3*y[x] + (1 + x + y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[((-2)^(2/3)*(1 + x + (-1 + 2*x)*Log[(-3*(-2)^(2/3)*(-2 + x))/(1 + x + y[x]
)] + Log[(3*(-2)^(2/3)*(-1 + 2*x + y[x]))/(1 + x + y[x])] - 2*x*Log[(3*(-2)^(2/3
)*(-1 + 2*x + y[x]))/(1 + x + y[x])] + (1 + Log[(-3*(-2)^(2/3)*(-2 + x))/(1 + x
+ y[x])] - Log[(3*(-2)^(2/3)*(-1 + 2*x + y[x]))/(1 + x + y[x])])*y[x]))/(9*(-1 +
 2*x + y[x])) == C[1] + ((-2)^(2/3)*Log[-2 + x])/9, y[x]]

Maple raw input

dsolve((1+x+y(x))*diff(y(x),x)+1+4*x+3*y(x) = 0, y(x))

Maple raw output

[y(x) = -3-(x-2)*(2*LambertW(_C1*(x-2))+1)/LambertW(_C1*(x-2))]