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))]