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 = 1.66141 (sec), leaf count = 159

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

Maple
cpu = 0.025 (sec), leaf count = 64

\[ \left \{ {\frac {1}{-1+y \relax (x ) +2\,x} \left (\left (-1+y \relax (x ) +2\,x \right ) \ln \left ({\frac {1-y \relax (x ) -2\,x}{x-2}} \right ) + \left (-1+y \relax (x ) +2\,x \right ) \ln \left (x-2 \right ) -{\it \_C1}\,y \relax (x ) + \left (-2\,{\it \_C1}+1 \right ) x+{\it \_C1}-2 \right ) }=0 \right \} \] 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),'implicit')

Maple raw output

((-1+y(x)+2*x)*ln((1-y(x)-2*x)/(x-2))+(-1+y(x)+2*x)*ln(x-2)-_C1*y(x)+(-2*_C1+1)*
x+_C1-2)/(-1+y(x)+2*x) = 0