##### 4.10.48 $$(9 y(x)+x+1) y'(x)+5 y(x)+x+1=0$$

ODE
$(9 y(x)+x+1) y'(x)+5 y(x)+x+1=0$ 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 = 43.0505 (sec), leaf count = 145

$\text {Solve}\left [\frac {(-2)^{2/3} \left ((x+1) \left (3 \log \left (-\frac {6 (-2)^{2/3} (x+1)}{9 y(x)+x+1}\right )-3 \log \left (\frac {9 (-2)^{2/3} (3 y(x)+x+1)}{9 y(x)+x+1}\right )+1\right )+9 y(x) \left (\log \left (-\frac {6 (-2)^{2/3} (x+1)}{9 y(x)+x+1}\right )-\log \left (\frac {9 (-2)^{2/3} (3 y(x)+x+1)}{9 y(x)+x+1}\right )+1\right )\right )}{27 (3 y(x)+x+1)}=\frac {1}{9} (-2)^{2/3} \log (x+1)+c_1,y(x)\right ]$

Maple
cpu = 0.126 (sec), leaf count = 29

$\left [y \left (x \right ) = -\frac {\left (x +1\right ) \left (2+3 \LambertW \left (\frac {2 \textit {\_C1} \left (x +1\right )}{3}\right )\right )}{9 \LambertW \left (\frac {2 \textit {\_C1} \left (x +1\right )}{3}\right )}\right ]$ Mathematica raw input

DSolve[1 + x + 5*y[x] + (1 + x + 9*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[((-2)^(2/3)*((1 + x)*(1 + 3*Log[(-6*(-2)^(2/3)*(1 + x))/(1 + x + 9*y[x])]
- 3*Log[(9*(-2)^(2/3)*(1 + x + 3*y[x]))/(1 + x + 9*y[x])]) + 9*(1 + Log[(-6*(-2)
^(2/3)*(1 + x))/(1 + x + 9*y[x])] - Log[(9*(-2)^(2/3)*(1 + x + 3*y[x]))/(1 + x +
 9*y[x])])*y[x]))/(27*(1 + x + 3*y[x])) == C[1] + ((-2)^(2/3)*Log[1 + x])/9, y[x
]]

Maple raw input

dsolve((1+x+9*y(x))*diff(y(x),x)+1+x+5*y(x) = 0, y(x))

Maple raw output

[y(x) = -1/9*(x+1)*(2+3*LambertW(2/3*_C1*(x+1)))/LambertW(2/3*_C1*(x+1))]