4.5.30 $(x+1)y^{\prime }(x)=(x+1)\sqrt{y(x)+1}+y(x)+1$

ODE
$$(x+1)y^{\prime }(x)=(x+1)\sqrt{y(x)+1}+y(x)+1$$ ODE Classification

[[_1st_order, _with_linear_symmetries]]

Book solution method
Change of Variable, new dependent variable

Mathematica ✓
cpu = 0.185196 (sec), leaf count = 57

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

Maple ✓
cpu = 0.097 (sec), leaf count = 81

$$\{\frac{1}{x^2+2x-y\left(x\right)}((-y\left(x\right)\mathit{\_}\mathit{C}\mathit{1}+1+\mathit{\_}\mathit{C}\mathit{1}{x}^2+(2\mathit{\_}\mathit{C}\mathit{1}+1)x)\sqrt{1+y\left(x\right)}-(1+x)(-y\left(x\right)\mathit{\_}\mathit{C}\mathit{1}-1+\mathit{\_}\mathit{C}\mathit{1}{x}^2+(2\mathit{\_}\mathit{C}\mathit{1}-1)x)){(-\sqrt{1+y\left(x\right)}+x+1)}^{-1}=0\}$$ Mathematica raw input

DSolve[(1 + x)*y'[x] == 1 + y[x] + (1 + x)*Sqrt[1 + y[x]],y[x],x]

Mathematica raw output

Solve[C[1] + Log[1 + x] == Log[-(x*(2 + x)) + y[x]] + (2*ArcTan[(1 + x)/Sqrt[-1 
- y[x]]]*Sqrt[1 + y[x]])/Sqrt[-1 - y[x]], y[x]]

Maple raw input

dsolve((1+x)*diff(y(x),x) = 1+y(x)+(1+x)*(1+y(x))^(1/2), y(x),'implicit')

Maple raw output

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