4.15.28 $(\sqrt{y(x)+x}+1)y^{\prime }(x)+1=0$

ODE
$$(\sqrt{y(x)+x}+1)y^{\prime }(x)+1=0$$ ODE Classification

[[_homogeneous, `class C`], _dAlembert]

Book solution method
Change of Variable, new dependent variable

Mathematica ✓
cpu = 0.0324476 (sec), leaf count = 39

$$\{\{y(x)\to -2\sqrt{c_1+x+1}+c_1+2\},\{y(x)\to 2\sqrt{c_1+x+1}+c_1+2\}\}$$

Maple ✓
cpu = 0.024 (sec), leaf count = 19

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

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

Mathematica raw output

{{y[x] -> 2 + C[1] - 2*Sqrt[1 + x + C[1]]}, {y[x] -> 2 + C[1] + 2*Sqrt[1 + x + C
[1]]}}

Maple raw input

dsolve((1+(x+y(x))^(1/2))*diff(y(x),x)+1 = 0, y(x),'implicit')

Maple raw output

-2*(x+y(x))^(1/2)-y(x)-_C1 = 0