ODE
\[ \left (\sqrt {y(x)+x}+1\right ) y'(x)+1=0 \] ODE Classification
[[_homogeneous, `class C`], _dAlembert]
Book solution method
Change of Variable, new dependent variable
Mathematica ✓
cpu = 0.0324476 (sec), leaf count = 39
\[\left \{\left \{y(x)\to -2 \sqrt {c_1+x+1}+c_1+2\right \},\left \{y(x)\to 2 \sqrt {c_1+x+1}+c_1+2\right \}\right \}\]
Maple ✓
cpu = 0.024 (sec), leaf count = 19
\[ \left \{ -2\,\sqrt {x+y \left ( x \right ) }-y \left ( x \right ) -{\it \_C1}=0 \right \} \] 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