ODE
\[ (x+1) y'(x)=(x+1) \sqrt {y(x)+1}+y(x)+1 \] ODE Classification
[[_1st_order, _with_linear_symmetries]]
Book solution method
Change of Variable, new dependent variable
Mathematica ✓
cpu = 0.52842 (sec), leaf count = 57
\[\text {Solve}\left [\log (x+1)+c_1=\log (y(x)-x (x+2))+\frac {2 \sqrt {y(x)+1} \tan ^{-1}\left (\frac {x+1}{\sqrt {-y(x)-1}}\right )}{\sqrt {-y(x)-1}},y(x)\right ]\]
Maple ✓
cpu = 0.078 (sec), leaf count = 160
\[\left [\frac {\sqrt {1+y \left (x \right )}}{\left (-x^{2}-2 x +y \left (x \right )\right ) \left (\sqrt {1+y \left (x \right )}-x -1\right )}+\frac {\sqrt {1+y \left (x \right )}\, x}{\left (-x^{2}-2 x +y \left (x \right )\right ) \left (\sqrt {1+y \left (x \right )}-x -1\right )}+\frac {1}{\left (-x^{2}-2 x +y \left (x \right )\right ) \left (\sqrt {1+y \left (x \right )}-x -1\right )}+\frac {2 x}{\left (-x^{2}-2 x +y \left (x \right )\right ) \left (\sqrt {1+y \left (x \right )}-x -1\right )}+\frac {x^{2}}{\left (-x^{2}-2 x +y \left (x \right )\right ) \left (\sqrt {1+y \left (x \right )}-x -1\right )}-\textit {\_C1} = 0\right ]\] 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))
Maple raw output
[1/(-x^2-2*x+y(x))/((1+y(x))^(1/2)-x-1)*(1+y(x))^(1/2)+1/(-x^2-2*x+y(x))/((1+y(x))^(1/2)-x-1)*(1+y(x))^(1/2)*x+1/(-x^2-2*x+y(x))/((1+y(x))^(1/2)-x-1)+2/(-x^2-2*x+y(x))/((1+y(x))^(1/2)-x-1)*x+1/(-x^2-2*x+y(x))/((1+y(x))^(1/2)-x-1)*x^2-_C1 = 0]