4.5.30 \((x+1) y'(x)=(x+1) \sqrt {y(x)+1}+y(x)+1\)

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 = 0.185196 (sec), leaf count = 57

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

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

\[ \left \{ {\frac {1}{{x}^{2}+2\,x-y \relax (x ) } \left (\left (-y \relax (x ) {\it \_C1}+1+{\it \_C1}\,{x}^{2}+ \left (2\,{\it \_C1}+1 \right ) x \right ) \sqrt {1+y \relax (x ) }- \left (1+x \right ) \left (-y \relax (x ) {\it \_C1}-1+{\it \_C1}\,{x}^{2}+ \left (2\,{\it \_C1}-1 \right ) x \right ) \right ) \left (-\sqrt {1+y \relax (x ) }+x+1 \right ) ^{-1}}=0 \right \} \] 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),'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