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