4.9.35 \((y(x)+1) y'(x)=y(x)+x\)

ODE
\[ (y(x)+1) y'(x)=y(x)+x \] ODE Classification

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type``class A`]]

Book solution method
Equation linear in the variables, \(y'(x)=f\left (\frac {X_1}{X_2} \right ) \)

Mathematica
cpu = 0.134524 (sec), leaf count = 71

\[\text {Solve}\left [\frac {1}{2} \log \left (\frac {x^2-y(x)^2+(x-3) y(x)-x-1}{(x-1)^2}\right )+\log (1-x)=c_1+\frac {\tanh ^{-1}\left (\frac {y(x)+2 x-1}{\sqrt {5} (y(x)+1)}\right )}{\sqrt {5}},y(x)\right ]\]

Maple
cpu = 0.021 (sec), leaf count = 66

\[ \left \{ -{\frac {1}{2}\ln \left ({\frac { \left (y \relax (x ) \right ) ^{2}+ \left (3-x \right ) y \relax (x ) -{x}^{2}+x+1}{ \left (-1+x \right ) ^{2}}} \right ) }-{\frac {\sqrt {5}}{5}{\it Artanh} \left ({\frac { \left (-3+x-2\,y \relax (x ) \right ) \sqrt {5}}{5\,x-5}} \right ) }-\ln \left (-1+x \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

Solve[Log[1 - x] + Log[(-1 - x + x^2 + (-3 + x)*y[x] - y[x]^2)/(-1 + x)^2]/2 == 
ArcTanh[(-1 + 2*x + y[x])/(Sqrt[5]*(1 + y[x]))]/Sqrt[5] + C[1], y[x]]

Maple raw input

dsolve((1+y(x))*diff(y(x),x) = x+y(x), y(x),'implicit')

Maple raw output

-1/2*ln((y(x)^2+(3-x)*y(x)-x^2+x+1)/(-1+x)^2)-1/5*5^(1/2)*arctanh((-3+x-2*y(x))*
5^(1/2)/(5*x-5))-ln(-1+x)-_C1 = 0