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 = 0.317264 (sec), leaf count = 71
\[\text {Solve}\left [\frac {1}{2} \log \left (\frac {x^2-y(x)^2+(x-3) y(x)-x-1}{(x-1)^2}\right )+\log (1-x)=\frac {\tanh ^{-1}\left (\frac {y(x)+2 x-1}{\sqrt {5} (y(x)+1)}\right )}{\sqrt {5}}+c_1,y(x)\right ]\]
Maple ✓
cpu = 0.911 (sec), leaf count = 73
\[\left [-\frac {\ln \left (-\frac {\left (x -1\right )^{2}-\left (x -1\right ) \left (-y \left (x \right )-1\right )-\left (-y \left (x \right )-1\right )^{2}}{\left (x -1\right )^{2}}\right )}{2}-\frac {\sqrt {5}\, \arctanh \left (\frac {\left (x -3-2 y \left (x \right )\right ) \sqrt {5}}{5 x -5}\right )}{5}-\ln \left (x -1\right )-\textit {\_C1} = 0\right ]\] 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))
Maple raw output
[-1/2*ln(-((x-1)^2-(x-1)*(-y(x)-1)-(-y(x)-1)^2)/(x-1)^2)-1/5*5^(1/2)*arctanh(1/5*(x-3-2*y(x))*5^(1/2)/(x-1))-ln(x-1)-_C1 = 0]