ODE
\[ (9 y(x)+x+1) y'(x)+5 y(x)+x+1=0 \] 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 = 22.0036 (sec), leaf count = 145
\[\text {Solve}\left [\frac {(-2)^{2/3} \left ((x+1) \left (3 \log \left (-\frac {6 (-2)^{2/3} (x+1)}{9 y(x)+x+1}\right )-3 \log \left (\frac {9 (-2)^{2/3} (3 y(x)+x+1)}{9 y(x)+x+1}\right )+1\right )+9 y(x) \left (\log \left (-\frac {6 (-2)^{2/3} (x+1)}{9 y(x)+x+1}\right )-\log \left (\frac {9 (-2)^{2/3} (3 y(x)+x+1)}{9 y(x)+x+1}\right )+1\right )\right )}{27 (3 y(x)+x+1)}=c_1+\frac {1}{9} (-2)^{2/3} \log (x+1),y(x)\right ]\]
Maple ✓
cpu = 0.025 (sec), leaf count = 72
\[ \left \{ {\frac {1}{9\,y \left ( x \right ) +3+3\,x} \left ( \left ( -3\,x-9\,y \left ( x \right ) -3 \right ) \ln \left ( {\frac {-3\,y \left ( x \right ) -1-x}{1+x}} \right ) + \left ( -3\,x-9\,y \left ( x \right ) -3 \right ) \ln \left ( 1+x \right ) -9\,{\it \_C1}\,y \left ( x \right ) + \left ( -3\,{\it \_C1}-2 \right ) x-3\,{\it \_C1}-2 \right ) }=0 \right \} \] Mathematica raw input
DSolve[1 + x + 5*y[x] + (1 + x + 9*y[x])*y'[x] == 0,y[x],x]
Mathematica raw output
Solve[((-2)^(2/3)*((1 + x)*(1 + 3*Log[(-6*(-2)^(2/3)*(1 + x))/(1 + x + 9*y[x])] - 3*Log[(9*(-2)^(2/3)*(1 + x + 3*y[x]))/(1 + x + 9*y[x])]) + 9*(1 + Log[(-6*(-2)^(2/3)*(1 + x))/(1 + x + 9*y[x])] - Log[(9*(-2)^(2/3)*(1 + x + 3*y[x]))/(1 + x + 9*y[x])])*y[x]))/(27*(1 + x + 3*y[x])) == C[1] + ((-2)^(2/3)*Log[1 + x])/9, y[x]]
Maple raw input
dsolve((1+x+9*y(x))*diff(y(x),x)+1+x+5*y(x) = 0, y(x),'implicit')
Maple raw output
((-3*x-9*y(x)-3)*ln((-3*y(x)-1-x)/(1+x))+(-3*x-9*y(x)-3)*ln(1+x)-9*_C1*y(x)+(-3*_C1-2)*x-3*_C1-2)/(9*y(x)+3+3*x) = 0