ODE
\[ y'(x) (a x+b y(x))+a y(x)+b x=0 \] ODE Classification
[[_homogeneous, `class A`], _exact, _rational, [_Abel, `2nd type`, `class A`]]
Book solution method
Exact equation
Mathematica ✓
cpu = 0.208761 (sec), leaf count = 83
\[\left \{\left \{y(x)\to -\frac {a x+\sqrt {a^2 x^2-b^2 x^2+b e^{2 c_1}}}{b}\right \},\left \{y(x)\to \frac {-a x+\sqrt {a^2 x^2-b^2 x^2+b e^{2 c_1}}}{b}\right \}\right \}\]
Maple ✓
cpu = 0.141 (sec), leaf count = 85
\[\left [y \left (x \right ) = -\frac {\textit {\_C1} a x -\sqrt {a^{2} x^{2} \textit {\_C1}^{2}-\textit {\_C1}^{2} b^{2} x^{2}+b}}{b \textit {\_C1}}, y \left (x \right ) = -\frac {\textit {\_C1} a x +\sqrt {a^{2} x^{2} \textit {\_C1}^{2}-\textit {\_C1}^{2} b^{2} x^{2}+b}}{b \textit {\_C1}}\right ]\] Mathematica raw input
DSolve[b*x + a*y[x] + (a*x + b*y[x])*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -((a*x + Sqrt[b*E^(2*C[1]) + a^2*x^2 - b^2*x^2])/b)}, {y[x] -> (-(a*x) + Sqrt[b*E^(2*C[1]) + a^2*x^2 - b^2*x^2])/b}}
Maple raw input
dsolve((a*x+b*y(x))*diff(y(x),x)+b*x+a*y(x) = 0, y(x))
Maple raw output
[y(x) = -(_C1*a*x-(_C1^2*a^2*x^2-_C1^2*b^2*x^2+b)^(1/2))/b/_C1, y(x) = -(_C1*a*x+(_C1^2*a^2*x^2-_C1^2*b^2*x^2+b)^(1/2))/b/_C1]