##### 4.11.6 $$\text {a1}+y'(x) (\text {a2}+b x+\text {c2} y(x))+b y(x)+\text {b1} x=0$$

ODE
$\text {a1}+y'(x) (\text {a2}+b x+\text {c2} y(x))+b y(x)+\text {b1} x=0$ ODE Classiﬁcation

[[_homogeneous, class C], _exact, _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 = 0.184532 (sec), leaf count = 106

$\left \{\left \{y(x)\to -\frac {\frac {\sqrt {-x (2 \text {a1}+\text {b1} x)+\frac {(\text {a2}+b x)^2}{\text {c2}}+\text {c2} c_1}}{\sqrt {\frac {1}{\text {c2}}}}+\text {a2}+b x}{\text {c2}}\right \},\left \{y(x)\to -\frac {\text {a2}+b x}{\text {c2}}+\sqrt {\frac {1}{\text {c2}}} \sqrt {-x (2 \text {a1}+\text {b1} x)+\frac {(\text {a2}+b x)^2}{\text {c2}}+\text {c2} c_1}\right \}\right \}$

Maple
cpu = 0.2 (sec), leaf count = 113

$\left [y \left (x \right ) = \frac {-\mathit {a1} b +\mathit {a2} \mathit {b1} +\frac {b \left (x \left (-b^{2}+\mathit {b1} \mathit {c2} \right )+\mathit {a1} \mathit {c2} -\mathit {a2} b \right ) \textit {\_C1} +\sqrt {b^{2} \left (x \left (-b^{2}+\mathit {b1} \mathit {c2} \right )+\mathit {a1} \mathit {c2} -\mathit {a2} b \right )^{2} \textit {\_C1}^{2}-\textit {\_C1}^{2} \mathit {b1} \mathit {c2} \left (x \left (-b^{2}+\mathit {b1} \mathit {c2} \right )+\mathit {a1} \mathit {c2} -\mathit {a2} b \right )^{2}+\mathit {c2}}}{\mathit {c2} \textit {\_C1}}}{b^{2}-\mathit {b1} \mathit {c2}}\right ]$ Mathematica raw input

DSolve[a1 + b1*x + b*y[x] + (a2 + b*x + c2*y[x])*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -((a2 + b*x + Sqrt[(a2 + b*x)^2/c2 - x*(2*a1 + b1*x) + c2*C[1]]/Sqrt[c
2^(-1)])/c2)}, {y[x] -> -((a2 + b*x)/c2) + Sqrt[c2^(-1)]*Sqrt[(a2 + b*x)^2/c2 -
x*(2*a1 + b1*x) + c2*C[1]]}}

Maple raw input

dsolve((a2+b*x+c2*y(x))*diff(y(x),x)+a1+b1*x+b*y(x) = 0, y(x))

Maple raw output

[y(x) = (-a1*b+a2*b1+(b*(x*(-b^2+b1*c2)+a1*c2-a2*b)*_C1+(b^2*(x*(-b^2+b1*c2)+a1*
c2-a2*b)^2*_C1^2-_C1^2*b1*c2*(x*(-b^2+b1*c2)+a1*c2-a2*b)^2+c2)^(1/2))/c2/_C1)/(b
^2-b1*c2)]