##### 4.11.7 $$y'(x) (\text {a2}+\text {b2} y(x)+\text {c2} y(x))=\text {a1}+\text {b1} x+\text {c1} y(x)$$

ODE
$y'(x) (\text {a2}+\text {b2} y(x)+\text {c2} y(x))=\text {a1}+\text {b1} x+\text {c1} y(x)$ 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 = 1.46234 (sec), leaf count = 194

$\text {Solve}\left [\frac {\text {c1}^2 \left (-\frac {2 \tan ^{-1}\left (\frac {-2 \text {a1} (\text {b2}+\text {c2})+\text {a2} \text {c1}-2 \text {b1} \text {b2} x-2 \text {b1} \text {c2} x-\text {c1} (\text {b2}+\text {c2}) y(x)}{\text {c1} \sqrt {-\frac {4 \text {b1} (\text {b2}+\text {c2})}{\text {c1}^2}-1} (\text {a2}+(\text {b2}+\text {c2}) y(x))}\right )}{\sqrt {-\frac {4 \text {b1} (\text {b2}+\text {c2})}{\text {c1}^2}-1}}+\log \left (\frac {(\text {a2}+(\text {b2}+\text {c2}) y(x))^2 \left (\frac {(\text {b2}+\text {c2}) (\text {a1}+\text {b1} x+\text {c1} y(x)) (\text {a1} (\text {b2}+\text {c2})-\text {a2} \text {c1}+\text {b1} x (\text {b2}+\text {c2}))}{(\text {a2}+(\text {b2}+\text {c2}) y(x))^2}-\text {b1} \text {b2}-\text {b1} \text {c2}\right )}{(\text {a1} (\text {b2}+\text {c2})-\text {a2} \text {c1}+\text {b1} x (\text {b2}+\text {c2}))^2}\right )+2 \log (\text {a1} (\text {b2}+\text {c2})-\text {a2} \text {c1}+\text {b1} x (\text {b2}+\text {c2}))\right )}{2 \text {b1} (\text {b2}+\text {c2})}=c_1,y(x)\right ]$

Maple
cpu = 0.727 (sec), leaf count = 264

$\left [y \left (x \right ) = \frac {\mathit {a2} \mathit {b1} +\RootOf \left (\left (\mathit {b2} +\mathit {c2} \right ) \textit {\_Z}^{2}-{\mathrm e}^{\RootOf \left (\left (\mathit {b1} \mathit {b2} x +\mathit {b1} \mathit {c2} x +\mathit {a1} \mathit {b2} +\mathit {a1} \mathit {c2} -\mathit {a2} \mathit {c1} \right )^{2} \left (-4 \left (\tanh ^{2}\left (\frac {\sqrt {4 \mathit {b1} \mathit {b2} +4 \mathit {b1} \mathit {c2} +\mathit {c1}^{2}}\, \left (2 \textit {\_C1} +\textit {\_Z} +2 \ln \left (\mathit {b1} \mathit {b2} x +\mathit {b1} \mathit {c2} x +\mathit {a1} \mathit {b2} +\mathit {a1} \mathit {c2} -\mathit {a2} \mathit {c1} \right )\right )}{2 \mathit {c1}}\right )\right ) \mathit {b1} \mathit {b2} -4 \left (\tanh ^{2}\left (\frac {\sqrt {4 \mathit {b1} \mathit {b2} +4 \mathit {b1} \mathit {c2} +\mathit {c1}^{2}}\, \left (2 \textit {\_C1} +\textit {\_Z} +2 \ln \left (\mathit {b1} \mathit {b2} x +\mathit {b1} \mathit {c2} x +\mathit {a1} \mathit {b2} +\mathit {a1} \mathit {c2} -\mathit {a2} \mathit {c1} \right )\right )}{2 \mathit {c1}}\right )\right ) \mathit {b1} \mathit {c2} -\left (\tanh ^{2}\left (\frac {\sqrt {4 \mathit {b1} \mathit {b2} +4 \mathit {b1} \mathit {c2} +\mathit {c1}^{2}}\, \left (2 \textit {\_C1} +\textit {\_Z} +2 \ln \left (\mathit {b1} \mathit {b2} x +\mathit {b1} \mathit {c2} x +\mathit {a1} \mathit {b2} +\mathit {a1} \mathit {c2} -\mathit {a2} \mathit {c1} \right )\right )}{2 \mathit {c1}}\right )\right ) \mathit {c1}^{2}+4 \,{\mathrm e}^{\textit {\_Z}} \mathit {b2} +4 \,{\mathrm e}^{\textit {\_Z}} \mathit {c2} +4 \mathit {b1} \mathit {b2} +4 \mathit {b1} \mathit {c2} +\mathit {c1}^{2}\right )\right )}-\mathit {b1} +\textit {\_Z} \mathit {c1} \right ) \left (x \left (\mathit {b1} \mathit {b2} +\mathit {b1} \mathit {c2} \right )+\mathit {a1} \mathit {b2} +\mathit {a1} \mathit {c2} -\mathit {a2} \mathit {c1} \right )}{-\mathit {b1} \mathit {b2} -\mathit {b1} \mathit {c2}}\right ]$ Mathematica raw input

DSolve[(a2 + b2*y[x] + c2*y[x])*y'[x] == a1 + b1*x + c1*y[x],y[x],x]

Mathematica raw output

Solve[(c1^2*((-2*ArcTan[(a2*c1 - 2*a1*(b2 + c2) - 2*b1*b2*x - 2*b1*c2*x - c1*(b2
+ c2)*y[x])/(c1*Sqrt[-1 - (4*b1*(b2 + c2))/c1^2]*(a2 + (b2 + c2)*y[x]))])/Sqrt[
-1 - (4*b1*(b2 + c2))/c1^2] + 2*Log[-(a2*c1) + a1*(b2 + c2) + b1*(b2 + c2)*x] +
Log[((a2 + (b2 + c2)*y[x])^2*(-(b1*b2) - b1*c2 + ((b2 + c2)*(-(a2*c1) + a1*(b2 +
c2) + b1*(b2 + c2)*x)*(a1 + b1*x + c1*y[x]))/(a2 + (b2 + c2)*y[x])^2))/(-(a2*c1
) + a1*(b2 + c2) + b1*(b2 + c2)*x)^2]))/(2*b1*(b2 + c2)) == C[1], y[x]]

Maple raw input

dsolve((a2+b2*y(x)+c2*y(x))*diff(y(x),x) = a1+b1*x+c1*y(x), y(x))

Maple raw output

[y(x) = (a2*b1+RootOf((b2+c2)*_Z^2-exp(RootOf((b1*b2*x+b1*c2*x+a1*b2+a1*c2-a2*c1
)^2*(-4*tanh(1/2*(4*b1*b2+4*b1*c2+c1^2)^(1/2)*(2*_C1+_Z+2*ln(b1*b2*x+b1*c2*x+a1*
b2+a1*c2-a2*c1))/c1)^2*b1*b2-4*tanh(1/2*(4*b1*b2+4*b1*c2+c1^2)^(1/2)*(2*_C1+_Z+2
*ln(b1*b2*x+b1*c2*x+a1*b2+a1*c2-a2*c1))/c1)^2*b1*c2-tanh(1/2*(4*b1*b2+4*b1*c2+c1
^2)^(1/2)*(2*_C1+_Z+2*ln(b1*b2*x+b1*c2*x+a1*b2+a1*c2-a2*c1))/c1)^2*c1^2+4*exp(_Z
)*b2+4*exp(_Z)*c2+4*b1*b2+4*b1*c2+c1^2)))-b1+_Z*c1)*(x*(b1*b2+b1*c2)+a1*b2+a1*c2
-a2*c1))/(-b1*b2-b1*c2)]