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

ODE
$$y^{\prime }(x)(\text{a2}+\text{b2}y(x)+\text{c2}y(x))=\text{a1}+\text{b1}x+\text{c1}y(x)$$ ODE Classification

[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]

Book solution method
Equation linear in the variables, $y^{\prime }(x)=f\left(\frac{X_1}{X_2}\right)$

Mathematica ✓
cpu = 1.4045 (sec), leaf count = 216

$$\text{Solve}[\frac{{\text{c1}}^2\log \left(\frac{(\text{a2}+(\text{b2}+\text{c2})y(x){)}^2(\frac{(\text{b2}+\text{c2})(\text{a1}+\text{b1}x+\text{c1}y(x))(\text{a1}(\text{b2}+\text{c2})-\text{a2}\text{c1}+\text{b1}x(\text{b2}+\text{c2}))}{(\text{a2}+(\text{b2}+\text{c2})y(x){)}^2}-\text{b1}\text{b2}-\text{b1}\text{c2})}{(\text{a1}(\text{b2}+\text{c2})-\text{a2}\text{c1}+\text{b1}x(\text{b2}+\text{c2}){)}^2}\right)-\frac{2{\text{c1}}^2{\tan }^{-1}\left(\frac{-2\text{a1}(\text{b2}+\text{c2})+\text{a2}\text{c1}-2\text{b1}\text{b2}x-2\text{b1}\text{c2}x-\text{c1}(\text{b2}+\text{c2})y(x)}{\text{c1}\sqrt{-\frac{4\text{b1}(\text{b2}+\text{c2})+{\text{c1}}^2}{{\text{c1}}^2}}(\text{a2}+(\text{b2}+\text{c2})y(x))}\right)}{\sqrt{-\frac{4\text{b1}(\text{b2}+\text{c2})+{\text{c1}}^2}{{\text{c1}}^2}}}+2{\text{c1}}^2\log (\text{a1}(\text{b2}+\text{c2})-\text{a2}\text{c1}+\text{b1}x(\text{b2}+\text{c2}))-2\text{b1}{c}_1(\text{b2}+\text{c2})}{2\text{b1}(\text{b2}+\text{c2})}=0,y(x)]$$

Maple ✓
cpu = 0.043 (sec), leaf count = 228

$$\{-\frac{1}{2}\ln (-\frac{\mathit{b}\mathit{1}(\mathit{b}\mathit{2}+\mathit{c}\mathit{2})(-\mathit{b}\mathit{1}{(\mathit{b}\mathit{2}+\mathit{c}\mathit{2})}^2{\left(y\left(x\right)\right)}^2+((\mathit{b}\mathit{2}+\mathit{c}\mathit{2})(x\mathit{c}\mathit{1}-2\mathit{a}\mathit{2})\mathit{b}\mathit{1}+\mathit{c}\mathit{1}(\mathit{a}\mathit{1}\mathit{b}\mathit{2}+\mathit{a}\mathit{1}\mathit{c}\mathit{2}-\mathit{a}\mathit{2}\mathit{c}\mathit{1}))y\left(x\right)+x^2(\mathit{b}\mathit{2}+\mathit{c}\mathit{2}){\mathit{b}\mathit{1}}^2+(2\mathit{a}\mathit{1}\mathit{b}\mathit{2}x+2\mathit{a}\mathit{1}\mathit{c}\mathit{2}x-\mathit{a}\mathit{2}\mathit{c}\mathit{1}x-{\mathit{a}\mathit{2}}^2)\mathit{b}\mathit{1}+\mathit{a}\mathit{1}(\mathit{a}\mathit{1}\mathit{b}\mathit{2}+\mathit{a}\mathit{1}\mathit{c}\mathit{2}-\mathit{a}\mathit{2}\mathit{c}\mathit{1}))}{{((\mathit{b}\mathit{2}+\mathit{c}\mathit{2})\mathit{a}\mathit{1}+x(\mathit{b}\mathit{2}+\mathit{c}\mathit{2})\mathit{b}\mathit{1}-\mathit{a}\mathit{2}\mathit{c}\mathit{1})}^2})-\mathit{c}\mathit{1}\mathit{A}\mathit{r}\mathit{t}\mathit{a}\mathit{n}\mathit{h}\left(\frac{-2\mathit{b}\mathit{1}{(\mathit{b}\mathit{2}+\mathit{c}\mathit{2})}^{2}y\left(x\right)+(\mathit{b}\mathit{2}+\mathit{c}\mathit{2})(x\mathit{c}\mathit{1}-2\mathit{a}\mathit{2})\mathit{b}\mathit{1}+\mathit{c}\mathit{1}(\mathit{a}\mathit{1}\mathit{b}\mathit{2}+\mathit{a}\mathit{1}\mathit{c}\mathit{2}-\mathit{a}\mathit{2}\mathit{c}\mathit{1})}{(\mathit{b}\mathit{2}+\mathit{c}\mathit{2})\mathit{a}\mathit{1}+x(\mathit{b}\mathit{2}+\mathit{c}\mathit{2})\mathit{b}\mathit{1}-\mathit{a}\mathit{2}\mathit{c}\mathit{1}}\frac{1}{\sqrt{4\mathit{b}\mathit{1}\mathit{b}\mathit{2}+4\mathit{b}\mathit{1}\mathit{c}\mathit{2}+{\mathit{c}\mathit{1}}^2}}\right)\frac{1}{\sqrt{4\mathit{b}\mathit{1}\mathit{b}\mathit{2}+4\mathit{b}\mathit{1}\mathit{c}\mathit{2}+{\mathit{c}\mathit{1}}^2}}-\ln ((\mathit{b}\mathit{2}+\mathit{c}\mathit{2})\mathit{a}\mathit{1}+x(\mathit{b}\mathit{2}+\mathit{c}\mathit{2})\mathit{b}\mathit{1}-\mathit{a}\mathit{2}\mathit{c}\mathit{1})-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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[((-2*c1^2*ArcTan[(a2*c1 - 2*a1*(b2 + c2) - 2*b1*b2*x - 2*b1*c2*x - c1*(b2 
+ c2)*y[x])/(c1*Sqrt[-((c1^2 + 4*b1*(b2 + c2))/c1^2)]*(a2 + (b2 + c2)*y[x]))])/S
qrt[-((c1^2 + 4*b1*(b2 + c2))/c1^2)] - 2*b1*(b2 + c2)*C[1] + 2*c1^2*Log[-(a2*c1)
 + a1*(b2 + c2) + b1*(b2 + c2)*x] + c1^2*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)) == 0, 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),'implicit')

Maple raw output

-1/2*ln(-(b2+c2)*b1*(-b1*(b2+c2)^2*y(x)^2+((b2+c2)*(c1*x-2*a2)*b1+c1*(a1*b2+a1*c
2-a2*c1))*y(x)+x^2*(b2+c2)*b1^2+(2*a1*b2*x+2*a1*c2*x-a2*c1*x-a2^2)*b1+a1*(a1*b2+
a1*c2-a2*c1))/((b2+c2)*a1+x*(b2+c2)*b1-a2*c1)^2)-1/(4*b1*b2+4*b1*c2+c1^2)^(1/2)*
arctanh((-2*b1*(b2+c2)^2*y(x)+(b2+c2)*(c1*x-2*a2)*b1+c1*(a1*b2+a1*c2-a2*c1))/(4*
b1*b2+4*b1*c2+c1^2)^(1/2)/((b2+c2)*a1+x*(b2+c2)*b1-a2*c1))*c1-ln((b2+c2)*a1+x*(b
2+c2)*b1-a2*c1)-_C1 = 0