4.12.49 $(y(x{)}^2+x)y^{\prime }(x)+y(x)=a+bx$

ODE
$$(y(x{)}^2+x)y^{\prime }(x)+y(x)=a+bx$$ ODE Classification

[_exact, _rational]

Book solution method
Exact equation

Mathematica ✓
cpu = 0.218048 (sec), leaf count = 420

$$\{\{y(x)\to \frac{\sqrt[3]{2}(\sqrt{9(2ax+b{x}^2+2c_1){}^2+16x^3}+6ax+3b{x}^2+6c_1){}^{2/3}-22^{2/3}x}{2\sqrt[3]{\sqrt{9(2ax+b{x}^2+2c_1){}^2+16x^3}+6ax+3b{x}^2+6c_1}}\},\{y(x)\to \frac{i(\sqrt{3}+i)(\sqrt{36a^2{x}^2+36ab{x}^3+72a{c}_{1}x+9b^2{x}^4+36b{c}_1{x}^2+36c_1^2+16x^3}+6ax+3b{x}^2+6c_1){}^{2/3}+2\sqrt[3]{2}(1+i\sqrt{3})x}{22^{2/3}\sqrt[3]{\sqrt{9(2ax+b{x}^2+2c_1){}^2+16x^3}+6ax+3b{x}^2+6c_1}}\},\{y(x)\to \frac{x-i\sqrt{3}x}{\sqrt[3]{2}\sqrt[3]{\sqrt{9(2ax+b{x}^2+2c_1){}^2+16x^3}+6ax+3b{x}^2+6c_1}}-\frac{i(\sqrt{3}-i)\sqrt[3]{\sqrt{9(2ax+b{x}^2+2c_1){}^2+16x^3}+6ax+3b{x}^2+6c_1}}{22^{2/3}}\}\}$$

Maple ✓
cpu = 0.014 (sec), leaf count = 25

$$\{-\frac{b{x}^2}{2}-ax+xy\left(x\right)+\frac{{\left(y\left(x\right)\right)}^3}{3}+\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

DSolve[y[x] + (x + y[x]^2)*y'[x] == a + b*x,y[x],x]

Mathematica raw output

{{y[x] -> (-2*2^(2/3)*x + 2^(1/3)*(6*a*x + 3*b*x^2 + 6*C[1] + Sqrt[16*x^3 + 9*(2
*a*x + b*x^2 + 2*C[1])^2])^(2/3))/(2*(6*a*x + 3*b*x^2 + 6*C[1] + Sqrt[16*x^3 + 9
*(2*a*x + b*x^2 + 2*C[1])^2])^(1/3))}, {y[x] -> (2*2^(1/3)*(1 + I*Sqrt[3])*x + I
*(I + Sqrt[3])*(6*a*x + 3*b*x^2 + 6*C[1] + Sqrt[36*a^2*x^2 + 16*x^3 + 36*a*b*x^3
 + 9*b^2*x^4 + 72*a*x*C[1] + 36*b*x^2*C[1] + 36*C[1]^2])^(2/3))/(2*2^(2/3)*(6*a*
x + 3*b*x^2 + 6*C[1] + Sqrt[16*x^3 + 9*(2*a*x + b*x^2 + 2*C[1])^2])^(1/3))}, {y[
x] -> (x - I*Sqrt[3]*x)/(2^(1/3)*(6*a*x + 3*b*x^2 + 6*C[1] + Sqrt[16*x^3 + 9*(2*
a*x + b*x^2 + 2*C[1])^2])^(1/3)) - ((I/2)*(-I + Sqrt[3])*(6*a*x + 3*b*x^2 + 6*C[
1] + Sqrt[16*x^3 + 9*(2*a*x + b*x^2 + 2*C[1])^2])^(1/3))/2^(2/3)}}

Maple raw input

dsolve((x+y(x)^2)*diff(y(x),x)+y(x) = b*x+a, y(x),'implicit')

Maple raw output

-1/2*b*x^2-a*x+x*y(x)+1/3*y(x)^3+_C1 = 0