4.7.45 $(a+bx{)}^2{y}^{\prime }(x)+y(x{)}^3(a+bx)+cy(x{)}^2=0$

ODE
$$(a+bx{)}^2{y}^{\prime }(x)+y(x{)}^3(a+bx)+cy(x{)}^2=0$$ ODE Classification

[_rational, _Abel]

Book solution method
Abel ODE, Second kind

Mathematica ✓
cpu = 2.50611 (sec), leaf count = 110

$$\text{Solve}[-\frac{c}{\sqrt{-b(a+bx{)}^2}}=\frac{2\exp (-\frac{(b(a+bx)+cy(x){)}^2}{2by(x{)}^2(a+bx{)}^2})}{2c_1-\sqrt{2\pi }\text{erfi}\left(\frac{b(a+bx)+cy(x)}{\sqrt{2}y(x)\sqrt{-b(a+bx{)}^2}}\right)},y(x)]$$

Maple ✓
cpu = 0.047 (sec), leaf count = 153

$$\{\frac{1}{2}((\mathit{E}\mathit{r}\mathit{f}\left(\frac{\sqrt{2}(cy\left(x\right)+b(bx+a))}{2(bx+a)y\left(x\right)}\frac{1}{\sqrt{b}}\right){\mathrm{e}}^{\frac{{(cy\left(x\right)+b(bx+a))}^2}{2{\left(y\left(x\right)\right)}^2{(bx+a)}^{2}b}}\sqrt{\pi }\sqrt{2}bc+2(bx+a)b^{3/2}){\mathrm{e}}^{-\frac{((-bx-a+c)y\left(x\right)+b(bx+a))((bx+a+c)y\left(x\right)+b(bx+a))}{2{\left(y\left(x\right)\right)}^2{(bx+a)}^{2}b}}+2\mathit{\_}\mathit{C}\mathit{1}{b}^{5/2})b^{-\frac{5}{2}}=0\}$$ Mathematica raw input

DSolve[c*y[x]^2 + (a + b*x)*y[x]^3 + (a + b*x)^2*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[-(c/Sqrt[-(b*(a + b*x)^2)]) == 2/(E^((b*(a + b*x) + c*y[x])^2/(2*b*(a + b*
x)^2*y[x]^2))*(2*C[1] - Sqrt[2*Pi]*Erfi[(b*(a + b*x) + c*y[x])/(Sqrt[2]*Sqrt[-(b
*(a + b*x)^2)]*y[x])])), y[x]]

Maple raw input

dsolve((b*x+a)^2*diff(y(x),x)+c*y(x)^2+(b*x+a)*y(x)^3 = 0, y(x),'implicit')

Maple raw output

1/2*((erf(1/2*2^(1/2)*(c*y(x)+b*(b*x+a))/b^(1/2)/y(x)/(b*x+a))*exp(1/2*(c*y(x)+b
*(b*x+a))^2/y(x)^2/(b*x+a)^2/b)*Pi^(1/2)*2^(1/2)*b*c+2*(b*x+a)*b^(3/2))*exp(-1/2
*((-b*x-a+c)*y(x)+b*(b*x+a))*((b*x+a+c)*y(x)+b*(b*x+a))/y(x)^2/(b*x+a)^2/b)+2*_C
1*b^(5/2))/b^(5/2) = 0