4.15.6 $x{y}^{\prime }(x)(a+bxy(x{)}^3)+y(x)(a+c{x}^{3}y(x))=0$

ODE
$$x{y}^{\prime }(x)(a+bxy(x{)}^3)+y(x)(a+c{x}^{3}y(x))=0$$ ODE Classification

[_rational]

Book solution method
Exact equation, integrating factor

Mathematica ✓
cpu = 0.0417136 (sec), leaf count = 491

$$\{\{y(x)\to \frac{\sqrt[3]{3}(\sqrt{3}\sqrt{b^3{x}^4(27a^{2}b+x^2(c{x}^2-2c_1){}^3)}+9a{b}^2{x}^2){}^{2/3}+3^{2/3}b{x}^2(2c_1-c{x}^2)}{3bx\sqrt[3]{\sqrt{3}\sqrt{b^3{x}^4(27a^{2}b+x^2(c{x}^2-2c_1){}^3)}+9a{b}^2{x}^2}}\},\{y(x)\to \frac{i\sqrt[6]{3}(\sqrt{3}+i)(\sqrt{3}\sqrt{b^3{x}^4(27a^{2}b+x^2(c{x}^2-2c_1){}^3)}+9a{b}^2{x}^2){}^{2/3}+(\sqrt{3}+3i)b{x}^2(c{x}^2-2c_1)}{23^{5/6}bx\sqrt[3]{\sqrt{3}\sqrt{b^3{x}^4(27a^{2}b+x^2(c{x}^2-2c_1){}^3)}+9a{b}^2{x}^2}}\},\{y(x)\to \frac{(\sqrt{3}-3i)b{x}^2(c{x}^2-2c_1)-i\sqrt[6]{3}(\sqrt{3}-i)(\sqrt{3}\sqrt{b^3{x}^4(27a^{2}b+x^2(c{x}^2-2c_1){}^3)}+9a{b}^2{x}^2){}^{2/3}}{23^{5/6}bx\sqrt[3]{\sqrt{3}\sqrt{b^3{x}^4(27a^{2}b+x^2(c{x}^2-2c_1){}^3)}+9a{b}^2{x}^2}}\}\}$$

Maple ✓
cpu = 0.114 (sec), leaf count = 27

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

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

Mathematica raw output

{{y[x] -> (3^(1/3)*(9*a*b^2*x^2 + Sqrt[3]*Sqrt[b^3*x^4*(27*a^2*b + x^2*(c*x^2 - 
2*C[1])^3)])^(2/3) + 3^(2/3)*b*x^2*(-(c*x^2) + 2*C[1]))/(3*b*x*(9*a*b^2*x^2 + Sq
rt[3]*Sqrt[b^3*x^4*(27*a^2*b + x^2*(c*x^2 - 2*C[1])^3)])^(1/3))}, {y[x] -> (I*3^
(1/6)*(I + Sqrt[3])*(9*a*b^2*x^2 + Sqrt[3]*Sqrt[b^3*x^4*(27*a^2*b + x^2*(c*x^2 -
 2*C[1])^3)])^(2/3) + (3*I + Sqrt[3])*b*x^2*(c*x^2 - 2*C[1]))/(2*3^(5/6)*b*x*(9*
a*b^2*x^2 + Sqrt[3]*Sqrt[b^3*x^4*(27*a^2*b + x^2*(c*x^2 - 2*C[1])^3)])^(1/3))}, 
{y[x] -> ((-I)*3^(1/6)*(-I + Sqrt[3])*(9*a*b^2*x^2 + Sqrt[3]*Sqrt[b^3*x^4*(27*a^
2*b + x^2*(c*x^2 - 2*C[1])^3)])^(2/3) + (-3*I + Sqrt[3])*b*x^2*(c*x^2 - 2*C[1]))
/(2*3^(5/6)*b*x*(9*a*b^2*x^2 + Sqrt[3]*Sqrt[b^3*x^4*(27*a^2*b + x^2*(c*x^2 - 2*C
[1])^3)])^(1/3))}}

Maple raw input

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

Maple raw output

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