4.15.13 $(-2y(x{)}^4-xy(x{)}^3+4x)y^{\prime }(x)=y(x)(y(x{)}^3+2)$

ODE
$$(-2y(x{)}^4-xy(x{)}^3+4x)y^{\prime }(x)=y(x)(y(x{)}^3+2)$$ ODE Classification

[_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

Book solution method
Exact equation, integrating factor

Mathematica ✓
cpu = 0.150154 (sec), leaf count = 2021

$$\{\{y(x)\to -\frac{x}{4}-\frac{1}{2}\sqrt{\frac{x^2}{4}+\frac{2c_1}{3}+\frac{\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}{3\sqrt[3]{2}}+\frac{\sqrt[3]{2}(c_1^2+24x)}{3\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}}-\frac{1}{2}\sqrt{\frac{x^2}{2}+\frac{(x^2+4c_1)x}{4\sqrt{\frac{x^2}{4}+\frac{2c_1}{3}+\frac{\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}{3\sqrt[3]{2}}+\frac{\sqrt[3]{2}(c_1^2+24x)}{3\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}}}+\frac{4c_1}{3}-\frac{\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}{3\sqrt[3]{2}}-\frac{\sqrt[3]{2}(c_1^2+24x)}{3\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}}\},\{y(x)\to -\frac{x}{4}-\frac{1}{2}\sqrt{\frac{x^2}{4}+\frac{2c_1}{3}+\frac{\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}{3\sqrt[3]{2}}+\frac{\sqrt[3]{2}(c_1^2+24x)}{3\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}}+\frac{1}{2}\sqrt{\frac{x^2}{2}+\frac{(x^2+4c_1)x}{4\sqrt{\frac{x^2}{4}+\frac{2c_1}{3}+\frac{\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}{3\sqrt[3]{2}}+\frac{\sqrt[3]{2}(c_1^2+24x)}{3\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}}}+\frac{4c_1}{3}-\frac{\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}{3\sqrt[3]{2}}-\frac{\sqrt[3]{2}(c_1^2+24x)}{3\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}}\},\{y(x)\to -\frac{x}{4}+\frac{1}{2}\sqrt{\frac{x^2}{4}+\frac{2c_1}{3}+\frac{\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}{3\sqrt[3]{2}}+\frac{\sqrt[3]{2}(c_1^2+24x)}{3\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}}-\frac{1}{2}\sqrt{\frac{x^2}{2}-\frac{(x^2+4c_1)x}{4\sqrt{\frac{x^2}{4}+\frac{2c_1}{3}+\frac{\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}{3\sqrt[3]{2}}+\frac{\sqrt[3]{2}(c_1^2+24x)}{3\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}}}+\frac{4c_1}{3}-\frac{\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}{3\sqrt[3]{2}}-\frac{\sqrt[3]{2}(c_1^2+24x)}{3\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}}\},\{y(x)\to -\frac{x}{4}+\frac{1}{2}\sqrt{\frac{x^2}{4}+\frac{2c_1}{3}+\frac{\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}{3\sqrt[3]{2}}+\frac{\sqrt[3]{2}(c_1^2+24x)}{3\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}}+\frac{1}{2}\sqrt{\frac{x^2}{2}-\frac{(x^2+4c_1)x}{4\sqrt{\frac{x^2}{4}+\frac{2c_1}{3}+\frac{\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}{3\sqrt[3]{2}}+\frac{\sqrt[3]{2}(c_1^2+24x)}{3\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}}}+\frac{4c_1}{3}-\frac{\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}{3\sqrt[3]{2}}-\frac{\sqrt[3]{2}(c_1^2+24x)}{3\sqrt[3]{54x^3+144c_{1}x-2c_1^3+\sqrt{(54x^3+144c_{1}x-2c_1^3){}^2-4(c_1^2+24x){}^3}}}}\}\}$$

Maple ✓
cpu = 0.016 (sec), leaf count = 27

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

DSolve[(4*x - x*y[x]^3 - 2*y[x]^4)*y'[x] == y[x]*(2 + y[x]^3),y[x],x]

Mathematica raw output

{{y[x] -> -x/4 - Sqrt[x^2/4 + (2*C[1])/3 + (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 
+ 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C
[1]^3)^2])^(1/3)) + (54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 
+ (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)/(3*2^(1/3))]/2 - Sqrt[x^2/2 + (4*C[
1])/3 - (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(
24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) - (54*x^3 + 144*x
*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^
2])^(1/3)/(3*2^(1/3)) + (x*(x^2 + 4*C[1]))/(4*Sqrt[x^2/4 + (2*C[1])/3 + (2^(1/3)
*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3
 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) + (54*x^3 + 144*x*C[1] - 2*C[1]^3
 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)/(3*2^(
1/3))])]/2}, {y[x] -> -x/4 - Sqrt[x^2/4 + (2*C[1])/3 + (2^(1/3)*(24*x + C[1]^2))
/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*
x*C[1] - 2*C[1]^3)^2])^(1/3)) + (54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x 
+ C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)/(3*2^(1/3))]/2 + Sqrt[x
^2/2 + (4*C[1])/3 - (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3
 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) - (54
*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] 
- 2*C[1]^3)^2])^(1/3)/(3*2^(1/3)) + (x*(x^2 + 4*C[1]))/(4*Sqrt[x^2/4 + (2*C[1])/
3 + (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x
 + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) + (54*x^3 + 144*x*C[1
] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^
(1/3)/(3*2^(1/3))])]/2}, {y[x] -> -x/4 + Sqrt[x^2/4 + (2*C[1])/3 + (2^(1/3)*(24*
x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (5
4*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) + (54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sq
rt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)/(3*2^(1/3))
]/2 - Sqrt[x^2/2 + (4*C[1])/3 - (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1
] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^
(1/3)) - (54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 +
 144*x*C[1] - 2*C[1]^3)^2])^(1/3)/(3*2^(1/3)) - (x*(x^2 + 4*C[1]))/(4*Sqrt[x^2/4
 + (2*C[1])/3 + (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3 + S
qrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) + (54*x^3
 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*
C[1]^3)^2])^(1/3)/(3*2^(1/3))])]/2}, {y[x] -> -x/4 + Sqrt[x^2/4 + (2*C[1])/3 + (
2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[
1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)) + (54*x^3 + 144*x*C[1] - 2
*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)
/(3*2^(1/3))]/2 + Sqrt[x^2/2 + (4*C[1])/3 - (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3
 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*
C[1]^3)^2])^(1/3)) - (54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3
 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3)/(3*2^(1/3)) - (x*(x^2 + 4*C[1]))/(
4*Sqrt[x^2/4 + (2*C[1])/3 + (2^(1/3)*(24*x + C[1]^2))/(3*(54*x^3 + 144*x*C[1] - 
2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144*x*C[1] - 2*C[1]^3)^2])^(1/3
)) + (54*x^3 + 144*x*C[1] - 2*C[1]^3 + Sqrt[-4*(24*x + C[1]^2)^3 + (54*x^3 + 144
*x*C[1] - 2*C[1]^3)^2])^(1/3)/(3*2^(1/3))])]/2}}

Maple raw input

dsolve((4*x-x*y(x)^3-2*y(x)^4)*diff(y(x),x) = (2+y(x)^3)*y(x), y(x),'implicit')

Maple raw output

x-(-y(x)^2+_C1)/(2+y(x)^3)*y(x)^2 = 0