4.23.4 $3y^{\prime }(x{)}^5-y(x)y^{\prime }(x)+1=0$

ODE
$$3y^{\prime }(x{)}^5-y(x)y^{\prime }(x)+1=0$$ ODE Classification

[_quadrature]

Book solution method
Missing Variables ODE, Independent variable missing, Solve for $y$

Mathematica ✓
cpu = 0.454499 (sec), leaf count = 171

$$\{\text{Solve}[c_1+x={\int }_1^{y(x)}\frac{1}{\text{Root}[-\mathrm{\#}\text{1}K[1]+3{\mathrm{\#}\text{1}}^5+1\mathrm{\&},1]}dK[1],y(x)],\text{Solve}[c_1+x={\int }_1^{y(x)}\frac{1}{\text{Root}[-\mathrm{\#}\text{1}K[2]+3{\mathrm{\#}\text{1}}^5+1\mathrm{\&},2]}dK[2],y(x)],\text{Solve}[c_1+x={\int }_1^{y(x)}\frac{1}{\text{Root}[-\mathrm{\#}\text{1}K[3]+3{\mathrm{\#}\text{1}}^5+1\mathrm{\&},3]}dK[3],y(x)],\text{Solve}[c_1+x={\int }_1^{y(x)}\frac{1}{\text{Root}[-\mathrm{\#}\text{1}K[4]+3{\mathrm{\#}\text{1}}^5+1\mathrm{\&},4]}dK[4],y(x)],\text{Solve}[c_1+x={\int }_1^{y(x)}\frac{1}{\text{Root}[-\mathrm{\#}\text{1}K[5]+3{\mathrm{\#}\text{1}}^5+1\mathrm{\&},5]}dK[5],y(x)]\}$$

Maple ✓
cpu = 0.174 (sec), leaf count = 50

$$\{x-\frac{{\left(y\left(x\right)\right)}^2}{2}-\frac{5{\left(\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(3{\mathit{\_}\mathit{Z}}^5-y\left(x\right)\mathit{\_}\mathit{Z}+1)\right)}^3}{2}+\frac{3y\left(x\right){\left(\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(3{\mathit{\_}\mathit{Z}}^5-y\left(x\right)\mathit{\_}\mathit{Z}+1)\right)}^4}{2}-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

DSolve[1 - y[x]*y'[x] + 3*y'[x]^5 == 0,y[x],x]

Mathematica raw output

{Solve[x + C[1] == Integrate[Root[1 - K[1]*#1 + 3*#1^5 & , 1]^(-1), {K[1], 1, y[
x]}], y[x]], Solve[x + C[1] == Integrate[Root[1 - K[2]*#1 + 3*#1^5 & , 2]^(-1), 
{K[2], 1, y[x]}], y[x]], Solve[x + C[1] == Integrate[Root[1 - K[3]*#1 + 3*#1^5 &
 , 3]^(-1), {K[3], 1, y[x]}], y[x]], Solve[x + C[1] == Integrate[Root[1 - K[4]*#
1 + 3*#1^5 & , 4]^(-1), {K[4], 1, y[x]}], y[x]], Solve[x + C[1] == Integrate[Roo
t[1 - K[5]*#1 + 3*#1^5 & , 5]^(-1), {K[5], 1, y[x]}], y[x]]}

Maple raw input

dsolve(3*diff(y(x),x)^5-y(x)*diff(y(x),x)+1 = 0, y(x),'implicit')

Maple raw output

x-1/2*y(x)^2-5/2*RootOf(3*_Z^5-y(x)*_Z+1)^3+3/2*y(x)*RootOf(3*_Z^5-y(x)*_Z+1)^4-
_C1 = 0