4.23.2 $a{y}^{\prime }(x{)}^3+b{y}^{\prime }(x{)}^2+y^{\prime }(x{)}^5=cy(x)$

ODE
$$a{y}^{\prime }(x{)}^3+b{y}^{\prime }(x{)}^2+y^{\prime }(x{)}^5=cy(x)$$ ODE Classification

[_quadrature]

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

Mathematica ✓
cpu = 0.502506 (sec), leaf count = 211

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

Maple ✓
cpu = 0.109 (sec), leaf count = 40

$$\{x-{\int }^{y\left(x\right)}{\left(\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}({\mathit{\_}\mathit{Z}}^5+a{\mathit{\_}\mathit{Z}}^3+b{\mathit{\_}\mathit{Z}}^2-\mathit{\_}\mathit{a}c)\right)}^{-1}d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0,y\left(x\right)=0\}$$ Mathematica raw input

DSolve[b*y'[x]^2 + a*y'[x]^3 + y'[x]^5 == c*y[x],y[x],x]

Mathematica raw output

{Solve[x + C[1] == Integrate[Root[-(c*K[1]) + b*#1^2 + a*#1^3 + #1^5 & , 1]^(-1)
, {K[1], 1, y[x]}], y[x]], Solve[x + C[1] == Integrate[Root[-(c*K[2]) + b*#1^2 +
 a*#1^3 + #1^5 & , 2]^(-1), {K[2], 1, y[x]}], y[x]], Solve[x + C[1] == Integrate
[Root[-(c*K[3]) + b*#1^2 + a*#1^3 + #1^5 & , 3]^(-1), {K[3], 1, y[x]}], y[x]], S
olve[x + C[1] == Integrate[Root[-(c*K[4]) + b*#1^2 + a*#1^3 + #1^5 & , 4]^(-1), 
{K[4], 1, y[x]}], y[x]], Solve[x + C[1] == Integrate[Root[-(c*K[5]) + b*#1^2 + a
*#1^3 + #1^5 & , 5]^(-1), {K[5], 1, y[x]}], y[x]]}

Maple raw input

dsolve(diff(y(x),x)^5+a*diff(y(x),x)^3+b*diff(y(x),x)^2 = c*y(x), y(x),'implicit')

Maple raw output

y(x) = 0, x-Intat(1/RootOf(_Z^5+_Z^3*a+_Z^2*b-_a*c),_a = y(x))-_C1 = 0