4.23.5 $y^{\prime }(x{)}^6=(y(x)-a{)}^4(y(x)-b{)}^3$

ODE
$$y^{\prime }(x{)}^6=(y(x)-a{)}^4(y(x)-b{)}^3$$ ODE Classification

[_quadrature]

Book solution method
Change of variable

Mathematica ✓
cpu = 0.999114 (sec), leaf count = 569

$$\{\{y(x)\to \text{InverseFunction}[-\frac{\sqrt[3]{a-\mathrm{\#}\text{1}}\sqrt{\frac{\mathrm{\#}\text{1}-b}{a-b}}{B}_{\frac{a-\mathrm{\#}\text{1}}{a-b}}(\frac{1}{3},\frac{1}{2})}{\sqrt{b-\mathrm{\#}\text{1}}\sqrt[3]{\frac{a-\mathrm{\#}\text{1}}{a-b}}}\mathrm{\&}][c_1-ix]\},\{y(x)\to \text{InverseFunction}[-\frac{\sqrt[3]{a-\mathrm{\#}\text{1}}\sqrt{\frac{\mathrm{\#}\text{1}-b}{a-b}}{B}_{\frac{a-\mathrm{\#}\text{1}}{a-b}}(\frac{1}{3},\frac{1}{2})}{\sqrt{b-\mathrm{\#}\text{1}}\sqrt[3]{\frac{a-\mathrm{\#}\text{1}}{a-b}}}\mathrm{\&}][c_1+ix]\},\{y(x)\to \text{InverseFunction}[-\frac{\sqrt[3]{a-\mathrm{\#}\text{1}}\sqrt{\frac{\mathrm{\#}\text{1}-b}{a-b}}{B}_{\frac{a-\mathrm{\#}\text{1}}{a-b}}(\frac{1}{3},\frac{1}{2})}{\sqrt{b-\mathrm{\#}\text{1}}\sqrt[3]{\frac{a-\mathrm{\#}\text{1}}{a-b}}}\mathrm{\&}][c_1-\sqrt[6]{-1}x]\},\{y(x)\to \text{InverseFunction}[-\frac{\sqrt[3]{a-\mathrm{\#}\text{1}}\sqrt{\frac{\mathrm{\#}\text{1}-b}{a-b}}{B}_{\frac{a-\mathrm{\#}\text{1}}{a-b}}(\frac{1}{3},\frac{1}{2})}{\sqrt{b-\mathrm{\#}\text{1}}\sqrt[3]{\frac{a-\mathrm{\#}\text{1}}{a-b}}}\mathrm{\&}][c_1+\sqrt[6]{-1}x]\},\{y(x)\to \text{InverseFunction}[-\frac{\sqrt[3]{a-\mathrm{\#}\text{1}}\sqrt{\frac{\mathrm{\#}\text{1}-b}{a-b}}{B}_{\frac{a-\mathrm{\#}\text{1}}{a-b}}(\frac{1}{3},\frac{1}{2})}{\sqrt{b-\mathrm{\#}\text{1}}\sqrt[3]{\frac{a-\mathrm{\#}\text{1}}{a-b}}}\mathrm{\&}][c_1-(-1{)}^{5/6}x]\},\{y(x)\to \text{InverseFunction}[-\frac{\sqrt[3]{a-\mathrm{\#}\text{1}}\sqrt{\frac{\mathrm{\#}\text{1}-b}{a-b}}{B}_{\frac{a-\mathrm{\#}\text{1}}{a-b}}(\frac{1}{3},\frac{1}{2})}{\sqrt{b-\mathrm{\#}\text{1}}\sqrt[3]{\frac{a-\mathrm{\#}\text{1}}{a-b}}}\mathrm{\&}][c_1+(-1{)}^{5/6}x]\}\}$$

Maple ✓
cpu = 0.413 (sec), leaf count = 250

$$\{x-{\int }^{y\left(x\right)}\frac{1}{\sqrt[6]{{(\mathit{\_}\mathit{a}-a)}^4{(\mathit{\_}\mathit{a}-b)}^3}}d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0,x-{\int }^{y\left(x\right)}\frac{-2i}{-\sqrt{3}+i}\frac{1}{\sqrt[6]{-{(-\mathit{\_}\mathit{a}+a)}^4{(-\mathit{\_}\mathit{a}+b)}^3}}d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0,x-{\int }^{y\left(x\right)}\frac{-2i}{\sqrt{3}+i}\frac{1}{\sqrt[6]{-{(-\mathit{\_}\mathit{a}+a)}^4{(-\mathit{\_}\mathit{a}+b)}^3}}d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0,x-{\int }^{y\left(x\right)}\frac{2i}{-\sqrt{3}+i}\frac{1}{\sqrt[6]{-{(-\mathit{\_}\mathit{a}+a)}^4{(-\mathit{\_}\mathit{a}+b)}^3}}d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0,x-{\int }^{y\left(x\right)}\frac{2i}{\sqrt{3}+i}\frac{1}{\sqrt[6]{-{(-\mathit{\_}\mathit{a}+a)}^4{(-\mathit{\_}\mathit{a}+b)}^3}}d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0,x-{\int }^{y\left(x\right)}-\frac{1}{\sqrt[6]{-{(-\mathit{\_}\mathit{a}+a)}^4{(-\mathit{\_}\mathit{a}+b)}^3}}d\mathit{\_}\mathit{a}-\mathit{\_}\mathit{C}\mathit{1}=0,y\left(x\right)=a,y\left(x\right)=b\}$$ Mathematica raw input

DSolve[y'[x]^6 == (-a + y[x])^4*(-b + y[x])^3,y[x],x]

Mathematica raw output

{{y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/3, 1/2]*(a - #1)^(1/3)*Sqr
t[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/3)*Sqrt[b - #1])) & ][(-I)*x + C[1]
]}, {y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/3, 1/2]*(a - #1)^(1/3)*
Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/3)*Sqrt[b - #1])) & ][I*x + C[1]
]}, {y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/3, 1/2]*(a - #1)^(1/3)*
Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/3)*Sqrt[b - #1])) & ][-((-1)^(1/
6)*x) + C[1]]}, {y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/3, 1/2]*(a 
- #1)^(1/3)*Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/3)*Sqrt[b - #1])) & 
][(-1)^(1/6)*x + C[1]]}, {y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/3,
 1/2]*(a - #1)^(1/3)*Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/3)*Sqrt[b -
 #1])) & ][-((-1)^(5/6)*x) + C[1]]}, {y[x] -> InverseFunction[-((Beta[(a - #1)/(
a - b), 1/3, 1/2]*(a - #1)^(1/3)*Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1
/3)*Sqrt[b - #1])) & ][(-1)^(5/6)*x + C[1]]}}

Maple raw input

dsolve(diff(y(x),x)^6 = (y(x)-a)^4*(y(x)-b)^3, y(x),'implicit')

Maple raw output

y(x) = a, y(x) = b, x-Intat(1/((_a-a)^4*(_a-b)^3)^(1/6),_a = y(x))-_C1 = 0, x-In
tat(-2*I/(-(-_a+a)^4*(-_a+b)^3)^(1/6)/(-3^(1/2)+I),_a = y(x))-_C1 = 0, x-Intat(-
2*I/(-(-_a+a)^4*(-_a+b)^3)^(1/6)/(3^(1/2)+I),_a = y(x))-_C1 = 0, x-Intat(2*I/(-(
-_a+a)^4*(-_a+b)^3)^(1/6)/(3^(1/2)+I),_a = y(x))-_C1 = 0, x-Intat(2*I/(-(-_a+a)^
4*(-_a+b)^3)^(1/6)/(-3^(1/2)+I),_a = y(x))-_C1 = 0, x-Intat(-1/(-(-_a+a)^4*(-_a+
b)^3)^(1/6),_a = y(x))-_C1 = 0