4.23.9 \(f(x) (y(x)-a)^5 (y(x)-b)^4 (y(x)-c)^3+y'(x)^6=0\)

ODE
\[ f(x) (y(x)-a)^5 (y(x)-b)^4 (y(x)-c)^3+y'(x)^6=0 \] ODE Classification

[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Book solution method
Binomial equation \((y')^m + F(x) G(y)=0\)

Mathematica
cpu = 2.57428 (sec), leaf count = 815

\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {6 \, _2F_1\left (\frac {1}{6},\frac {2}{3};\frac {7}{6};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right ) \sqrt [6]{a-\text {$\#$1}} \left (\frac {(a-c) (b-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )^{2/3} \sqrt {c-\text {$\#$1}}}{(a-c) (b-\text {$\#$1})^{2/3}}\& \right ]\left [c_1+\int _1^x-i \sqrt [6]{f(K[1])}dK[1]\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {6 \, _2F_1\left (\frac {1}{6},\frac {2}{3};\frac {7}{6};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right ) \sqrt [6]{a-\text {$\#$1}} \left (\frac {(a-c) (b-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )^{2/3} \sqrt {c-\text {$\#$1}}}{(a-c) (b-\text {$\#$1})^{2/3}}\& \right ]\left [c_1+\int _1^xi \sqrt [6]{f(K[2])}dK[2]\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {6 \, _2F_1\left (\frac {1}{6},\frac {2}{3};\frac {7}{6};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right ) \sqrt [6]{a-\text {$\#$1}} \left (\frac {(a-c) (b-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )^{2/3} \sqrt {c-\text {$\#$1}}}{(a-c) (b-\text {$\#$1})^{2/3}}\& \right ]\left [c_1+\int _1^x-\sqrt [6]{-1} \sqrt [6]{f(K[3])}dK[3]\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {6 \, _2F_1\left (\frac {1}{6},\frac {2}{3};\frac {7}{6};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right ) \sqrt [6]{a-\text {$\#$1}} \left (\frac {(a-c) (b-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )^{2/3} \sqrt {c-\text {$\#$1}}}{(a-c) (b-\text {$\#$1})^{2/3}}\& \right ]\left [c_1+\int _1^x\sqrt [6]{-1} \sqrt [6]{f(K[4])}dK[4]\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {6 \, _2F_1\left (\frac {1}{6},\frac {2}{3};\frac {7}{6};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right ) \sqrt [6]{a-\text {$\#$1}} \left (\frac {(a-c) (b-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )^{2/3} \sqrt {c-\text {$\#$1}}}{(a-c) (b-\text {$\#$1})^{2/3}}\& \right ]\left [c_1+\int _1^x-(-1)^{5/6} \sqrt [6]{f(K[5])}dK[5]\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {6 \, _2F_1\left (\frac {1}{6},\frac {2}{3};\frac {7}{6};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right ) \sqrt [6]{a-\text {$\#$1}} \left (\frac {(a-c) (b-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )^{2/3} \sqrt {c-\text {$\#$1}}}{(a-c) (b-\text {$\#$1})^{2/3}}\& \right ]\left [c_1+\int _1^x(-1)^{5/6} \sqrt [6]{f(K[6])}dK[6]\right ]\right \}\right \}\]

Maple
cpu = 1.608 (sec), leaf count = 92

\[\left [\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a} -c}\, \left (\textit {\_a} -a \right )^{\frac {5}{6}} \left (\textit {\_a} -b \right )^{\frac {2}{3}}}d \textit {\_a} +\int _{}^{x}-\frac {\left (-f \left (\textit {\_a} \right ) \left (c -y \left (x \right )\right )^{3} \left (b -y \left (x \right )\right )^{4} \left (a -y \left (x \right )\right )^{5}\right )^{\frac {1}{6}}}{\sqrt {y \left (x \right )-c}\, \left (y \left (x \right )-a \right )^{\frac {5}{6}} \left (y \left (x \right )-b \right )^{\frac {2}{3}}}d \textit {\_a} +\textit {\_C1} = 0\right ]\] Mathematica raw input

DSolve[f[x]*(-a + y[x])^5*(-b + y[x])^4*(-c + y[x])^3 + y'[x]^6 == 0,y[x],x]

Mathematica raw output

{{y[x] -> InverseFunction[(6*Hypergeometric2F1[1/6, 2/3, 7/6, ((-b + c)*(a - #1)
)/((a - b)*(c - #1))]*(a - #1)^(1/6)*(((a - c)*(b - #1))/((a - b)*(c - #1)))^(2/
3)*Sqrt[c - #1])/((a - c)*(b - #1)^(2/3)) & ][C[1] + Inactive[Integrate][(-I)*f[
K[1]]^(1/6), {K[1], 1, x}]]}, {y[x] -> InverseFunction[(6*Hypergeometric2F1[1/6,
 2/3, 7/6, ((-b + c)*(a - #1))/((a - b)*(c - #1))]*(a - #1)^(1/6)*(((a - c)*(b -
 #1))/((a - b)*(c - #1)))^(2/3)*Sqrt[c - #1])/((a - c)*(b - #1)^(2/3)) & ][C[1] 
+ Inactive[Integrate][I*f[K[2]]^(1/6), {K[2], 1, x}]]}, {y[x] -> InverseFunction
[(6*Hypergeometric2F1[1/6, 2/3, 7/6, ((-b + c)*(a - #1))/((a - b)*(c - #1))]*(a 
- #1)^(1/6)*(((a - c)*(b - #1))/((a - b)*(c - #1)))^(2/3)*Sqrt[c - #1])/((a - c)
*(b - #1)^(2/3)) & ][C[1] + Inactive[Integrate][-((-1)^(1/6)*f[K[3]]^(1/6)), {K[
3], 1, x}]]}, {y[x] -> InverseFunction[(6*Hypergeometric2F1[1/6, 2/3, 7/6, ((-b 
+ c)*(a - #1))/((a - b)*(c - #1))]*(a - #1)^(1/6)*(((a - c)*(b - #1))/((a - b)*(
c - #1)))^(2/3)*Sqrt[c - #1])/((a - c)*(b - #1)^(2/3)) & ][C[1] + Inactive[Integ
rate][(-1)^(1/6)*f[K[4]]^(1/6), {K[4], 1, x}]]}, {y[x] -> InverseFunction[(6*Hyp
ergeometric2F1[1/6, 2/3, 7/6, ((-b + c)*(a - #1))/((a - b)*(c - #1))]*(a - #1)^(
1/6)*(((a - c)*(b - #1))/((a - b)*(c - #1)))^(2/3)*Sqrt[c - #1])/((a - c)*(b - #
1)^(2/3)) & ][C[1] + Inactive[Integrate][-((-1)^(5/6)*f[K[5]]^(1/6)), {K[5], 1, 
x}]]}, {y[x] -> InverseFunction[(6*Hypergeometric2F1[1/6, 2/3, 7/6, ((-b + c)*(a
 - #1))/((a - b)*(c - #1))]*(a - #1)^(1/6)*(((a - c)*(b - #1))/((a - b)*(c - #1)
))^(2/3)*Sqrt[c - #1])/((a - c)*(b - #1)^(2/3)) & ][C[1] + Inactive[Integrate][(
-1)^(5/6)*f[K[6]]^(1/6), {K[6], 1, x}]]}}

Maple raw input

dsolve(diff(y(x),x)^6+f(x)*(y(x)-a)^5*(y(x)-b)^4*(y(x)-c)^3 = 0, y(x))

Maple raw output

[Intat(1/(_a-c)^(1/2)/(_a-a)^(5/6)/(_a-b)^(2/3),_a = y(x))+Intat(-(-f(_a)*(c-y(x
))^3*(b-y(x))^4*(a-y(x))^5)^(1/6)/(y(x)-c)^(1/2)/(y(x)-a)^(5/6)/(y(x)-b)^(2/3),_
a = x)+_C1 = 0]