ODE
\[ f(x) (y(x)-a)^5 (y(x)-b)^3+y'(x)^6=0 \] 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 = 1.27147 (sec), leaf count = 557
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\& \right ]\left [\int _1^x-i \sqrt [6]{f(K[1])}dK[1]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\& \right ]\left [\int _1^xi \sqrt [6]{f(K[2])}dK[2]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\& \right ]\left [\int _1^x-\sqrt [6]{-1} \sqrt [6]{f(K[3])}dK[3]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\& \right ]\left [\int _1^x\sqrt [6]{-1} \sqrt [6]{f(K[4])}dK[4]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\& \right ]\left [\int _1^x-(-1)^{5/6} \sqrt [6]{f(K[5])}dK[5]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\& \right ]\left [\int _1^x(-1)^{5/6} \sqrt [6]{f(K[6])}dK[6]+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 0.966 (sec), leaf count = 69
\[\left [\int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a} -a \right )^{\frac {5}{6}} \sqrt {\textit {\_a} -b}}d \textit {\_a} +\int _{}^{x}-\frac {\left (-f \left (\textit {\_a} \right ) \left (b -y \left (x \right )\right )^{3} \left (a -y \left (x \right )\right )^{5}\right )^{\frac {1}{6}}}{\left (y \left (x \right )-a \right )^{\frac {5}{6}} \sqrt {y \left (x \right )-b}}d \textit {\_a} +\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[f[x]*(-a + y[x])^5*(-b + y[x])^3 + y'[x]^6 == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[(-6*Hypergeometric2F1[1/6, 1/2, 7/6, (a - #1)/(a - b)]*(a - #1)^(1/6)*Sqrt[(-b + #1)/(a - b)])/Sqrt[b - #1] & ][C[1] + Inactive[Integrate][(-I)*f[K[1]]^(1/6), {K[1], 1, x}]]}, {y[x] -> InverseFunction[(-6*Hypergeometric2F1[1/6, 1/2, 7/6, (a - #1)/(a - b)]*(a - #1)^(1/6)*Sqrt[(-b + #1)/(a - b)])/Sqrt[b - #1] & ][C[1] + Inactive[Integrate][I*f[K[2]]^(1/6), {K[2], 1, x}]]}, {y[x] -> InverseFunction[(-6*Hypergeometric2F1[1/6, 1/2, 7/6, (a - #1)/(a - b)]*(a - #1)^(1/6)*Sqrt[(-b + #1)/(a - b)])/Sqrt[b - #1] & ][C[1] + Inactive[Integrate][-((-1)^(1/6)*f[K[3]]^(1/6)), {K[3], 1, x}]]}, {y[x] -> InverseFunction[(-6*Hypergeometric2F1[1/6, 1/2, 7/6, (a - #1)/(a - b)]*(a - #1)^(1/6)*Sqrt[(-b + #1)/(a - b)])/Sqrt[b - #1] & ][C[1] + Inactive[Integrate][(-1)^(1/6)*f[K[4]]^(1/6), {K[4], 1, x}]]}, {y[x] -> InverseFunction[(-6*Hypergeometric2F1[1/6, 1/2, 7/6, (a - #1)/(a - b)]*(a - #1)^(1/6)*Sqrt[(-b + #1)/(a - b)])/Sqrt[b - #1] & ][C[1] + Inactive[Integrate][-((-1)^(5/6)*f[K[5]]^(1/6)), {K[5], 1, x}]]}, {y[x] -> InverseFunction[(-6*Hypergeometric2F1[1/6, 1/2, 7/6, (a - #1)/(a - b)]*(a - #1)^(1/6)*Sqrt[(-b + #1)/(a - b)])/Sqrt[b - #1] & ][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)^3 = 0, y(x))
Maple raw output
[Intat(1/(_a-a)^(5/6)/(_a-b)^(1/2),_a = y(x))+Intat(-(-f(_a)*(b-y(x))^3*(a-y(x))^5)^(1/6)/(y(x)-a)^(5/6)/(y(x)-b)^(1/2),_a = x)+_C1 = 0]