ODE
\[ f(x) (y(x)-a)^2 (y(x)-b)^2 (y(x)-c)^2+y'(x)^3=0 \] 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 = 1.22529 (sec), leaf count = 406
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt [3]{c-\text {$\#$1}} \left (\frac {(b-\text {$\#$1}) (a-c)}{(c-\text {$\#$1}) (a-b)}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{2/3} (a-c)}\& \right ]\left [\int _1^x-\sqrt [3]{f(K[1])}dK[1]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt [3]{c-\text {$\#$1}} \left (\frac {(b-\text {$\#$1}) (a-c)}{(c-\text {$\#$1}) (a-b)}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{2/3} (a-c)}\& \right ]\left [\int _1^x\sqrt [3]{-1} \sqrt [3]{f(K[2])}dK[2]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt [3]{c-\text {$\#$1}} \left (\frac {(b-\text {$\#$1}) (a-c)}{(c-\text {$\#$1}) (a-b)}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{2/3} (a-c)}\& \right ]\left [\int _1^x-(-1)^{2/3} \sqrt [3]{f(K[3])}dK[3]+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 1.339 (sec), leaf count = 275
\[\left [\int _{}^{y \left (x \right )}\frac {1}{\left (-\left (-\textit {\_a} +c \right ) \left (-\textit {\_a} +b \right ) \left (-\textit {\_a} +a \right )\right )^{\frac {2}{3}}}d \textit {\_a} +\int _{}^{x}-\frac {\left (-f \left (\textit {\_a} \right ) \left (c -y \left (x \right )\right )^{2} \left (b -y \left (x \right )\right )^{2} \left (a -y \left (x \right )\right )^{2}\right )^{\frac {1}{3}}}{\left (-\left (c -y \left (x \right )\right ) \left (b -y \left (x \right )\right ) \left (a -y \left (x \right )\right )\right )^{\frac {2}{3}}}d \textit {\_a} +\textit {\_C1} = 0, \int _{}^{y \left (x \right )}\frac {1}{\left (-\left (-\textit {\_a} +c \right ) \left (-\textit {\_a} +b \right ) \left (-\textit {\_a} +a \right )\right )^{\frac {2}{3}}}d \textit {\_a} +\int _{}^{x}\frac {\left (-f \left (\textit {\_a} \right ) \left (c -y \left (x \right )\right )^{2} \left (b -y \left (x \right )\right )^{2} \left (a -y \left (x \right )\right )^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{2 \left (-\left (c -y \left (x \right )\right ) \left (b -y \left (x \right )\right ) \left (a -y \left (x \right )\right )\right )^{\frac {2}{3}}}d \textit {\_a} +\textit {\_C1} = 0, \int _{}^{y \left (x \right )}\frac {1}{\left (-\left (-\textit {\_a} +c \right ) \left (-\textit {\_a} +b \right ) \left (-\textit {\_a} +a \right )\right )^{\frac {2}{3}}}d \textit {\_a} +\int _{}^{x}-\frac {\left (-f \left (\textit {\_a} \right ) \left (c -y \left (x \right )\right )^{2} \left (b -y \left (x \right )\right )^{2} \left (a -y \left (x \right )\right )^{2}\right )^{\frac {1}{3}} \left (-1+i \sqrt {3}\right )}{2 \left (-\left (c -y \left (x \right )\right ) \left (b -y \left (x \right )\right ) \left (a -y \left (x \right )\right )\right )^{\frac {2}{3}}}d \textit {\_a} +\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[f[x]*(-a + y[x])^2*(-b + y[x])^2*(-c + y[x])^2 + y'[x]^3 == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[(3*Hypergeometric2F1[1/3, 2/3, 4/3, ((-b + c)*(a - #1))/((a - b)*(c - #1))]*(a - #1)^(1/3)*(((a - c)*(b - #1))/((a - b)*(c - #1)))^(2/3)*(c - #1)^(1/3))/((a - c)*(b - #1)^(2/3)) & ][C[1] + Inactive[Integrate][-f[K[1]]^(1/3), {K[1], 1, x}]]}, {y[x] -> InverseFunction[(3*Hypergeometric2F1[1/3, 2/3, 4/3, ((-b + c)*(a - #1))/((a - b)*(c - #1))]*(a - #1)^(1/3)*(((a - c)*(b - #1))/((a - b)*(c - #1)))^(2/3)*(c - #1)^(1/3))/((a - c)*(b - #1)^(2/3)) & ][C[1] + Inactive[Integrate][(-1)^(1/3)*f[K[2]]^(1/3), {K[2], 1, x}]]}, {y[x] -> InverseFunction[(3*Hypergeometric2F1[1/3, 2/3, 4/3, ((-b + c)*(a - #1))/((a - b)*(c - #1))]*(a - #1)^(1/3)*(((a - c)*(b - #1))/((a - b)*(c - #1)))^(2/3)*(c - #1)^(1/3))/((a - c)*(b - #1)^(2/3)) & ][C[1] + Inactive[Integrate][-((-1)^(2/3)*f[K[3]]^(1/3)), {K[3], 1, x}]]}}
Maple raw input
dsolve(diff(y(x),x)^3+f(x)*(y(x)-a)^2*(y(x)-b)^2*(y(x)-c)^2 = 0, y(x))
Maple raw output
[Intat(1/(-(-_a+c)*(-_a+b)*(-_a+a))^(2/3),_a = y(x))+Intat(-(-f(_a)*(c-y(x))^2*(b-y(x))^2*(a-y(x))^2)^(1/3)/(-(c-y(x))*(b-y(x))*(a-y(x)))^(2/3),_a = x)+_C1 = 0, Intat(1/(-(-_a+c)*(-_a+b)*(-_a+a))^(2/3),_a = y(x))+Intat(1/2*(-f(_a)*(c-y(x))^2*(b-y(x))^2*(a-y(x))^2)^(1/3)*(1+I*3^(1/2))/(-(c-y(x))*(b-y(x))*(a-y(x)))^(2/3),_a = x)+_C1 = 0, Intat(1/(-(-_a+c)*(-_a+b)*(-_a+a))^(2/3),_a = y(x))+Intat(-1/2*(-f(_a)*(c-y(x))^2*(b-y(x))^2*(a-y(x))^2)^(1/3)*(-1+I*3^(1/2))/(-(c-y(x))*(b-y(x))*(a-y(x)))^(2/3),_a = x)+_C1 = 0]