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 = 1.51623 (sec), leaf count = 397
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \sqrt [3]{\frac {\text {$\#$1}-c}{a-c}} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{2/3} (c-\text {$\#$1})^{2/3}}\& \right ]\left [\int _1^x -\sqrt [3]{f(K[1])} \, dK[1]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \sqrt [3]{\frac {\text {$\#$1}-c}{a-c}} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{2/3} (c-\text {$\#$1})^{2/3}}\& \right ]\left [\int _1^x \sqrt [3]{-1} \sqrt [3]{f(K[2])} \, dK[2]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \sqrt [3]{\frac {\text {$\#$1}-c}{a-c}} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{2/3} (c-\text {$\#$1})^{2/3}}\& \right ]\left [\int _1^x -(-1)^{2/3} \sqrt [3]{f(K[3])} \, dK[3]+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 0.567 (sec), leaf count = 275
\[ \left \{ \int ^{y \left ( x \right ) }\! \left ( - \left ( -{\it \_a}+c \right ) \left ( -{\it \_a}+b \right ) \left ( -{\it \_a}+a \right ) \right ) ^{-{\frac {2}{3}}}{d{\it \_a}}+\int ^{x}\!-{1\sqrt [3]{-f \left ( {\it \_a} \right ) \left ( c-y \left ( x \right ) \right ) ^{2} \left ( b-y \left ( x \right ) \right ) ^{2} \left ( a-y \left ( x \right ) \right ) ^{2}} \left ( - \left ( c-y \left ( x \right ) \right ) \left ( b-y \left ( x \right ) \right ) \left ( a-y \left ( x \right ) \right ) \right ) ^{-{\frac {2}{3}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\! \left ( - \left ( -{\it \_a}+c \right ) \left ( -{\it \_a}+b \right ) \left ( -{\it \_a}+a \right ) \right ) ^{-{\frac {2}{3}}}{d{\it \_a}}+\int ^{x}\!-{\frac {i\sqrt {3}-1}{2}\sqrt [3]{-f \left ( {\it \_a} \right ) \left ( c-y \left ( x \right ) \right ) ^{2} \left ( b-y \left ( x \right ) \right ) ^{2} \left ( a-y \left ( x \right ) \right ) ^{2}} \left ( - \left ( c-y \left ( x \right ) \right ) \left ( b-y \left ( x \right ) \right ) \left ( a-y \left ( x \right ) \right ) \right ) ^{-{\frac {2}{3}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\! \left ( - \left ( -{\it \_a}+c \right ) \left ( -{\it \_a}+b \right ) \left ( -{\it \_a}+a \right ) \right ) ^{-{\frac {2}{3}}}{d{\it \_a}}+\int ^{x}\!{\frac {i\sqrt {3}+1}{2}\sqrt [3]{-f \left ( {\it \_a} \right ) \left ( c-y \left ( x \right ) \right ) ^{2} \left ( b-y \left ( x \right ) \right ) ^{2} \left ( a-y \left ( x \right ) \right ) ^{2}} \left ( - \left ( c-y \left ( x \right ) \right ) \left ( b-y \left ( x \right ) \right ) \left ( a-y \left ( x \right ) \right ) \right ) ^{-{\frac {2}{3}}}}{d{\it \_a}}+{\it \_C1}=0 \right \} \] 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)*((-b + #1)/(a - b))^(2/3)*((-c + #1)/(a - c))^(1/3))/((b - #1)^(2/3)*(c - #1)^(2/3)) & ][C[1] + 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)*((-b + #1)/(a - b))^(2/3)*((-c + #1)/(a - c))^(1/3))/((b - #1)^(2/3)*(c - #1)^(2/3)) & ][C[1] + 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)*((-b + #1)/(a - b))^(2/3)*((-c + #1)/(a - c))^(1/3))/((b - #1)^(2/3)*(c - #1)^(2/3)) & ][C[1] + 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),'implicit')
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)*(I*3^(1/2)+1)/(-(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)*(I*3^(1/2)-1)/(-(c-y(x))*(b-y(x))*(a-y(x)))^(2/3),_a = x)+_C1 = 0