ODE
\[ f(x) (y(x)-a)^3 (y(x)-b)^3 (y(x)-c)^2+y'(x)^4=0 \] 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 = 1.93796 (sec), leaf count = 535
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \sqrt [4]{\frac {\text {$\#$1}-c}{a-c}} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{3/4} \sqrt {c-\text {$\#$1}}}\& \right ]\left [\int _1^x -\sqrt [4]{-1} \sqrt [4]{f(K[1])} \, dK[1]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \sqrt [4]{\frac {\text {$\#$1}-c}{a-c}} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{3/4} \sqrt {c-\text {$\#$1}}}\& \right ]\left [\int _1^x \sqrt [4]{-1} \sqrt [4]{f(K[2])} \, dK[2]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \sqrt [4]{\frac {\text {$\#$1}-c}{a-c}} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{3/4} \sqrt {c-\text {$\#$1}}}\& \right ]\left [\int _1^x -(-1)^{3/4} \sqrt [4]{f(K[3])} \, dK[3]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \sqrt [4]{\frac {\text {$\#$1}-c}{a-c}} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{3/4} \sqrt {c-\text {$\#$1}}}\& \right ]\left [\int _1^x (-1)^{3/4} \sqrt [4]{f(K[4])} \, dK[4]+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 0.746 (sec), leaf count = 366
\[ \left \{ \int ^{y \left ( x \right ) }\!{1 \left ( {\it \_a}-b \right ) ^{-{\frac {3}{4}}} \left ( {\it \_a}-a \right ) ^{-{\frac {3}{4}}}{\frac {1}{\sqrt {{\it \_a}-c}}}}{d{\it \_a}}+\int ^{x}\!{1\sqrt [4]{-f \left ( {\it \_a} \right ) \left ( c-y \left ( x \right ) \right ) ^{2} \left ( b-y \left ( x \right ) \right ) ^{3} \left ( a-y \left ( x \right ) \right ) ^{3}} \left ( y \left ( x \right ) -b \right ) ^{-{\frac {3}{4}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {3}{4}}}{\frac {1}{\sqrt {y \left ( x \right ) -c}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\!{1 \left ( {\it \_a}-b \right ) ^{-{\frac {3}{4}}} \left ( {\it \_a}-a \right ) ^{-{\frac {3}{4}}}{\frac {1}{\sqrt {{\it \_a}-c}}}}{d{\it \_a}}+\int ^{x}\!{-i\sqrt [4]{-f \left ( {\it \_a} \right ) \left ( c-y \left ( x \right ) \right ) ^{2} \left ( b-y \left ( x \right ) \right ) ^{3} \left ( a-y \left ( x \right ) \right ) ^{3}} \left ( y \left ( x \right ) -b \right ) ^{-{\frac {3}{4}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {3}{4}}}{\frac {1}{\sqrt {y \left ( x \right ) -c}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\!{1 \left ( {\it \_a}-b \right ) ^{-{\frac {3}{4}}} \left ( {\it \_a}-a \right ) ^{-{\frac {3}{4}}}{\frac {1}{\sqrt {{\it \_a}-c}}}}{d{\it \_a}}+\int ^{x}\!{i\sqrt [4]{-f \left ( {\it \_a} \right ) \left ( c-y \left ( x \right ) \right ) ^{2} \left ( b-y \left ( x \right ) \right ) ^{3} \left ( a-y \left ( x \right ) \right ) ^{3}} \left ( y \left ( x \right ) -b \right ) ^{-{\frac {3}{4}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {3}{4}}}{\frac {1}{\sqrt {y \left ( x \right ) -c}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\!{1 \left ( {\it \_a}-b \right ) ^{-{\frac {3}{4}}} \left ( {\it \_a}-a \right ) ^{-{\frac {3}{4}}}{\frac {1}{\sqrt {{\it \_a}-c}}}}{d{\it \_a}}+\int ^{x}\!-{1\sqrt [4]{-f \left ( {\it \_a} \right ) \left ( c-y \left ( x \right ) \right ) ^{2} \left ( b-y \left ( x \right ) \right ) ^{3} \left ( a-y \left ( x \right ) \right ) ^{3}} \left ( y \left ( x \right ) -b \right ) ^{-{\frac {3}{4}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {3}{4}}}{\frac {1}{\sqrt {y \left ( x \right ) -c}}}}{d{\it \_a}}+{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[f[x]*(-a + y[x])^3*(-b + y[x])^3*(-c + y[x])^2 + y'[x]^4 == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[(-4*Hypergeometric2F1[1/4, 3/4, 5/4, ((-b + c)*(a - #1))/((a - b)*(c - #1))]*(a - #1)^(1/4)*((-b + #1)/(a - b))^(3/4)*((-c + #1)/(a - c))^(1/4))/((b - #1)^(3/4)*Sqrt[c - #1]) & ][C[1] + Integrate[-((-1)^(1/4)*f[K[1]]^(1/4)), {K[1], 1, x}]]}, {y[x] -> InverseFunction[(-4*Hypergeometric2F1[1/4, 3/4, 5/4, ((-b + c)*(a - #1))/((a - b)*(c - #1))]*(a - #1)^(1/4)*((-b + #1)/(a - b))^(3/4)*((-c + #1)/(a - c))^(1/4))/((b - #1)^(3/4)*Sqrt[c - #1]) & ][C[1] + Integrate[(-1)^(1/4)*f[K[2]]^(1/4), {K[2], 1, x}]]}, {y[x] -> InverseFunction[(-4*Hypergeometric2F1[1/4, 3/4, 5/4, ((-b + c)*(a - #1))/((a - b)*(c - #1))]*(a - #1)^(1/4)*((-b + #1)/(a - b))^(3/4)*((-c + #1)/(a - c))^(1/4))/((b - #1)^(3/4)*Sqrt[c - #1]) & ][C[1] + Integrate[-((-1)^(3/4)*f[K[3]]^(1/4)), {K[3], 1, x}]]}, {y[x] -> InverseFunction[(-4*Hypergeometric2F1[1/4, 3/4, 5/4, ((-b + c)*(a - #1))/((a - b)*(c - #1))]*(a - #1)^(1/4)*((-b + #1)/(a - b))^(3/4)*((-c + #1)/(a - c))^(1/4))/((b - #1)^(3/4)*Sqrt[c - #1]) & ][C[1] + Integrate[(-1)^(3/4)*f[K[4]]^(1/4), {K[4], 1, x}]]}}
Maple raw input
dsolve(diff(y(x),x)^4+f(x)*(y(x)-a)^3*(y(x)-b)^3*(y(x)-c)^2 = 0, y(x),'implicit')
Maple raw output
Intat(1/(_a-b)^(3/4)/(_a-a)^(3/4)/(_a-c)^(1/2),_a = y(x))+Intat(-(-f(_a)*(c-y(x))^2*(b-y(x))^3*(a-y(x))^3)^(1/4)/(y(x)-b)^(3/4)/(y(x)-a)^(3/4)/(y(x)-c)^(1/2),_a = x)+_C1 = 0, Intat(1/(_a-b)^(3/4)/(_a-a)^(3/4)/(_a-c)^(1/2),_a = y(x))+Intat(I*(-f(_a)*(c-y(x))^2*(b-y(x))^3*(a-y(x))^3)^(1/4)/(y(x)-b)^(3/4)/(y(x)-a)^(3/4)/(y(x)-c)^(1/2),_a = x)+_C1 = 0, Intat(1/(_a-b)^(3/4)/(_a-a)^(3/4)/(_a-c)^(1/2),_a = y(x))+Intat(-I*(-f(_a)*(c-y(x))^2*(b-y(x))^3*(a-y(x))^3)^(1/4)/(y(x)-b)^(3/4)/(y(x)-a)^(3/4)/(y(x)-c)^(1/2),_a = x)+_C1 = 0, Intat(1/(_a-b)^(3/4)/(_a-a)^(3/4)/(_a-c)^(1/2),_a = y(x))+Intat((-f(_a)*(c-y(x))^2*(b-y(x))^3*(a-y(x))^3)^(1/4)/(y(x)-b)^(3/4)/(y(x)-a)^(3/4)/(y(x)-c)^(1/2),_a = x)+_C1 = 0