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 = 2.96344 (sec), leaf count = 797
\[\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 {\text {$\#$1}-b}{a-b}\right )^{2/3} \sqrt [3]{\frac {\text {$\#$1}-c}{a-c}}}{(b-\text {$\#$1})^{2/3} \sqrt {c-\text {$\#$1}}}\& \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 {\text {$\#$1}-b}{a-b}\right )^{2/3} \sqrt [3]{\frac {\text {$\#$1}-c}{a-c}}}{(b-\text {$\#$1})^{2/3} \sqrt {c-\text {$\#$1}}}\& \right ]\left [c_1+\int _1^x i \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 {\text {$\#$1}-b}{a-b}\right )^{2/3} \sqrt [3]{\frac {\text {$\#$1}-c}{a-c}}}{(b-\text {$\#$1})^{2/3} \sqrt {c-\text {$\#$1}}}\& \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 {\text {$\#$1}-b}{a-b}\right )^{2/3} \sqrt [3]{\frac {\text {$\#$1}-c}{a-c}}}{(b-\text {$\#$1})^{2/3} \sqrt {c-\text {$\#$1}}}\& \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 {\text {$\#$1}-b}{a-b}\right )^{2/3} \sqrt [3]{\frac {\text {$\#$1}-c}{a-c}}}{(b-\text {$\#$1})^{2/3} \sqrt {c-\text {$\#$1}}}\& \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 {\text {$\#$1}-b}{a-b}\right )^{2/3} \sqrt [3]{\frac {\text {$\#$1}-c}{a-c}}}{(b-\text {$\#$1})^{2/3} \sqrt {c-\text {$\#$1}}}\& \right ]\left [c_1+\int _1^x (-1)^{5/6} \sqrt [6]{f(K[6])} \, dK[6]\right ]\right \}\right \}\]
Maple ✓
cpu = 2.018 (sec), leaf count = 578
\[ \left \{ \int ^{y \left ( x \right ) }\!{1 \left ( {\it \_a}-b \right ) ^{-{\frac {2}{3}}} \left ( {\it \_a}-a \right ) ^{-{\frac {5}{6}}}{\frac {1}{\sqrt {{\it \_a}-c}}}}{d{\it \_a}}+\int ^{x}\!{1\sqrt [6]{-f \left ( {\it \_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}} \left ( y \left ( x \right ) -b \right ) ^{-{\frac {2}{3}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {5}{6}}}{\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 {2}{3}}} \left ( {\it \_a}-a \right ) ^{-{\frac {5}{6}}}{\frac {1}{\sqrt {{\it \_a}-c}}}}{d{\it \_a}}+\int ^{x}\!-{1\sqrt [6]{-f \left ( {\it \_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}} \left ( y \left ( x \right ) -b \right ) ^{-{\frac {2}{3}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {5}{6}}}{\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 {2}{3}}} \left ( {\it \_a}-a \right ) ^{-{\frac {5}{6}}}{\frac {1}{\sqrt {{\it \_a}-c}}}}{d{\it \_a}}+\int ^{x}\!-{\frac {i\sqrt {3}-1}{2}\sqrt [6]{-f \left ( {\it \_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}} \left ( y \left ( x \right ) -b \right ) ^{-{\frac {2}{3}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {5}{6}}}{\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 {2}{3}}} \left ( {\it \_a}-a \right ) ^{-{\frac {5}{6}}}{\frac {1}{\sqrt {{\it \_a}-c}}}}{d{\it \_a}}+\int ^{x}\!{\frac {i\sqrt {3}-1}{2}\sqrt [6]{-f \left ( {\it \_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}} \left ( y \left ( x \right ) -b \right ) ^{-{\frac {2}{3}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {5}{6}}}{\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 {2}{3}}} \left ( {\it \_a}-a \right ) ^{-{\frac {5}{6}}}{\frac {1}{\sqrt {{\it \_a}-c}}}}{d{\it \_a}}+\int ^{x}\!-{\frac {i\sqrt {3}+1}{2}\sqrt [6]{-f \left ( {\it \_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}} \left ( y \left ( x \right ) -b \right ) ^{-{\frac {2}{3}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {5}{6}}}{\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 {2}{3}}} \left ( {\it \_a}-a \right ) ^{-{\frac {5}{6}}}{\frac {1}{\sqrt {{\it \_a}-c}}}}{d{\it \_a}}+\int ^{x}\!{\frac {i\sqrt {3}+1}{2}\sqrt [6]{-f \left ( {\it \_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}} \left ( y \left ( x \right ) -b \right ) ^{-{\frac {2}{3}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {5}{6}}}{\frac {1}{\sqrt {y \left ( x \right ) -c}}}}{d{\it \_a}}+{\it \_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)*((-b + #1)/(a - b))^(2/3)*((-c + #1)/(a - c))^(1/3))/((b - #1)^(2/3)*Sqrt[c - #1]) & ][C[1] + 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)*((-b + #1)/(a - b))^(2/3)*((-c + #1)/(a - c))^(1/3))/((b - #1)^(2/3)*Sqrt[c - #1]) & ][C[1] + 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)*((-b + #1)/(a - b))^(2/3)*((-c + #1)/(a - c))^(1/3))/((b - #1)^(2/3)*Sqrt[c - #1]) & ][C[1] + 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)*((-b + #1)/(a - b))^(2/3)*((-c + #1)/(a - c))^(1/3))/((b - #1)^(2/3)*Sqrt[c - #1]) & ][C[1] + Integrate[(-1)^(1/6)*f[K[4]]^(1/6), {K[4], 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)*((-b + #1)/(a - b))^(2/3)*((-c + #1)/(a - c))^(1/3))/((b - #1)^(2/3)*Sqrt[c - #1]) & ][C[1] + 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)*((-b + #1)/(a - b))^(2/3)*((-c + #1)/(a - c))^(1/3))/((b - #1)^(2/3)*Sqrt[c - #1]) & ][C[1] + 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),'implicit')
Maple raw output
Intat(1/(_a-b)^(2/3)/(_a-a)^(5/6)/(_a-c)^(1/2),_a = y(x))+Intat(-(-f(_a)*(c-y(x))^3*(b-y(x))^4*(a-y(x))^5)^(1/6)/(y(x)-b)^(2/3)/(y(x)-a)^(5/6)/(y(x)-c)^(1/2),_a = x)+_C1 = 0, Intat(1/(_a-b)^(2/3)/(_a-a)^(5/6)/(_a-c)^(1/2),_a = y(x))+Intat(1/2*(I*3^(1/2)+1)*(-f(_a)*(c-y(x))^3*(b-y(x))^4*(a-y(x))^5)^(1/6)/(y(x)-b)^(2/3)/(y(x)-a)^(5/6)/(y(x)-c)^(1/2),_a = x)+_C1 = 0, Intat(1/(_a-b)^(2/3)/(_a-a)^(5/6)/(_a-c)^(1/2),_a = y(x))+Intat(-1/2*(I*3^(1/2)-1)*(-f(_a)*(c-y(x))^3*(b-y(x))^4*(a-y(x))^5)^(1/6)/(y(x)-b)^(2/3)/(y(x)-a)^(5/6)/(y(x)-c)^(1/2),_a = x)+_C1 = 0, Intat(1/(_a-b)^(2/3)/(_a-a)^(5/6)/(_a-c)^(1/2),_a = y(x))+Intat(1/2*(I*3^(1/2)-1)*(-f(_a)*(c-y(x))^3*(b-y(x))^4*(a-y(x))^5)^(1/6)/(y(x)-b)^(2/3)/(y(x)-a)^(5/6)/(y(x)-c)^(1/2),_a = x)+_C1 = 0, Intat(1/(_a-b)^(2/3)/(_a-a)^(5/6)/(_a-c)^(1/2),_a = y(x))+Intat(-1/2*(I*3^(1/2)+1)*(-f(_a)*(c-y(x))^3*(b-y(x))^4*(a-y(x))^5)^(1/6)/(y(x)-b)^(2/3)/(y(x)-a)^(5/6)/(y(x)-c)^(1/2),_a = x)+_C1 = 0, Intat(1/(_a-b)^(2/3)/(_a-a)^(5/6)/(_a-c)^(1/2),_a = y(x))+Intat((-f(_a)*(c-y(x))^3*(b-y(x))^4*(a-y(x))^5)^(1/6)/(y(x)-b)^(2/3)/(y(x)-a)^(5/6)/(y(x)-c)^(1/2),_a = x)+_C1 = 0