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 = 1.54843 (sec), leaf count = 641
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt [6]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} B_{\frac {a-\text {$\#$1}}{a-b}}\left (\frac {1}{6},\frac {1}{2}\right )}{\sqrt {b-\text {$\#$1}} \sqrt [6]{\frac {a-\text {$\#$1}}{a-b}}}\& \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 {\sqrt [6]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} B_{\frac {a-\text {$\#$1}}{a-b}}\left (\frac {1}{6},\frac {1}{2}\right )}{\sqrt {b-\text {$\#$1}} \sqrt [6]{\frac {a-\text {$\#$1}}{a-b}}}\& \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 {\sqrt [6]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} B_{\frac {a-\text {$\#$1}}{a-b}}\left (\frac {1}{6},\frac {1}{2}\right )}{\sqrt {b-\text {$\#$1}} \sqrt [6]{\frac {a-\text {$\#$1}}{a-b}}}\& \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 {\sqrt [6]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} B_{\frac {a-\text {$\#$1}}{a-b}}\left (\frac {1}{6},\frac {1}{2}\right )}{\sqrt {b-\text {$\#$1}} \sqrt [6]{\frac {a-\text {$\#$1}}{a-b}}}\& \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 {\sqrt [6]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} B_{\frac {a-\text {$\#$1}}{a-b}}\left (\frac {1}{6},\frac {1}{2}\right )}{\sqrt {b-\text {$\#$1}} \sqrt [6]{\frac {a-\text {$\#$1}}{a-b}}}\& \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 {\sqrt [6]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} B_{\frac {a-\text {$\#$1}}{a-b}}\left (\frac {1}{6},\frac {1}{2}\right )}{\sqrt {b-\text {$\#$1}} \sqrt [6]{\frac {a-\text {$\#$1}}{a-b}}}\& \right ]\left [\int _1^x (-1)^{5/6} \sqrt [6]{f(K[6])} \, dK[6]+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 0.666 (sec), leaf count = 440
\[ \left \{ \int ^{y \left ( x \right ) }\!{1{\frac {1}{\sqrt {{\it \_a}-b}}} \left ( {\it \_a}-a \right ) ^{-{\frac {5}{6}}}}{d{\it \_a}}+\int ^{x}\!{1\sqrt [6]{-f \left ( {\it \_a} \right ) \left ( b-y \left ( x \right ) \right ) ^{3} \left ( a-y \left ( x \right ) \right ) ^{5}}{\frac {1}{\sqrt {y \left ( x \right ) -b}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {5}{6}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\!{1{\frac {1}{\sqrt {{\it \_a}-b}}} \left ( {\it \_a}-a \right ) ^{-{\frac {5}{6}}}}{d{\it \_a}}+\int ^{x}\!-{1\sqrt [6]{-f \left ( {\it \_a} \right ) \left ( b-y \left ( x \right ) \right ) ^{3} \left ( a-y \left ( x \right ) \right ) ^{5}}{\frac {1}{\sqrt {y \left ( x \right ) -b}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {5}{6}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\!{1{\frac {1}{\sqrt {{\it \_a}-b}}} \left ( {\it \_a}-a \right ) ^{-{\frac {5}{6}}}}{d{\it \_a}}+\int ^{x}\!-{\frac {i\sqrt {3}-1}{2}\sqrt [6]{-f \left ( {\it \_a} \right ) \left ( b-y \left ( x \right ) \right ) ^{3} \left ( a-y \left ( x \right ) \right ) ^{5}}{\frac {1}{\sqrt {y \left ( x \right ) -b}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {5}{6}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\!{1{\frac {1}{\sqrt {{\it \_a}-b}}} \left ( {\it \_a}-a \right ) ^{-{\frac {5}{6}}}}{d{\it \_a}}+\int ^{x}\!{\frac {i\sqrt {3}-1}{2}\sqrt [6]{-f \left ( {\it \_a} \right ) \left ( b-y \left ( x \right ) \right ) ^{3} \left ( a-y \left ( x \right ) \right ) ^{5}}{\frac {1}{\sqrt {y \left ( x \right ) -b}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {5}{6}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\!{1{\frac {1}{\sqrt {{\it \_a}-b}}} \left ( {\it \_a}-a \right ) ^{-{\frac {5}{6}}}}{d{\it \_a}}+\int ^{x}\!-{\frac {i\sqrt {3}+1}{2}\sqrt [6]{-f \left ( {\it \_a} \right ) \left ( b-y \left ( x \right ) \right ) ^{3} \left ( a-y \left ( x \right ) \right ) ^{5}}{\frac {1}{\sqrt {y \left ( x \right ) -b}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {5}{6}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\!{1{\frac {1}{\sqrt {{\it \_a}-b}}} \left ( {\it \_a}-a \right ) ^{-{\frac {5}{6}}}}{d{\it \_a}}+\int ^{x}\!{\frac {i\sqrt {3}+1}{2}\sqrt [6]{-f \left ( {\it \_a} \right ) \left ( b-y \left ( x \right ) \right ) ^{3} \left ( a-y \left ( x \right ) \right ) ^{5}}{\frac {1}{\sqrt {y \left ( x \right ) -b}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {5}{6}}}}{d{\it \_a}}+{\it \_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[-((Beta[(a - #1)/(a - b), 1/6, 1/2]*(a - #1)^(1/6)*Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/6)*Sqrt[b - #1])) & ][C[1] + Integrate[(-I)*f[K[1]]^(1/6), {K[1], 1, x}]]}, {y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/6, 1/2]*(a - #1)^(1/6)*Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/6)*Sqrt[b - #1])) & ][C[1] + Integrate[I*f[K[2]]^(1/6), {K[2], 1, x}]]}, {y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/6, 1/2]*(a - #1)^(1/6)*Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/6)*Sqrt[b - #1])) & ][C[1] + Integrate[-((-1)^(1/6)*f[K[3]]^(1/6)), {K[3], 1, x}]]}, {y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/6, 1/2]*(a - #1)^(1/6)*Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/6)*Sqrt[b - #1])) & ][C[1] + Integrate[(-1)^(1/6)*f[K[4]]^(1/6), {K[4], 1, x}]]}, {y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/6, 1/2]*(a - #1)^(1/6)*Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/6)*Sqrt[b - #1])) & ][C[1] + Integrate[-((-1)^(5/6)*f[K[5]]^(1/6)), {K[5], 1, x}]]}, {y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/6, 1/2]*(a - #1)^(1/6)*Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/6)*Sqrt[b - #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)^3 = 0, y(x),'implicit')
Maple raw output
Intat(1/(_a-b)^(1/2)/(_a-a)^(5/6),_a = y(x))+Intat(-(-f(_a)*(b-y(x))^3*(a-y(x))^5)^(1/6)/(y(x)-b)^(1/2)/(y(x)-a)^(5/6),_a = x)+_C1 = 0, Intat(1/(_a-b)^(1/2)/(_a-a)^(5/6),_a = y(x))+Intat(1/2*(I*3^(1/2)+1)*(-f(_a)*(b-y(x))^3*(a-y(x))^5)^(1/6)/(y(x)-b)^(1/2)/(y(x)-a)^(5/6),_a = x)+_C1 = 0, Intat(1/(_a-b)^(1/2)/(_a-a)^(5/6),_a = y(x))+Intat(-1/2*(I*3^(1/2)-1)*(-f(_a)*(b-y(x))^3*(a-y(x))^5)^(1/6)/(y(x)-b)^(1/2)/(y(x)-a)^(5/6),_a = x)+_C1 = 0, Intat(1/(_a-b)^(1/2)/(_a-a)^(5/6),_a = y(x))+Intat(1/2*(I*3^(1/2)-1)*(-f(_a)*(b-y(x))^3*(a-y(x))^5)^(1/6)/(y(x)-b)^(1/2)/(y(x)-a)^(5/6),_a = x)+_C1 = 0, Intat(1/(_a-b)^(1/2)/(_a-a)^(5/6),_a = y(x))+Intat(-1/2*(I*3^(1/2)+1)*(-f(_a)*(b-y(x))^3*(a-y(x))^5)^(1/6)/(y(x)-b)^(1/2)/(y(x)-a)^(5/6),_a = x)+_C1 = 0, Intat(1/(_a-b)^(1/2)/(_a-a)^(5/6),_a = y(x))+Intat((-f(_a)*(b-y(x))^3*(a-y(x))^5)^(1/6)/(y(x)-b)^(1/2)/(y(x)-a)^(5/6),_a = x)+_C1 = 0