4.8.33 \(x^7 y'(x)+5 x^3 y(x)^2+2 \left (x^2+1\right ) y(x)^3=0\)

ODE
\[ x^7 y'(x)+5 x^3 y(x)^2+2 \left (x^2+1\right ) y(x)^3=0 \] ODE Classification

[_rational, _Abel]

Book solution method
Abel ODE, First kind

Mathematica
cpu = 0.457795 (sec), leaf count = 108

\[\text {Solve}\left [c_1=\frac {i \left (\sqrt [4]{\frac {x^4}{y(x)^2}+\frac {1}{x^2}+\frac {2 x}{y(x)}+1} \left (x^3+y(x)\right ) \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {3}{2};-\frac {\left (x^3+y(x)\right )^2}{x^2 y(x)^2}\right )+2 x^2 y(x)\right )}{2 x y(x) \sqrt [4]{-\frac {\left (x^3+y(x)\right )^2}{x^2 y(x)^2}-1}},y(x)\right ]\]

Maple
cpu = 0.065 (sec), leaf count = 63

\[ \left \{ {\it \_C1}+{x{\frac {1}{\sqrt [4]{ \left ({x}^{-1}+{\frac {{x}^{2}}{y \relax (x ) }} \right ) ^{2}+1}}}}+{\frac {{x}^{3}+y \relax (x ) }{2\,xy \relax (x ) }{\mbox {$_2$F$_1$}({\frac {1}{2}},{\frac {5}{4}};\,{\frac {3}{2}};\,-{\frac { \left ({x}^{3}+y \relax (x ) \right ) ^{2}}{{x}^{2} \left (y \relax (x ) \right ) ^{2}}})}}=0 \right \} \] Mathematica raw input

DSolve[5*x^3*y[x]^2 + 2*(1 + x^2)*y[x]^3 + x^7*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[C[1] == ((I/2)*(2*x^2*y[x] + Hypergeometric2F1[1/2, 5/4, 3/2, -((x^3 + y[x
])^2/(x^2*y[x]^2))]*(1 + x^(-2) + x^4/y[x]^2 + (2*x)/y[x])^(1/4)*(x^3 + y[x])))/
(x*y[x]*(-1 - (x^3 + y[x])^2/(x^2*y[x]^2))^(1/4)), y[x]]

Maple raw input

dsolve(x^7*diff(y(x),x)+5*x^3*y(x)^2+2*(x^2+1)*y(x)^3 = 0, y(x),'implicit')

Maple raw output

_C1+x/((1/x+x^2/y(x))^2+1)^(1/4)+1/2*(x^3+y(x))*hypergeom([1/2, 5/4],[3/2],-(x^3
+y(x))^2/x^2/y(x)^2)/x/y(x) = 0