#### 2.249   ODE No. 249

$y'(x) \left (a x y(x)+b x^n\right )+\alpha y(x)^3+\beta y(x)^2=0$ Mathematica : cpu = 3.52738 (sec), leaf count = 115

$\text {Solve}\left [\frac {(a (-n)+a+\alpha y(x)) y(x)^{\frac {a-a n}{\beta }-1} (\alpha y(x)+\beta )^{\frac {a (n-1)}{\beta }}}{a^2 (n-1)^2 (a (n-1)+\beta )}+\frac {x^{1-n} \exp \left (-\frac {a (n-1) (\log (y(x))-\log (\alpha y(x)+\beta ))}{\beta }\right )}{a b (1-n) (n-1)}=c_1,y(x)\right ]$ Maple : cpu = 0.171 (sec), leaf count = 202

$\left \{ y \left ( x \right ) ={\beta \left ( {\it RootOf} \left ( -{x}^{1-n}{{\it \_Z}}^{{\frac {a \left ( n-1 \right ) }{\beta }}}{a}^{2}\beta \,n+{\it \_C1}\,{a}^{2}b{n}^{2}+{x}^{1-n}{{\it \_Z}}^{{\frac {a \left ( n-1 \right ) }{\beta }}}{a}^{2}\beta -{x}^{1-n}{{\it \_Z}}^{{\frac {a \left ( n-1 \right ) }{\beta }}}a{\beta }^{2}-{{\it \_Z}}^{{\frac {an-a+\beta }{\beta }}}\beta \,abn+{{\it \_Z}}^{{\frac {a \left ( n-1 \right ) }{\beta }}}a\alpha \,bn-2\,{\it \_C1}\,{a}^{2}bn+{\it \_C1}\,ab\beta \,n+{{\it \_Z}}^{{\frac {an-a+\beta }{\beta }}}\beta \,ab-{{\it \_Z}}^{{\frac {a \left ( n-1 \right ) }{\beta }}}a\alpha \,b+{{\it \_Z}}^{{\frac {a \left ( n-1 \right ) }{\beta }}}\alpha \,b\beta +{\it \_C1}\,{a}^{2}b-{\it \_C1}\,ab\beta \right ) \beta -\alpha \right ) ^{-1}} \right \}$