If you can find a differential equation of the form
$$y'=F(y)$$ for a quintic $y$, its inverse would be expressed as
$$x=\text{const.}+\int\frac{dy}{F(y)}.$$
Differentiating any quintic yields a quartic, and by eliminating between the quintic and its derivative, we may have it such that the coefficients work out nicely in order to isolate such an explicit function as $F(y)$ from some $\Phi(y,y')=0$. For example, in the quadratic case
$$y=x^2+ax+b$$ we have
$$y'=2x+a\iff x=\frac{y'-a}{2}$$ and therefore,
$$4y=y'^2+4b-a^2$$ so
$$x=\int \frac{dy}{\sqrt{4y+a^2-4b}}$$ very conveniently. Thus my question: can the coefficients be manipulated for this to work. Are there enough "degrees of freedom" so to speak?