Cfrgtkky

I am trying to solve the equation

$$
frac{dy}{dt}=sqrt{(gamma-1+frac{2alphabeta}{2alpha-1})e^{-2alpha y}-frac{2alphabeta}{2alpha-1}e^{-y}+1}tag{1}
$$

$y(0) = 0$; $t_{0}=0$; $alpha$, $beta$ and $gamma$ are positive constants ($beta$ is also less than 1). There is also a special form for $alpha=frac{1}{2}$ but I will not dwell on this right now.

Provided I am not mistaken, letting $x = e^{-y}$ and rearranging leads to

$$
int_{e^{-y(t)}}^{1}{frac{dx}{xsqrt{(gamma-1+frac{2 alphabeta}{2alpha-1})x^{2alpha}-frac{2 alphabeta}{2alpha-1}x+1}}}=ttag{2}
$$

This integral is solvable for particular values of $alpha$ (ones that allow a partial fraction decomposition of the integrand(?))

My questions are:

1) Is there a relatively compact way that may determine if the
integral $(2)$ can be expressed in closed form?

2) Can $x(t)$ be expressed in terms of relatively well-studied special
functions?

3) Even if a general method for solving this equation (which I believe
does not exist), is there a way of obtaining an approximate solution
for $x(t)$ in closed form?

I have considered taking the Taylor series of the radical, but unfortunately this method does not work very well because the integrand does contain a region of rapid change within the integration region.

Any ideas on how to treat this problem will be greatly appreciated!