# Separable non-linear ODE (with radicals)

I am trying to solve the equation

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

$$y(0) = 0$$; $$t_{0}=0$$; $$\alpha$$, $$\beta$$ 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

$$\int_{e^{-y(t)}}^{1}{\frac{dx}{x\sqrt{(\gamma-1+\frac{2 \alpha\beta}{2\alpha-1})x^{2\alpha}-\frac{2 \alpha\beta}{2\alpha-1}x+1}}}=t\tag{2}$$

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!