I have developed a differential equation for the variation of a star's semi-major axis with respect to its eccentricity.

It is as follows:

$$\frac{dy}{dx}=\frac{12}{19}\frac{y\left(1+\left(\frac{73}{24}x^2\right)+\left(\frac{37}{26}x^4\right)\right)}{x\left(1+\left(\frac{121}{304}x^2\right)\right)}$$

Where $y$ is the semi-major axis and $x$ is the eccentricity.

Where $y$ is the semi-major axis and $x$ is the eccentricity. The 3-D plots of this equation can be found [here](http://www.wolframalpha.com/input/?i=3D+plot++%5Cfrac%7B12%7D%7B19%7D%5Cfrac%7By%5Cleft(1%2B%5Cleft(%5Cfrac%7B73%7D%7B24%7Dx%5E2%5Cright)%2B%5Cleft(%5Cfrac%7B37%7D%7B26%7Dx%5E4%5Cright)%5Cright)%7D%7Bx%5Cleft(1%2B%5Cleft(%5Cfrac%7B121%7D%7B304%7Dx%5E2%5Cright)%5Cright)%7D)

And this is the solution to the above DE [here](http://www.wolframalpha.com/input/?i=solve+y'%3D+%5Cfrac%7B12%7D%7B19%7D%5Cfrac%7By%5Cleft(1%2B%5Cleft(%5Cfrac%7B73%7D%7B24%7Dx%5E2%5Cright)%2B%5Cleft(%5Cfrac%7B37%7D%7B26%7Dx%5E4%5Cright)%5Cright)%7D%7Bx%5Cleft(1%2B%5Cleft(%5Cfrac%7B121%7D%7B304%7Dx%5E2%5Cright)%5Cright)%7D)

The decay time of stars can be found by solving the following integral:

$$T(a_{0},e_{0})=\frac{12(c_{0}^4)}{19\gamma}\int_{0}^{e_0}{\frac{e^{29/19}[1+(121/304)e^2]^{1181/2299}}{(1-e^2)^{3/2}}}de\tag1$$

Where $$\gamma=\frac{64G^3}{5c^5}m_{1}m_{2}(m_{1}+m_{2})$$

For $e_{0}$ close to $1$ the equation becomes:

$$T(a_{0},e_{0})\approx\frac{768}{425}T_{f}a_{0}(1-e_{0}^2)^{7/2}\tag2$$

Where $$T_{f}=\frac{a_{0}^4}{4\gamma}$$

I used Appell's hypergeometric functions to solve integral (1), but is there any way in which I can express the solutions in terms of few special functions with simpler symmetries, so that the analysis becomes easier.There is a well defined symmetry for the above equation from the plot. Hence, is it possible to express this in terms of other special function (which have different symmetries).

EDIT: I was suggested that since the powers in the integrand in equation (1) are very non-trivial, probably the hypergeometric function can't be further simplified. But I fail to understand why this might seem to pose a problem. Can't this D.E. be solved by Lie symmetry methods? Or can this solution's field be treated using Frobenius' theorem and the dimensions of it analysed?