Gamma Functions – How to Solve This Integral Using Gamma Functions

Question

I have to evaluate following integral.

$$ \int_{-c}^c \sqrt{b+\frac{c^6}{x^6-c^6}}dx$$

I know that final answer includes gamma function entries like ($\frac{\Gamma(1/6)\Gamma(1/2)}{\Gamma(2/3)})$.

But, I'm having difficulty figuring out how to integrate this. Any help will be highly appreciated.

Thank You,

Answer 1

The result (below) as expected includes $ \frac{\Gamma(1/6)\Gamma(1/2)}{\Gamma(2/3)} = \sqrt{\pi}\frac{\Gamma(1/6)}{\Gamma(2/3)}$ and an hypergeometric function.

See Eq.(9) in : http://mathworld.wolfram.com/AppellHypergeometricFunction.html

For some particular values of $b$, the hypergeometric function reduces to functions of lower level.