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I've been having some difficulty solving a three-term integral of the form:

[tex]\int^{\beta_{r}}_{\beta_{0}}\frac{y^{\tau-3+\frac{\eta}{3-\kappa}}}{\left(1-y^{2}\right)^{3-\frac{\tau}{2}}}\left(1+\left(\frac{1}{(3-\kappa)N}-\frac{1}{\beta_{0}}\right)y\right)^{-\frac{\eta}{3-\kappa}}dy[/tex]

Where [tex]y < 1[/tex] and [tex]\beta_{0}, \tau, \eta, \kappa, N[/tex] are constants.

I've tried expanding the term:

[tex]\left(1-y^{2}\right)^{3-\frac{\tau}{2}}}[/tex]

As a taylor series and then integrating term by term, although for this to be accurate this tends to require an expansion up to an order of 40 which cannot be done using Mathematica. It is also required that this integral be done non-numerically and expressed in terms of the limits of the integral.

It is also quite difficult expressing the solution using hypergeometric functions (eg 2F1) using these limits.

Any help would be greatly appreciated!

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# Three-term integral

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