Integral including Legendre and radical I would appreciate if you help me with the integral $\int\sqrt{\beta^2 - x^2} P_n(x) dx$. As simple as it may look, I could not find it in the table of integrals of different handbooks.
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$\ds{\int\root{\beta^{2} - x^{2}}{\rm P}_n\pars{x}\,\dd x:\ {\large ?}}$

\begin{align}
&{1 \over \root{1 - 2xh + h^{2}}}=\sum_{n = 0}^{\infty}h^{n}{\rm P}_{n}\pars{x}
\\[3mm]&\imp\quad
\int{\root{\beta^{2} - x^{2}} \over \root{1 - 2xh + h^{2}}}\,\dd x
=\sum_{n = 0}^{\infty}h^{n}\int\root{\beta^{2} - x^{2}}{\rm P}_{n}\pars{x}\,\dd x
\end{align}
  Integrates the left hand side and expands in power of $\ds{h}$. Mathematica can do that !!!.

