Given $1<p<\infty,$ Mathematica gave me the following result: $$\int_0^1\frac{x^2}{(1-x^2)^{\frac{p}{2}}}dx=\frac{\Gamma\left(1-\frac{p}{2}\right)\sqrt{\pi}}{4\Gamma\left(\frac{5}{2}-\frac{p}{2}\right)}$$
as long as $p<2.$ Although I am quite interested how this was determined and would appreciate the solution in your response, I'm more interested in an estimate for this integral simply to determine that it is a convergent integral for $1<p<2.$
Near zero, no problem.
Near $1$, the integrand $$\frac {x^2}{(1+x)^{p/2}(1-x)^{p/2}} $$
is equivalent $(\sim ) $ to
$$\frac {1}{2^{p/2}}\frac {1}{(1-x)^{p/2}} .$$
thus, the integral converges if and only if $$\frac {p}{2}<1$$ or $p <2$.