Approximation of an integral involving x and 1-x

Question

I am looking for an approximation of the integral

$F(k,R)=\displaystyle\int_0^1\frac{\mathrm{d}x}{(Rx)^{-2}+(1-x)^{-k}}$,

that is valid to within 1% over the range $2<k<10$ and $R>1$. Is there a standard approach to problems like this?

This integral came about in a research problem where I am trying to smoothly connect the functions $x^{2}$ and $(1-x)^{k}$. If someone can provide an approximation (or a closed form solution!) I'll gladly acknowledge them in the research paper this result is needed for.

Answer 1

The final solution was to use Gauss-Laguerre quadrature, with an order 10 scheme being sufficient to achieve the accuracy I required.