The Maple 18 code $$ A := eval(convert(hypergeom([-a, k-a-1/2], [k-a], y), FPS, y), y = 4*x*t*(-t*x+1))\,assuming k::posint:$$ $$int(A*t^{-a}*(1-t)^N*(-t*x+1){-a}, t = 0 .. 1)\,assuming a>0,a<1,N::posint,x>0,x<1 $$ outputs the integral under consideration as the series
$$\Gamma(N+1)\times$$
 $$\sum\limits_{k1=0}^\infty \frac{\mathop{\rm pochhammer}(-a,k1)x^{k1}2^{2k1}{\mbox{$_2$F$_1$}(- k1+a,1-a+ k1;-a+ k1+2+N;x)}}{\mathop{\rm pochhammer}(k-a,k1)\Gamma(-a+k1+2+N)\mathop{\rm pochhammer}(k-a-1/2,k1)^{-1}k1!},$$
 where $\mathop{\rm pochhammer}$ is described [here](http://www.maplesoft.com/support/help/Maple/view.aspx?path=pochhammer). See [here](https://www.dropbox.com/s/r1dhmwtr8fvbanv/hypergeom.pdf) for the Maple output.