I am trying to evaluate following integral \begin{equation} F(x,i,k)=\int_{-1}^{1} \frac{t^i}{(x_0+x_1t+\ldots +x_nt^n)^k} dt, \end{equation} where $x(t)=x_0+x_1t+\ldots +x_nt^n$ is positive polynomial and $i,k$ are as followings: \begin{alignat*}{2} i = \begin{cases} 0,\ldots,2n-1 \quad &k =2, \\ 0,\ldots ,3n-1 \quad &k=3, \\ 0,\ldots ,4n-1 \quad &k=4. \end{cases} \end{alignat*}
The questions are
1) Is there any way that the integrals can be computed analytically?
2) Is there any way that the integrals can be computed inductively, this is, given fixed $x(t)$ is there any way $F(x,i,k)$ can be computed base on knowing $F(x,i-1,k)$, $F(x,i,k-1)$ and $F(x,i-1,k-1)$ or other previous terms?