In short you should first decompose rational function into a sum of partial fractions, and after integrating that sum should be relatively easy.

For more information search the internet for the method of integration by partial fractions. 

Thanks for your suggestion.

I am familiar with partial fraction method. For polynomials of lower degree (2 or 3), this method is good. How about polynomials of higher degree? Factorizing is not an easy task.

Here you can find a method for finding roots of higher level polynominals, and therefore factorization (take note on: finding eigenvalues(λ) of given matrix, and roots of polynominal ri=1/λi):

http://mathworld.wolfram.com/PolynomialRoots.html

Thanks for the  information. I forgot to mention something in my question. The coefficients of the polynomials are from real, i.e., they are not assigned any specific numeric values. If it were some numbers, then it make sense to use partial fraction method with the help of computer.