I came across this problem in the practice of an old book. I looked into it, and find with surprise that it's really hard.
This problem is in a Complex Analysis book, and it definitely uses some technique of complex integral. My tentative idea is to split it into partial fractions, and apply the residue thm to each part; however, partial fraction is really hard here... I guess there must be some fast way to do it.
$$ I = \int_0^\infty \frac{x^{2m}}{\prod_{j=1}^{n}(a_j^{2m_j}+x^{2m_j})}dx. $$ where $$ \sum_{j=1}^{n} 2m_j > 2m + 2 $$.