I've been solving problems from my Complex Analysis course, and got stuck in this one that asks me to find the value of the integral $$\int_0^\infty\frac{x}{(1+x)^6}dx$$ using the Residues Theorem. It's easy to see that there's just one isolated singularity in this function, which is $x=-1$, and it's clearly a pole of order 6. I've checked all the different types of functions that appear in my course notes to identify which lines must I use to find the solution but I don't find a way to make this one. The only thing I can think of is using 4 curves: one that goes from $-1+\varepsilon$ to $R$ in the real axis, then one semicircumference from $R$ to $-R-1$, then a line from $-R-1$ to $-1-\varepsilon$ and finally a little semicurcumference from $-1-\varepsilon$ to $-1+\varepsilon$, but I'm not sure if this would even work.
Are the curves I proposed correct for doing this one? If not, how can I solve this using residues? any help will be appreciated, thanks in advance.