Brezis excercise 4.1: I am asked to show under what conditions does $f(x)=\{1+|x|^\alpha\}^{-1}\{1+|\log|x||^\beta\}^{-1}$ belong to $L_p(\mathbb{R}^n)$ where $\alpha,\beta>0$..
I know that for $f$ to belong is must show that $\int_{(\mathbb{R^n})}|f|^p<\infty$, so far I got this:
$$|f(x)|^p=\dfrac{1}{|1+|x|^\alpha|^p|1+|\log|x||^\beta|^p}$$ however I don't know what to do after this. Thanks in advance.