Find $p$ such that $f$ defined by $f(x)=(|x|^\frac{d}{2}+|x|^d)^{-1}$ is in $L^p(R^d)$

Question

The function f is definited in $\Bbb R^d$ by $$f(x)=\frac{1}{(|x|^\frac{d}{2}+|x|^d)}$$

How do you find $p\in[0.+\infty]$ such that $f(x)\in L^p$ ?

Any help is appreciated.

My attempt:

$r=|x| $, $r\in \Bbb R_+^d$:

$$||f||_p^p=\int_0^{+\infty} \frac{dr}{r^\frac{dp}{2}(1+ r^\frac{d}{2})^p}$$

In the neighbourhood of $x=0$: $\frac{dp}{2}<1$

In the neighbourhood of $x=+\infty$ : $p>1$.