Counter example to $L^{p}$ , $L^{q}$ inclusion on infinite measure spaces

Question

I want to construct a counter example to the inclusions between $L^{p}$ and $L^{q}$ when considering an infinite measure set such as $\Bbb R^{d}$.

I am trying to find a power function such that it would be integrable for $p$ but not for $q$.

if I consider the subset $(1, \infty)$, then I can choose the function $$f(x)=\frac{x^{-1/2}}{1+\ln{x}}$$

which would be integrable for $p=2$ but not for any other $q$, but I want to do the counterexample for the whole $\Bbb R^d$ (or $\Bbb R$).

Answer 1

Just replace $x$ by $1+|x|$ ;)