Is it true that $\left(\int |xf(x)|^2dx\right)^{1/2}\left(\int |f(x)|^2dx\right)^{1/2} \leq C \left(\int (x^2+1) |f(x)|^2dx\right)^{1/2}$?

Question

This is probably true. I'd like to know why it is true. Can we find the value of $C$?

$$\left(\int |xf(x)|^2dx\right)^{1/2}\left(\int |f(x)|^2dx\right)^{1/2} \leq C \left(\int (x^2+1) |f(x)|^2dx\right)^{1/2}$$ for some constant $C$.