Proving the number of solutions to an equation in $p$-adic numbers

Question

I'd like to show that the equation $x^3+5x+1=0$ has exactly one solution over $\mathbb{Q}_7$, i.e., the 7-adic numbers.

By Hensel's lemma, one sees that the quation has at least one solution since $1^3+5\cdot1+1\equiv0\mod7$ and $3\cdot1^2+5\not\equiv0\mod7$. But how can I make sure that this one solution is the only solution?