Use AM-GM's ineq:

\(\dfrac{b+c}{\sqrt{a}}+\dfrac{c+a}{\sqrt{b}}+\dfrac{a+b}{\sqrt{c}}\ge\dfrac{2\sqrt{bc}}{\sqrt{a}}+\dfrac{2\sqrt{ca}}{\sqrt{b}}+\dfrac{2\sqrt{ab}}{\sqrt{c}}\)

\(=\left(\sqrt{\dfrac{bc}{a}}+\sqrt{\dfrac{ca}{b}}\right)+\left(\sqrt{\dfrac{ca}{b}}+\sqrt{\dfrac{ab}{c}}\right)+\left(\sqrt{\dfrac{ab}{c}}+\sqrt{\dfrac{bc}{a}}\right)\)

\(\ge\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{a}+\sqrt{b}+\sqrt{c}\)

\(\ge\sqrt{a}+\sqrt{b}+\sqrt{c}+3\sqrt[3]{abc}\)\(=\sqrt{a}+\sqrt{b}+\sqrt{c}+3\)