A problem on minimisation

Find $$\min \bigg(\sqrt[4]{\frac{a}{b+c}}+\sqrt[4]{\frac{b}{a+c}}+\sqrt[4]{\frac{c}{b+a}}+\sqrt{\frac{b+c}{a}}+\sqrt{\frac{a+c}{b}}+\sqrt{\frac{a+b}{c}}\bigg)\quad a,b,c\in\Bbb R^+.$$

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We can use AM-GM inequality just like the previous problem. Using AM-GM inequality for the first three terms with the equality case when $a=b=c$, $$\begin{align}\sqrt[4]{\frac{a}{b+c}}+\sqrt[4]{\frac{b}{a+c}}+\sqrt[4]{\frac{c}{b+a}}&\geq 3\sqrt[3]{\sqrt[4]{\frac{a}{b+c}\frac{b}{a+c}\frac{c}{b+a}}}\\&=3\bigg(\frac{1}{8}\bigg)^{\frac{1}{4}}.\end{align}$$ Similarly, for the last three terms: $$\begin{align}\sqrt{\frac{b+c}{a}}+\sqrt{\frac{a+c}{b}}+\sqrt{\frac{a+b}{c}}&\geq 3\sqrt[3]{\sqrt{\frac{b+c}{a}\frac{a+c}{b}\frac{a+b}{c}}}\\&=3(8)^{\frac{1}{6}}.\end{align}$$ Adding the two inequalities gives $$3\bigg((\frac{1}{2})^{\frac{1}{4}}+2^{\frac{1}{2}}\bigg) \leq \bigg(\sqrt[4]{\frac{a}{b+c}}+\sqrt[4]{\frac{b}{a+c}}+\sqrt[4]{\frac{c}{b+a}}+\sqrt{\frac{b+c}{a}}+\sqrt{\frac{a+c}{b}}+\sqrt{\frac{a+b}{c}}\bigg).$$
Alternative solution
Let $M$ be the expression. Since $M$ is a homogeneous expression (same degree for each term), we can set $a+b+c=1$. Then we have $$M=\sum_{\text{cyc}}\bigg( \sqrt[4]{\frac{a}{1-a}}+\sqrt{\frac{1-a}{a}}\bigg).$$ Now $$f(x)=\sqrt[4]{\frac{x}{1-x}}+\sqrt{\frac{1-x}{x}}$$ is a convex function. We can use Jensen's inequality $\frac{1}{3}\bigg(f(a)+f(b)+f(c)\bigg)\geq f\bigg(\frac{1}{3}(a+b+c)\bigg)$: $$\begin{align}\sqrt[4]{\frac{a}{b+c}}+\sqrt{\frac{b+c}{a}}+\sqrt[4]{\frac{b}{a+c}}+\sqrt{\frac{a+c}{b}}+\sqrt[4]{\frac{c}{b+a}}+\sqrt{\frac{a+b}{c}}&\geq 3f(\frac{1}{3})\\&= 3\bigg((\frac{1}{2})^{\frac{1}{4}}+2^{\frac{1}{2}}\bigg).\end{align}$$