Proving $\Big(a+\frac{1}{b}\Big)^2 + \Big(b+\frac{1}{c}\Big)^2 + \Big(c+\frac{1}{a}\Big)^2 \geq 3(a+b+c+1)$ 
Prove for every $a, b, c \in\mathbb{R^+}$, given that $abc=1$ :
  $$\Big(a+\frac{1}{b}\Big)^2 + \Big(b+\frac{1}{c}\Big)^2 + \Big(c+\frac{1}{a}\Big)^2 \geq 3(a+b+c+1)$$

I tried using AM-GM or AM-HM but I can't figure it out.
$$\Big(a+\frac{1}{b}\Big)^2 + \Big(b+\frac{1}{c}\Big)^2 + \Big(c+\frac{1}{a}\Big)^2 \geq 3\root3\of{\Big(a+\frac{1}{b}\Big)^2\Big(b+\frac{1}{c}\Big)^2\Big(c+\frac{1}{a}\Big)^2}$$
$$\Big(a+\frac{1}{b}\Big)^2 + \Big(b+\frac{1}{c}\Big)^2 + \Big(c+\frac{1}{a}\Big)^2 \geq 3\bigg(\Big(a+\frac{1}{b}\Big)\Big(b+\frac{1}{c}\Big)\Big(c+\frac{1}{a}\Big)\bigg)^{2/3}$$
$$\Big(a+\frac{1}{b}\Big)^2 + \Big(b+\frac{1}{c}\Big)^2 + \Big(c+\frac{1}{a}\Big)^2 \geq 3\bigg(abc+a+b+c+\frac1a+\frac1b+\frac1c+\frac{1}{abc}\bigg)^{2/3}$$
$$\Big(a+\frac{1}{b}\Big)^2 + \Big(b+\frac{1}{c}\Big)^2 + \Big(c+\frac{1}{a}\Big)^2 \geq 3\bigg(2+a+b+c+\frac1a+\frac1b+\frac1c\bigg)^{2/3}$$

Let us show stronger, with an intermediate step:
$$
\begin{aligned}
\left(a+\frac{1}{b}\right)^2 + 
\left(b+\frac{1}{c}\right)^2 + 
\left(c+\frac{1}{a}\right)^2
&\geq 4(a+b+c)
\\
&\geq 3(a+b+c+1)
\ .
\end{aligned}
$$
The last inequality, compactly written $a+b+c\ge3\sqrt[3]{abc}=3$  is simple, so let us show the first one. We write 
$$
a=A^3\ ,\qquad 
b=B^3\ ,\qquad 
c=C^3\ ,\qquad A,B,C>0\ ABC=1\ 
$$
and the given inequality, written first as $\sum(a+ac)^2\ge \dots$, becomes the homogeneous inequality:
$$
A^6(C^3+ABC)^2+
B^6(A^3+ABC)^2+
C^6(B^3+ABC)^2
\ge
4(A^3+B^3+C^3)A^3B^3C^3\ .
$$
At this point we can forget the constraint $ABC=1$. 
Let us see how to dominate with the terms on the left side the ones on the right side.
First, let us expand $(C^3+ABC)^2 = C^6 + 2\,C^3\cdot ABC + (ABC)^2$. Then:
$$
\begin{aligned}
A^6C^6+B^6A^6+C^6B^6
&\ge (A^3+B^3+C^3)A^3B^3C^3\ ,
\\
&\qquad\text{which follows from}
\\
&\qquad\frac 12A^6C^6+\frac 12B^6A^6\ge A^6B^3C^3
\\
&\qquad\text{and the cycled versions of it.}
\\[2mm]
2(A^6C^3+B^6A^3+C^6B^3)ABC
&\ge 2(A^3+B^3+C^3)A^3B^3C^3\ ,
\\
&\qquad\text{which follows from}
\\
&\qquad\frac 23A^6C^3+\frac 13B^6A^3\ge (A^6C^3)^{2/3}(B^6A^3)^{1/3}=A^5B^2C^2
\\
&\qquad\text{and the cycled versions of it. Finally:}
\\
(A^6+B^6+C^6)A^2B^2C^2
&\ge
(A^3+B^3+C^3)A^3B^3C^3
\\
&\qquad\text{follows from}
\\
&\qquad\frac 46A^6+\frac 16B^6+\frac 16C^6\ge A^4BC
\\
&\qquad\text{and the cycled versions of it.}
\end{aligned}
$$
We have used the generalized form of the AM-GM inequality with more general weights $\lambda_1,\lambda_2,\dots,\lambda_n\ge 0$, total weight being one, $$\sum \lambda_kx_k\ge \prod x_k^{\lambda_k}\ .$$
(This is the Jensen inequality for the concave function $\ln$.)

Note: The idea was simple, it is the standard way to show such inequalities. Typing all the stuff...
