Logarithmic irrational inequality

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From the following inequality: $$\left({\sqrt{x^2-4x+3}}+1\right)\cdot(\log_5x-1)+\frac{1}{x}\cdot\left(\sqrt{8x-2x^2-6}+1\right)\le0$$ I found the domain: $x^2-4x+3\ge0$ and $8x-2x^2-6\ge0$ and $x\gt0$ from which I obtained $x\in(0,+\infty)$ But from there I can't seem to do anything else. Is this domain even correct and where do I start from here?