\(x\ge1\)
\(\sqrt{x-1}+\sqrt{\left(x^2+1\right)\left(x+1\right)}=1+\sqrt{\left(x^2+1\right)\left(x+1\right)\left(x-1\right)}\)
\(\Leftrightarrow\sqrt{\left(x^2+1\right)\left(x+1\right)\left(x-1\right)}-\sqrt{\left(x^2+1\right)\left(x+1\right)}-\left(\sqrt{x-1}-1\right)=0\)
\(\Leftrightarrow\sqrt{\left(x^2+1\right)\left(x+1\right)}\left(\sqrt{x-1}-1\right)-\left(\sqrt{x-1}-1\right)=0\)
\(\Leftrightarrow\left(\sqrt{\left(x^2+1\right)\left(x+1\right)}-1\right)\left(\sqrt{x-1}-1\right)=0\)
\(\Leftrightarrow\sqrt{x-1}-1=0\) (do \(x\ge1\Rightarrow\sqrt{\left(x^2+1\right)\left(x+1\right)}>1\Rightarrow\sqrt{\left(x^2+1\right)\left(x+1\right)}-1>0\))
\(\Leftrightarrow\sqrt{x-1}=1\)
\(\Rightarrow x=2\)
