E=\(\left(\frac{\sqrt{x^3}-1}{\sqrt{x}\left(\sqrt{x}-1\right)}-\frac{\sqrt{x^3}+1}{\sqrt{x}\left(\sqrt{x}+1\right)}\right)+\left(\frac{x-1}{\sqrt{x}}\right)\cdot\left(\frac{\left(\sqrt{x}+1\right)^2+\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\right)\)
= \(\left(\frac{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}-\frac{\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}+1\right)}\right)\) +\(\left(\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\sqrt{x}}\right)\cdot\left(\frac{x+2\sqrt{x}+1+x-2\sqrt{x}+1}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\right)\)
=\(\left(\frac{x+\sqrt{x}+1}{\sqrt{x}}-\frac{x-\sqrt{x}+1}{\sqrt{x}}\right)+\frac{1}{\sqrt{x}}\cdot\left(2x+2\right)\)
=\(\frac{x+\sqrt{x}+1-x+\sqrt{x}-1}{\sqrt{x}}+\frac{2x+2}{\sqrt{x}}\)
=\(\frac{2\sqrt{x}+2x+2}{\sqrt{x}}\)
ĐKXĐ:\(\left\{{}\begin{matrix}x>0\\x\ne1\end{matrix}\right.\)
Ta có:
+)\(\frac{x\sqrt{x}-1}{x-\sqrt{x}}-\frac{x\sqrt{x}+1}{x+\sqrt{x}}=\frac{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}-\frac{\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}+1\right)}=\frac{x+\sqrt{x}+1-x+\sqrt{x}-1}{\sqrt{x}}=\frac{2\sqrt{x}}{\sqrt{x}}=2\)
+)\(\frac{\sqrt{x}+1}{\sqrt{x}-1}+\frac{\sqrt{x}-1}{\sqrt{x}+1}=\frac{x+2\sqrt{x}+1+x-2\sqrt{x}+1}{x-1}=\frac{2\left(x+1\right)}{x-1}\)
\(\Rightarrow\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)\left(\frac{\sqrt{x}+1}{\sqrt{x}-1}+\frac{\sqrt{x}-1}{\sqrt{x}+1}\right)=\frac{x-1}{\sqrt{x}}.\frac{2\left(x+1\right)}{x-1}=\frac{2\left(x+1\right)\sqrt{x}}{x}\)
Thay vào E ta được: \(E=2+\frac{2\left(x+1\right)\sqrt{x}}{x}\)
