3. The equation $\frac{1}{x^{2}+1}+\frac{x+1}{x^{2}}=\frac{10}{3 x}$ is conceived as follows

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3. Answer: $\frac{3 \pm \sqrt{5}}{2}$.

Let $y=x^{2}+1$, the equation is $\frac{1}{y}+\frac{y}{x^{2}}=\frac{10}{3 x}$. Clearing the denominators: $3 x^{2}-10 x y+3 y^{2}=0$.
We get $y_{1}=3 x, y_{2}=\frac{1}{3} x$.
From $y=3 x, x^{2}-1=3 x$, we get $x=\frac{3 \pm}{2}$.
From $y=\frac{1}{3} x, x^{2}+1=\frac{x}{3}, \Delta<0$ no real roots.