Simplify: $\;$ $\left[x \sqrt[3]{\dfrac{x - 1}{\left(x + 1\right)^2}} + \dfrac{x - 1}{\sqrt[3]{\left(x^2 - 1\right)^2}}\right]^{\frac{-3}{5}} : \left(x^2 - 1\right)^{\frac{4}{5}}$
$\left[x \sqrt[3]{\dfrac{x - 1}{\left(x + 1\right)^2}} + \dfrac{x - 1}{\sqrt[3]{\left(x^2 - 1\right)^2}}\right]^{\frac{-3}{5}} : \left(x^2 - 1\right)^{\frac{4}{5}}$
$= \left[\dfrac{x \left(x - 1\right)^{\frac{1}{3}}}{\left(x + 1\right)^{\frac{2}{3}}} + \dfrac{x - 1}{\left(x^2 - 1\right)^{\frac{2}{3}}}\right]^{\frac{-3}{5}} : \left(x + 1\right)^{\frac{4}{5}} \left(x - 1\right)^{\frac{4}{5}}$
$= \left[\dfrac{x \left(x - 1\right)^{\frac{1}{3}}}{\left(x + 1\right)^{\frac{2}{3}}} + \dfrac{x - 1}{\left(x + 1\right)^{\frac{2}{3}} \left(x - 1\right)^{\frac{2}{3}}}\right]^{\frac{-3}{5}} : \left(x + 1\right)^{\frac{4}{5}} \left(x - 1\right)^{\frac{4}{5}}$
$= \left[\dfrac{x \left(x - 1\right)^{\frac{1}{3}} + \left(x - 1\right)^{\frac{1}{3}}}{\left(x + 1\right)^{\frac{2}{3}}}\right]^{\frac{-3}{5}} : \left(x + 1\right)^{\frac{4}{5}} \left(x - 1\right)^{\frac{4}{5}}$
$= \left[\dfrac{\left(x + 1\right)^{\frac{2}{3}}}{\left(x - 1\right)^{\frac{1}{3}} \left(x + 1\right)}\right]^{\frac{3}{5}} \times \dfrac{1}{\left(x + 1\right)^{\frac{4}{5}} \left(x - 1\right)^{\frac{4}{5}}}$
$= \dfrac{1}{\left[\left(x - 1\right)^{\frac{1}{3}}\right]^{\frac{3}{5}} \left[\left(x + 1\right)^{\frac{1}{3}}\right]^{\frac{3}{5}}} \times \dfrac{1}{\left(x + 1\right)^{\frac{4}{5}} \left(x - 1\right)^{\frac{4}{5}}}$
$= \dfrac{1}{\left(x - 1\right)^{\frac{1}{5}} \left(x + 1\right)^{\frac{1}{5}} \left(x + 1\right)^{\frac{4}{5}} \left(x - 1\right)^{\frac{4}{5}}}$
$= \dfrac{1}{\left(x - 1\right) \left(x + 1\right)}$
$= \dfrac{1}{x^2 - 1}$