\begin{array}{rcl} 
\displaystyle\int\limits_{-1}^{1}\frac{\textbf{d}x}{\sqrt[3]{(1-x)(1+x)^{2}}}
&=& \displaystyle\int\limits_{0}^{1}\frac{2 \textbf{d}t}{\sqrt[3]{(2-2t)(2t)^{2}}}\\
&=&\displaystyle\int\limits_{0}^{1}t^{-\frac{2}{3}}(1-t)^{-\frac{1}{3}}\textbf{d}t \\
&=& \displaystyle\textbf B\left(\frac{1}{3},\frac{2}{3}\right) = \frac{\Gamma\left(\frac{1}{3}\right)\Gamma\left(\frac{2}{3}\right)}{\Gamma(1)}\\
&=&\displaystyle\frac{\pi}{\sin\frac{\pi}{3}}=\frac{2\pi}{\sqrt{3}}