Calculate using Euler integrals $$\int \limits_{-1}^{1} \frac{dx}{\sqrt[3]{(1+x)^2(1-x)}}$$

What substitution is better to use here?

I have tried something like $t = \frac{1-x}{1+x}$ but it doesn't look good.


If you set $x=2t-1$ you get

$$ 2\int_{0}^{1}\frac{dt}{\sqrt[3]{(2t)^2(2-2t)}}=\int_{0}^{1} t^{-2/3}(1-t)^{-1/3}\,dt = \frac{\Gamma(1/3)\,\Gamma(2/3)}{\Gamma(1)} = \frac{\pi}{\sin\frac{\pi}{3}}=\color{red}{\frac{2\pi}{\sqrt{3}}}.$$


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