Prove $\int_0^1 \frac{dx}{(x-2) \sqrt[5]{x^2{(1-x)}^3}} = -\frac{2^{\frac{11}{10}} \pi}{\sqrt{5+\sqrt{5}}}$

calculus definite-integrals integration special-functions

Solution 1:

Hint:

Substitute $x \rightarrow\frac{1}{x-1}$. We'll get: $$-\int_0^\infty \dfrac{x^{-3/5}dx}{(2x+1)}$$ Can you continue from here?

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