According to my calculations

$$ \int\_0^\infty \frac{\mathrm dx}{(1+x^3)^n}=\frac{(3n-4)\times(3n-7)\times\cdots\times5\times2}{3^{n+1/2}(n-1)!}2\pi$$

How can an equivalent of $$ \int\_0^\infty \frac{\mathrm dx}{(1+x^3)^n}$$ be derived from this formula?

(Given that my objective is to study the nature of the series $ \sum \int\_0^\infty \frac{\mathrm dx}{(1+x^3)^n} $)

So my question is simply: is there a simple equivalent for $(3n-4)\times(3n-7)\times\cdots\times5\times2$ ?