An integral **exhibits $3$ interesting constants.**

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> $$8\int\_{0}^{\infty}{\ln x\over x}\left(e^{-x}-{1\over \sqrt[4]{1+8x}}\right)\mathrm dx=-32\color{red}C+4\color{blue}\gamma^2-5\color{green}\pi^2\tag1$$
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I am only interested in $(1)$, because **rarely** [Catalan's constant](http://mathworld.wolfram.com/CatalansConstant.html) and [Euler-Masheroni's constant](http://mathworld.wolfram.com/Euler-MascheroniConstant.html) they appear together!

**Making an attempt:**

It is too difficult here to make an attempt, apart from differentiating under the integral

$$I(a)=8\int\_{0}^{\infty}{\ln x\over x^{a}}\left(e^{-x}-{1\over \sqrt[4]{1+8x}}\right)\mathrm dx\tag2$$

$$I{'}(a)=8\int\_{0}^{\infty}{1\over x^{a}}\left(e^{-x}-{1\over \sqrt[4]{1+8x}}\right)\mathrm dx\tag3$$

I guess $(3)$ diverges

**How may we prove $(1)?$**