\[\begin{align} & \large I_1 = \int_0^{\infty} \dfrac{\ln^2 (x)}{e^x} x^{{1}/{4}} \mathrm{d}x \\ & \large I_2 = \int_0^{\infty} \dfrac{x^{-{1}/{4}}}{e^x} \mathrm{d}x \end{align}\]

Let \(I_1\) and \(I_2\) be as defined above. If the closed form of \(I_1 I_2\) can be represented as

\[\dfrac{a}{b \sqrt c} \pi^3 - \sqrt{c} \pi^2 + \dfrac{d}{c \sqrt{c}} \pi^2 \ln(c) + \dfrac{\pi^2 \gamma}{c \sqrt c} + \dfrac{\pi \gamma^2}{c \sqrt c} + \gamma \pi \left( \dfrac{d \ln(c)}{\sqrt c} - c \sqrt c \right) - e \sqrt{c} \pi \ln(c) + \dfrac{f \pi \ln^2 (c)}{c \sqrt{c}} + c \sqrt{c} \pi G \]

where alphabets in lowercase from \(a\) to \(f\) are distinct positive integers with coprime-pairs \((a,b) , (c,d)\) and \((c,f)\) and \(c\) is not a perfect power, evaluate \(a+b+c+d+e+f\).

**Notations:**

- \(G\) denotes the Catalan's constant.
- \(\gamma\) denotes the Euler-Mascheroni constant.
- \(\ln^n (\cdot) = {(\ln (\cdot))}^n\)

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