\[\large \int_0^{\infty}\frac{x^6}{e^{3x^2}}\,dx\]

The integral above can be written as \(\dfrac{a\sqrt{\pi}}{b^4\cdot c^{d/2}}\), where \(a\), \(b\), \(c\), and \(d\) are prime numbers.

Find \(a+b+c+d\).

This is problem for Daily challenge of Greek letters.

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