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

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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