Strange propery(possibly fake) of Laplace Transform

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I was walking around by the CS hang out places at my school and saw the following equation on a white board.

$\mathscr {L} (f) = f+f'+f"+...f^{n} =e^{-x} \int e^x f dx$

Where $n $ means $n $th derivative.

I proved the middle = the right by integration by parts, but after 2 hours of sitting there trying to prove the rest I realized the left can't equal the middle or right. I assume the left means Laplace transform.

Can this statement be proven or is it meaningless?

Thanks