Hot Integral - 12

$\displaystyle \int\limits_{-\infty}^{\infty} e^{-2\pi i \Psi x}f(x)dx =\frac{A-B\pi^C\Psi^D - E\pi^F\Psi^Gsgn(\Psi)}{H\pi^I\Psi^J}i - \frac{M\delta''(\Psi)}{K\pi^L}$

where

$\displaystyle f(x)=\frac{x^3 + x^2 + 2(1+xsgn(x))}{2x}$

Calculate $A+B+C+D+E+F+G+H+I+J+K+L+M$

Details and Assumptions

$\bullet \delta(\cdot)$ represents delta function.

$\bullet i=\sqrt{-1}$

$\bullet sgn(x)=\frac{|x|}{x}$ represents sign function.

$\bullet A,B,C,D,E,F,G,H,I,J,K,L,M$ are all positive integers.

$\blacksquare$ This is a part of Hot Integrals