# Hot Integral - 10

**Calculus**Level 5

Consider the integral given by

\(\qquad \qquad \displaystyle f(t) = \int\limits_0^{\infty}\frac{dx}{\sqrt{x^4+4x^3+(5-t)x^2+2(1-t)x}}\)

Then ,

\(\displaystyle \int\limits_0^{\infty} z^{n-1}\log z f(-z) dz = \frac{\Gamma(\frac{A}{B}-n)^C\Gamma(n)^D\left(-E\psi(\frac{F}{G}-n) + H\psi(K-n) + \psi(n)\right)}{L\Gamma(P-n)^Q}\)

Calculate

\(A+B+C+D+E+F+G+H+K+L+P+Q\)

**Details and Assumptions**

\( A,B,C,D,E,F,G,H,K,L,P,Q\) are all positive integers.

\( \Gamma\) represents the Gamma function.

\( \psi\) represents the Polygamma function.

\( n \in (1/64,1/4)\)

\( \gcd(A,B)=\gcd(F,G)=1\)