Integral of The Rational Function

Calculus Level 4

Given:

01x3+x+2x4+2x2+1dx=pq+rπs+lntu,\int_0^1 \frac{x^3+x+2}{x^4+2x^2+1}dx=\frac{p}{q}+\frac{r\pi}{s}+\frac{\ln t}{u},

where t\,t and u\,u are perfect squares. Find p+q+r+s+t+u\,p+q+r+s+t+u.

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