Consider the integral $$\int \frac{x^{21}-6 x^{14}+6 x^7-37}{\left(x^3-6 x^2+5 x\right)^3 \left(x^4-625\right)^2} \,dx$$ Find the general form of the complete partial fraction decomposition of the integrand. Use it to determine whether or not a term of each type listed below occurs in the complete partial fraction decomposition. Select True if it does occur and False if it does not.

Below $A_1, A_2, A_3, \ldots$ and $B_1, B_2, B_3, \ldots$ denote constants. (You may have been previously using $A, B, C, \ldots$ to denote these constants instead.) You must answer all parts correctly to earn credit for this problem.

? True False   1.   $\displaystyle\frac{B_5}{\left(x- 5\right)^2}$
? True False   2.   $\displaystyle\frac{A_2 x + B_2}{\left(x^2 + 25\right)^2}$
? True False   3.   $\displaystyle\frac{B_7}{\left(x+ 5\right)^2}$
? True False   4.   $\displaystyle\frac{B_4}{\left(x+ 5\right)^3}$
? True False   5.   $\displaystyle\frac{B_6}{\left(x- 5\right)^3}$
? True False   6.   $\displaystyle\frac{A_3 x + B_3}{\left(x^2 - 25\right)^2}$
? True False   7.   $\displaystyle\frac{A_8 x + B_8}{\left(x- 5\right)^2}$
? True False   8.   $\displaystyle\frac{B_1}{x + 1}$