To solve the integral

$\int \frac{x^4 - 9x^3 + 3x^2 - 87x}{(x+1)(x^2+9)^2} \, dx$

we can approach this using partial fraction decomposition and then integrate each resulting term separately.

Step-by-Step Breakdown:

1. Factor the denominator: The denominator is already factored as $(x + 1)(x^2 + 9)^2$.

2. Set up partial fractions: We need to express the integrand as a sum of partial fractions. The general form would be:

   $\frac{A}{x+1} + \frac{Bx + C}{x^2 + 9} + \frac{Dx + E}{(x^2 + 9)^2}$

   We aim to find constants $A$, $B$, $C$, $D$, and $E$ by solving for them.

3. Solve for the constants: We now equate the original expression to this sum of partial fractions:

   $\frac{x^4 - 9x^3 + 3x^2 - 87x}{(x+1)(x^2+9)^2} = \frac{A}{x+1} + \frac{Bx + C}{x^2 + 9} + \frac{Dx + E}{(x^2 + 9)^2}$

   Multiply both sides by $(x + 1)(x^2 + 9)^2$ to get rid of the denominator:

   $x^4 - 9x^3 + 3x^2 - 87x = A(x^2 + 9)^2 + (Bx + C)(x+1)(x^2 + 9) + (Dx + E)(x+1)$

   Now expand both sides and collect terms based on powers of $x$ to solve for $A$, $B$, $C$, $D$, and $E$.

Would you like me to continue with solving the partial fraction decomposition explicitly or prefer another method?