Is this integral unfit for partial fraction decomposition?

Question

I've been tasked with computing the following integral:

$$\int_0^4\frac{x^3+10x^2+3x+36}{(x-1)(x^2+4)^2}\text dx$$

The issue I have here is that the numerator is of a higher degree than the denominator; however when consulting some online resources it seems they go ahead with the decomposition anyway.

Is the degree of the polynomial on the denominator taken when it's fully expanded out (i.e, for this question, $x^5 - x^4 + 8x^3 - 8x^2 + 16x - 16$) (And a factored form can actually be of a lesser degree) or are these calculators doing something I'm not aware of?

Just really confused as to this.

Answer 1

make the Ansatz $$\frac{x^3+10x^2+3x+36}{(x-1)(x^2+4)^2}=\frac{A}{x-1}+\frac{Bx+C}{x^2+4}+\frac{Dx+E}{(x^2+4)^2}$$

Answer 2

The integrand can be split as $$\frac{2}{x - 1} + \frac{x}{ \left(x^{2} + 4 \right)^{2}} + \frac{-2 \; x - 1}{x^{2} + 4}$$