Partial Fractions Decomposition, Differentiating between linear and nonlinear

Question

My confusion is between $$\dfrac{(x^2+1)}{(x^3)(x+1)}= A/(x) + (Ax+b)/(x^2) + (Cx+d)/(x^3)$$

or

$$\dfrac{(x^2+1)}{(x^3)(x+1)}= A/(x) + B/(x^2) + C/(x^3)$$

Also in the following case, I think the first version is correct but I could use some clarification:

$$\dfrac{(x^2+1)}{(x)(x^2+1)^2}= (Ax+B)/(x^2+1) + (Bx+C)/(x^2+1)^2 + D/(x)$$

Or

$$\dfrac{(x^2+1)}{(x)(x^2+1)^2}= (Ax+B)/(x^2+1) + (Bx^2+Cx+D)/(x^2+1)^2 + E/(x)$$

Obviously those two things are related, and what I really need is someone to explain how it should be in the general case so I can figure out what the right way should be logically. From what I've seen the exponents on the outside of the parentheses don't matter and it's the inside ones tat count, but I don't see why it would be like this.

Answer 1

Hint:

Multiply both sides of equality by $(x^3)(x+1)$ the common denominator and simplify. Built a system of linear variables for each coefficient of the polynomial You've found ans substitute . You need this fractions:

$$\dfrac{(x^2+1)}{(x^3)(x+1)}= A/x + Bx/x^2 + Cx/x^3+(Dx+f)/(x+1)$$

for the first case

Answer 2

This is the right beginning: $$\frac{x^2+1}{x^3(x+1)}=\frac ax+\frac{b}{x^2}+\frac{c}{x^3}+\frac{d}{x+1}.$$ For your second case your first attempt is the right one.