At first I had no doubt that I will have to use partial fractions on this integral: \begin{equation} \int \frac{3x^2+4}{x^5+x^3}dx \end{equation} I split it into two integrals and one of them give me back this equation: \begin{equation} \ A(x^2+1)+Bx(x^2+1)+Cx^2(x^2+1)+Dx^4+Ex^3 =1 \end{equation} Then if I check what happens if $x = 0$ I find out that $A = 1$ after that I checked what happens then $ x =1 \ \ x=-1 \ \ x=2 $ my results there : \begin{equation} \ 2B+E=0 \\ \ 2C+D=-1 \\ \ E+2D=2 \end{equation} I am realizing that I can joggle variables in any fashion I like I will not be get their values. So using partial fractions is not effective for this particular case. What other way could you suggest of handling this equation?
P.S. I may have made a calculation mistake. In that case I am sorry for wasting your time but I would appreciate if you could point out my mistake.
EDIT: I wrote 'I split it into two integrals' it seams it needs to be shown: \begin{equation} 3\int \frac{dx}{x^2+1}dx +4 \int \frac{dx}{x^5+x^3}dx = \\ \ \arctan x +4(\int \frac{A}{x^3}dx+\int \frac{B}{x^2}dx+\int \frac{C}{x}dx+\int \frac{Dx+E}{x^2+1}dx) \end{equation}