<s>[INST] Integral question: $\int \frac{x^{5}+x+3}{x{^3}-5x^{2}}\mathrm dx$ i have integral $\int \frac{x^{5}+x+3}{x{^3}-5x^{2}}\mathrm  dx$, 
so first step is to divide polynoms, and i get:
$\frac{x^{5}+x+3}{x{^3}-5x^{2}}$ = $x^{2} + 5x + 25$ with remainder $125x^2 + x + 3$
is it corrent to divide this polynomial division into two integrals:
int 1: $\int x^{2}+5x+25 \mathrm dx$
int 2: $\int \frac{125x^{2}+x+3}{x{^2}(x-5)}\mathrm dx$
first integral solve directly from tables, and make partial fractions from second integral, then just merge solutions into one expression.
 [/INST] Yes, it is all correct. 
$$
   x^5 + x^2 + 3 = 125 x^2 + x + 3 + (x^3-5 x^2)(x^2 + 5 x + 25)
$$
Hence
$$
 \begin{eqnarray}
   \frac{x^5 + x + 3 }{x^3-5 x^2} &=& x^2 + 5 x + 25 + \frac{125 x^2 +x+ 3}{x^3-5 x} \\ &=& 
   x^2 + 5 x + 25 -\frac{3}{5 \, x^2} - \frac{8}{25 x} + \frac{3133}{25} \frac{x - 5}{x^2-5} \end{eqnarray}
$$
Therefore 
$$
 \int \frac{x^5 + x + 3 }{x^3-5 x^2} \mathrm{d} x = 
   \left(\frac{x^3}{3} + \frac{5}{2} x^2 + 25 x\right) + \left( \frac{3}{5} \frac{1}{x} - \frac{8}{25} \log x + \frac{3133}{25} \log(x-5)  \right) + C
$$
</s>