<s>[INST] Indefinite integral of rational expression involving cubic polynomials I was given the following exercise:

$$\int \frac{x^3}{(x^2+1)^3}dx$$

As a tip, my professor suggested using the following substitution: $t=x^2+1$. 
Notice that if  $t=x^2+1$, then $x^2=t-1$ and therefore $x=(t-1)^\frac{1}{2}$. Then 
$$x^3=x^2\cdot x=(t-1)(t-1)^\frac{1}{2}=(t-1)^\frac{3}{2}$$
and we have that
$$\int \frac{x^3}{(x^2+1)^3}dx=\int \frac{(t-1)^\frac{3}{2}}{t^3}dt.$$
But now that I have applied the suggested substitution, I don't really see how to continue with this integral. Integrating by parts doesn't seem to get me nowhere, and I can't seem to find any way to apply the substitution method. Am I missing something, or perhaps I made a mistake changing my integration variable from $x$ to $t=x^2+1$?
Thanks in advance.
 [/INST] Partial integration may also work as follows:
\begin{eqnarray*} \int \frac{x^3}{(x^2+1)^3}dx
 & = & \int x^2\frac{x}{(x^2+1)^3}dx \\
& = & x^2\left(-\frac{1}{4}\frac{1}{(x^2+1)^2}\right) + \frac{1}{4}\int \frac{2x}{(x^2+1)^2}dx \\
& = & -\frac{1}{4}\frac{x^2}{(x^2+1)^2} - \frac{1}{4}\frac{1}{x^2+1} (+C)
\end{eqnarray*}
</s>