# Can you integrate a quadratic raised to an arbitrary integer power?

I would like to integrate: $\int_0^\infty (ax^2 + bx + c)^n dx$, where $n$ is a negative integer. Does anyone know a method to do this?

My obvious first try has been using Mathematica, but it doesn't find a solution. And, I've not found it by going through Abramowitz and Stegun (although I may have missed a similar solution).

• A change of variables could go in one of two ways... either as the solutions of $ax^2+bx+c$, or as $u=ax^2+bx+c$, $du=2axdx+bdx$ and replace. Of course, the integration over $[0, \infty)$ of a quadratic (for positive $n$) will always be $\pm\infty$, so the actual integration may be moot... Sep 16, 2013 at 13:50
• It will exist if $n<-1/2$. Partial fractions may help if $n$ is an integer. Sep 16, 2013 at 13:52
• @Michael: true, but I'm assuming that you mean $n \le -1$ given that $n\in \mathbb Z-\{0\}$. Sep 16, 2013 at 13:54
• In fact in my real problem it will always be the case that $n < 1$. Sep 16, 2013 at 13:56
• Now we're getting somewhere! Just put that info plus what you have tried in your question statement and we'll be able to help you properly. :-) Sep 16, 2013 at 13:57

Use simple fractions: $$\frac{1}{(ax^2+bx+c)^n}=\frac{1}{a^n}\frac{1}{(x-x_1)^n(x-x_2)^n}=\frac{1}{a^n}\sum_{k=1}^n\frac{A_k}{(x-x_1)^k} + \frac{B_k}{(x-x_2)^k}$$