<s>[INST] Integral of a Polynomial in Square Root 
I need to solve the indefinite integral: 
  $$\int \frac{\sqrt{x^4+x^{-4}+2}}{x^3} dx,$$
  but I can not find any technique which could solve it.

Can you help me please?
 [/INST] It is : 
$$(x^2+x^{-2})^2=x^4+x^{-4}+2$$
which means that your initial integral can be substituted to : 
$$\int \frac{\sqrt{x^4+x^{-4}+2}}{x^3} dx = \int\frac{\sqrt{(x^2+x^{-2})^2}}{x^3}dx = \int \frac{x^2 + x^{-2}}{x^3}dx$$
since $x^2 + x^{-2} \geq 0 \space \forall x \in \mathbb R$.
So : 
$$\int \frac{x^2 + x^{-2}}{x^3}dx= \int \bigg(\frac{1}{x} + \frac{1}{x^5}\bigg)dx=\int x^{-1}dx + \int x^{-5}dx= \ln|x| +-4x^{-4}+ C$$
</s>