Integral of a Polynomial in Square Root

Question

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?

Answer 1

Hint: $$(x^2+x^{-2})^2=x^4+x^{-4}+2.$$

Answer 2

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 :

$$\int \frac{x^2 + x^{-2}}{x^3}dx= \int \bigg(\frac{1}{x} + \frac{1}{x^5}\bigg)dx=\int x^{-1}dx + \int x^{-5}dx= \ln|x| +-4x^{-4}+ C$$