TITLE: $\int\dfrac{dx}{x^2(x^4+1)^{3/4}}$
QUESTION [5 upvotes]: Evaluate $$\large{\int\dfrac{dx}{x^2(x^4+1)^{3/4}}}$$

I thought of rewriting this as $$\large{\int\dfrac{dx}{x^5(1+\frac{1}{x^4})^{3/4}}}$$ and substituting $$u^4=\left(1+\frac{1}{x^4}\right)\Rightarrow u=\left(1+\frac{1}{x^4}\right)^{1/4}$$ and subsequently I got $$du=\dfrac{1}{4}\left(1+\frac{1}{x^4}\right)^{-3/4}\times (-4x^{-5})dx$$
However, I can not think of how to proceed further. Any help would be truly appreciated. Many thanks in advance!

REPLY [2 votes]: Just taking up where you left off:
$$
-\int du\, = -u+C = -(1+\frac{1}{x^4})^{\frac{1}{4}}+C.
$$
That should be it.