<s>[INST] Finding the length of the arc for function $8y = x^4 +2x^{-2}$ 
Find the length of the arc of curve $8y = x^4 +2x^{-2}$ from $x=1$ to $x=2$

I first isolated for $y$ and derived:
$$ f(x) = {1 \over 8 }x^{4} + {1 \over 4}x^{-2} $$
$$f'(x) = {1 \over 2} x^3 - {1 \over 2}x^{-3}$$
Then found the arc length:
$$L = \int ^2 _1 \sqrt{1+[f'(x)]^2} dx
\\ = \int ^2 _1 \sqrt{1+({1\over 2}x^3 -{1 \over 2}x^{-3})^2} dx
\\ = \int ^2 _1 \sqrt{{1 \over 4} x^6 + {1\over 4}x^{-6} +{1 \over 2}} dx$$
Let $u = {1 \over 4} x^6 + {1\over 4}x^{-6} +{1 \over 2}$
$du = {{3 \over 2} x^5 - {3\over 2}x^{-7} dx}$
However, I cannot seem to integrate this
How can I integrate this equation?
 [/INST] Hint. Note that
$${1 \over 4} x^6 + {1\over 4}x^{-6} +{1 \over 2}=\frac{(x^3+x^{-3})^2}{4}.$$
so it remains to evaluate
$$L=\frac{1}{2}\int ^2 _1(x^3+x^{-3})\,dx.$$
</s>