Evaluating the integral $\int \frac{(x^{10}+x^8+1)^{\frac{1}{4}}(3 x^{10}+2 x^8-2)}{x^6}\,dx$

Question

I recently came across the integral $$\int \frac{(x^{10}+x^8+1)^{\frac{1}{4}}(3 x^{10}+2 x^8-2)}{x^6}\,dx$$ With a bit of imagination one can guess what the answer should look like. But can the answer be derived without having an idea a priori about the solution?

Answer 1

Rewrite the integral as $$I=\int \left(\frac{x^{10}+x^8+1}{x^4} \right)^{1/4} (3x^5+2x^3-2x^{-5}) dx= \int(x^6+x^4+x^{-4})(3x^5+2x^3-2x^{-5}) dx.$$ Let $x^6+x^4+x^{-4}=t.$ Then $$I=\int t^{1/4}\frac{dt}{2}=\frac{4}{5}t^{5/4}\frac{1}{2}=\frac{2}{5}(x^{6}+x^4+x^{-4})^{5/4}+C.$$

The same as told by Dr. Sonnhard.