Hard and Intriguing Indefinite Integral Problem

Compute the following indefinite integral:

[MATH]\int \dfrac{(3x^{10}+2x^8-2)\sqrt[4]{x^{10}+x^8+1}}{x^6} \,dx[/MATH]

Oh boy! Good mathematical Olympiad or any contest problems are hard to come by and this one particularly helps to cultivate analytical thinking and this is also one completely unpredictable and not straightforward problem.

This problem is by all means designed specifically for students to apply the substitution method.

The hardest part is to decide which part of the integrand should be replaced by the substitution. But we have to try, diligently and persistently as we know perseverance will paid off in the end.