int (X^2)^2+1/(X^2)^3+1

APLY FORMULA

A^2+B^2((A+B)^2-2AB)IN NUMURATOR

AND A^3+B^3 IN DENOMRATOR(A+B)(A^2+B^2-AB)

NOW U GET INT X^2+1/X^4-X^2+1 - INT 2 X^2/(X^2+1)(X^4+1-X^2)

PUT X^2=T

INT T+1/T^2-T+1 -INT2T/(T+1)(T^2-T+1)

NOW U CAN APPLY PARTIAL IF CAN'T APPLY TELL ME I DO

 integrate this question and tell whether it is integrable or not?

 it is integrable but a very long answer. see wolfram integrator.

cannot be integrated without a calculator(I think so)