can anyone integrate this :::: (3(x^2) +2 )^0.25?

Answer 1

I'm sorry to say that this integral is not elementary.
Let's start with a substitution:
u = 3x² + 2
x = 1/√3* (u-2)^1/2
dx = 1/(2*√3)*(u-2)^(-1/2)du.
Then the integral becomes 1/(2*√3)*∫ u^(1/4)*(u-2)^(-1/2) du.
This integral is nearly a beta function and I think
it can be put in that form with more manipulation.
Then we can apply a theorem of Chebyshev
to conclude that the integral is not elementary,
because neither of the exponents is an integer
and they do not add up to an integer.
I ran this integral through the Wolfram integrator
and it gave me the answer in terms of a hypergeometric
function.

Answer 2

(6x^2+6)^0.25