To integrate the function x^(3/4) + x^(2/3) + x^(1/2) / x dx, we first simplify the expression:

x^(3/4) + x^(2/3) + x^(1/2) / x
= x^(3/4) + x^(2/3) + x^(-1/2)

Now we integrate term by term:

∫x^(3/4) dx = (4/7)x^(7/4) + C

∫x^(2/3) dx = (3/5)x^(5/3) + C

∫x^(-1/2) dx = 2x^(1/2) + C

So the final integral of the function is:

(4/7)x^(7/4) + (3/5)x^(5/3) + 2x^(1/2) + C

where C is the constant of integration.