It is straightforward to verify that both are correct antiderivatives.

integrand = (4*x-1)/Sqrt[x^4 - 2*x^3 + 3*x^2 + 2*x + 1];
primWL = -Log[-2*x + 5*x^2 - 3*x^3 + x^4 + (-2 + 2*x - x^2)*
    Sqrt[1 + 2*x + 3*x^2 - 2*x^3 + x^4]];
primFC = Log[(x^2 - 2*x + 2)*Sqrt[x^4 - 2*x^3 + 3*x^2 + 2*x + 1] + x^4 -
    3*x^3 + 5*x^2 - 2*x];  

In[8]:= Simplify[D[primWL,x]-integrand]

(* Out[8]= 0 *)

In[9]:= Simplify[D[primFC,x]-integrand]

(* Out[9]= 0 *)

As for why they differ, it will amount to differences in paths taken in the respective codes.

Can Wolfram calculate the indefinite integral of: integrate (((-4x-8)log(x)+(-2x^2-4x))/(3xexp(2*log(x)+x)^2-x), x)

When I enter it, it gives me a definite integral result: NIntegrate[(-4 x - 2 x^2 + (-8 - 4 x) Log[x])/(-x + 3 E^(2 x) x^5), {x, 1, Infinity}]

Thanks

Why doesn't Mathematica calculate the indefinite integral? Is it outside the abilities of the software? If so, can you guess why? Thanks

"As you can see, these two results are different in form, although they are supposed to be equivalent."

FriCAS one is more correct, and it is not surprising, as it has almost complete Risch- Davenport-Trager-Bronstein-Miller algorithm. It even has the mistakes Davenport did (see paper that corrects Davenport https://www.intlpress.com/site/pub/pages/journals/items/acta/content/vols/0225/0002/a002/) fixed.

Indeed, if you do Plot

only

Plot[-Log[(x^2 - 2 x + 2) Sqrt[x^4 - 2 x^3 + 3 x^2 + 2 x + 1] + x^4 - 3 x^3 + 5 x^2 - 2 x], {x, -10, 10}]

plots. The other does not Plot because it is complex (and not just that it has some complex numbers inside, those can be plotted sometimes).

That was already discussed on stack.

FriCAS claims to have this feature: integration (most complete implementation of the Risch algorithm), what about wolfram?

That claimed difference is correct. This can be seen by noting that the product of the log arguments in the respective results (which are algebraic conjugates), once expanded, is -4. Since one factor is positive the law of logarithms applies (log(a*b)=log(a)+log(b)). So -log(b) is equivalent to log(a) as an antiderivative, and the difference is log(-4) (or its negative, depending on which is subtracted from the other).