There is a pole at x=0 (the integrand is infinity at 0)

    Limit[x^3*BesselK[2, x] BesselK[2, x], x -> 0]
    (*infinity*)

You can also see this by writing

Clear[a]
Integrate[x^3*BesselK[2, x] BesselK[2, x], {x, a, Infinity}]

Notice the Re[a]>0 in the above.

Now, why GenerateConditions -> False made it work? I do not know. I think this is a bug. But I am no expert on this. I do not think the value generated is even correct. Compare

Integrate[x^3*BesselK[2, x] BesselK[2, x], {x, 0, Infinity}, GenerateConditions -> False] // N
(*.797059*)

to

    Integrate[x^3*BesselK[2, x] BesselK[2, x], {x, 10^(-64), Infinity}] // N
    (*590.259*)

Compare also what happens when using PrincipalValue -> True which tells it to ignore the simple pole at x=0

       Integrate[x^3*BesselK[2, x] BesselK[2, x], {x, 0, Infinity}, PrincipalValue -> True]

it returns unevaluated. So I have no idea where this result

        Integrate[x^3*BesselK[2, x] BesselK[2, x], {x, 0, Infinity}, GenerateConditions -> False]
        (*1/3 - 4 EulerGamma + Log[16]*)

came from.

With the setting GenerateConditions -> False, Integrate gives a regularized result (a result with a "singular part" removed).

Hello Daniel; In this case, can you please explain why then

   Integrate[x^3*BesselK[2, x] BesselK[2, x], {x, 0, Infinity}, PrincipalValue -> True]

returned unevaluated? I thought the above is supposed to do pretty much what you said. In addition, I looked at help on GenerateConditions, and there is no mention of this side effect of its use. All what it says is that specifies that explicit conditions on parameters should be generated

Thanks for your answer.

Have you noticed that the result of the forth line just equals to 1/3 - 4 EulerGamma + Log[16] ?

How I find this?Using the following code.

Integrate[x^3*BesselK[2, x] BesselK[2, x], {x, m, Infinity}]/ Log[m] /. m -> 10^-16 // N

the answer is (-4.02163).(Of cousre,if you change 10^-16 to 10^-64,you'll get -4.00541).

It shows that the integral and Log[m] are infinites of the same order when m=0.

I'm a rookie on this.I'm wondering that "GenerateConditions -> False" can eliminate the divergent terms automatically.I do not know.

For further check.I have do the integral of x,x^2,x^-1,Log[x],and so on,which are obviously divergent. But "GenerateConditions -> False" makes them equal to zero.

Integrate[x, {x, 0, Infinity},GenerateConditions -> False]

Thanks again.