I think I have misunderstood your expression. Please edit your post and paste the InputForm of your expression with a blank line before and after and with four spaces at the beginning of the line. Or just attach your notebook file to your post.

Thank you.

I am guessing that you wanted to change all

\(40)

to

(

and

\(41)

to

)

and

e

to

E

and

Integrate[...,{x3,x2,3000},{x2,x1,3000},{x1,0,3000}]

to

Integrate[...,{x1,0,3000},{x2,x1,3000},{x3,x2,3000}]

If all that is correct then your input becomes

Integrate[(E^((-1/9*(-3 + x1)^2 - (-8 + x2)^2/16 - (-12 + x3)^2/36)/2)*
  (7.5*x1 + 6*(-x1 + x2) + 4.5*(-x2 + x3)))/(144*Sqrt[2]*Pi^(3/2)),
  {x1,0,3000},{x2,x1,3000},{x3,x2,3000}]

Then this

Table[x1=RandomReal[{0,3000}];x2=RandomReal[{x1,3000}];x3=RandomReal[{x2,3000}];
  (E^((-1/9*(-3 + x1)^2 - (-8 + x2)^2/16 - (-12 + x3)^2/36)/2)*
  (7.5*x1 + 6*(-x1 + x2) + 4.5*(-x2 + x3)))/(144*Sqrt[2]*Pi^(3/2)),
  {1000}]

returns this

{0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0....

seeming to show that your integrand is very small, almost zero, almost everywhere.

It appears that only for small values of x1,x2,x3, less than 50, is your integrand slightly larger than zero.That appears to be confusing the integration algorithm, where it looks at values of your integrand hundreds or thousands of times and finds the value is almost exactly zero every time.

If all of my changes and guesses are correct then this

NIntegrate[(E^((-1/9*(-3 + x1)^2 - (-8 + x2)^2/16 - (-12 + x3)^2/36)/2)*
  (7.5*x1 + 6*(-x1 + x2) + 4.5*(-x2 + x3)))/(144*Sqrt[2]*Pi^(3/2)),
  {x1,0,40},{x2,x1,40},{x3,x2,40}]

estimates your full integral is approximately 38.2249 but I do not know how much error is associated with that.

Please check every detail of this and try to make certain that you have found and corrected all my mistakes.

I am sorry. I had no idea that you were trying to do this using WolframAlpha.

In WolframAlpha If I simplify your integral to

Integrate[(E^((-(-3000+x1)^2/9-(-8000+x2)^2/16-(-12000+x3)^2/36)/(2*10^6))*(1.5(x1+x2+3x3))),{x1,0,3000},{x2,x1,3000},{x3,x2,3000}]

WolframAlphacalculation

that barely completes within the time allowed for the free version of Wolfram Alpha and gives a result

3.90247*10^12.

For the final step I calculate

3.90247*10^12/(144*10^9*Sqrt[2]*Pi^(3/2))

and WolframAlpha gives a result of 3.44142 which equals the result from Wolfram Mathematica if I do the complete problem in one step.

Are you able to complete this calculation now?

Please check every step of this very carefully to find and fix any errors that I have made.

The integral from the question/first post takes only a few milliseconds with NIntegrate[]:

NIntegrate[(E^((-1/3000^2*(-3000 + x1)^2 - (-8000 + x2)^2/
          4000^2 - (-12000 + x3)^2/6000^2)/2)*(7.5*x1 + 
      6*(-x1 + x2) + 4.5*(-x2 + x3)))/(3000*4000*6000*2*Sqrt[2]*
    Pi^(3/2)), {x1, 0, 3000}, {x2, x1, 3000}, {x3, x2, 3000}]

(*  3.44142  *)