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This is the integration I intended to do.

lw = 8;
    psi = x^2;
    Proietto[a_, b_] := 
     Integrate[a b, {x, 0, lw}, 
      Assumptions -> {u > 0, phi \[Element] Reals, w \[Element] Reals, 
        beta \[Element] Reals, rp \[Element] Reals, rw \[Element] Reals, 
        rb \[Element] Reals}]
    theolift = (-HeavisideTheta[-38/5 + x] + 
          HeavisideTheta[-32/5 + 
            x])*((763*Pi*(6/5 - x/20)^2*
             Cos[21375/(7*Pi*
                 u^2)]*(-((u*((7*
                        Sqrt[1 - (49*(6/5 - x/20)^2)/400]*(6/5 - x/20))/
                       20 - ArcCos[(7*(6/5 - x/20))/20])*
                    Derivative[1][beta][t])/
                  Pi) - ((6/5 - 
                    x/20)*(-(Sqrt[
                        1 - (49*(6/5 - x/20)^2)/
                        400]*(2 + (49*(6/5 - x/20)^2)/400))/
                     3 + (7*(6/5 - x/20)*ArcCos[(7*(6/5 - x/20))/20])/20)*
                  Derivative[2][beta][t])/(2*Pi)))/
           4000 + (763*Pi*u*(6/5 - x/20)*
             Cos[21375/(7*Pi*
                 u^2)]*((17577*u^2*x^2*
                  rb[t])/(50000*(6/5 - x/20)^2) - (393*u*x^2*
                  Derivative[1][rb][t])/(250*(6/5 - x/20)) + 
               x^2*Derivative[2][rb][t]))/2000) + 
       HeavisideTheta[
         x]*(0.8400759670307388*u^2*(1 - x^2/64)^0.5*
           Sin[21375/(7*Pi*u^2)]*(21375/(7*Pi*u^2) - 
              3.990764000117581*^-9*x + 
              x*phi[t] + (3*(6/5 - x/20)*x*Derivative[1][phi][t])/4)^2 + 
          Cos[21375/(7*Pi*
               u^2)]*((763*Pi*
                u*(6/5 - 
                  x/20)*((17577*u^2*x*
                     rp[t])/(50000*(6/5 - x/20)^2) + (17577*u^2*x^2*
                     rw[t])/(50000*(6/5 - x/20)^2) - (393*u*x*
                     Derivative[1][rp][t])/(250*(6/5 - x/20)) - (393*u*
                     x^2*Derivative[1][rw][t])/(250*(6/5 - x/20)) + 
                  x*Derivative[2][rp][t] + x^2*Derivative[2][rw][t]))/
              2000 + (763*
                Pi*(6/5 - x/20)^2*(u*x*
                   Derivative[1][phi][
                    t] + ((6/5 - x/20)*x*Derivative[2][phi][t])/4 - 
                  x^2*Derivative[2][w][t]))/4000));
    Proietto[theolift, psi]

This seamed to not get solved, until I mystically thought of this workaround.

 expan = Expand[theolift];
Proietto[expan, psi]

I'm happy to have solved this, yet why does it work? Finding out could be helpful.

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