> restart:
> with(LinearAlgebra):
> dim:=3;
> nu:=2;
> Sigma:=<<1,1/2,1/4>|<1/2,1,1/2>|<1/4,1/2,1>>;
> x:=<x1,x2,x3>;
> p:=GAMMA((dim+nu)/2)/(GAMMA(nu/2)*nu^(dim/2)*Pi^(dim/2)*Determinant(Si
> gma)^(1/2)*(1+1/nu*Transpose(x).MatrixInverse(Sigma).x)^((dim+nu)/2));

                               dim := 3


                               nu := 2


                              [ 1     1/2    1/4]
                              [                 ]
                     Sigma := [1/2     1     1/2]
                              [                 ]
                              [1/4    1/2     1 ]


                                   [x1]
                                   [  ]
                              x := [x2]
                                   [  ]
                                   [x3]


        1/2  1/2   1/2   / /      /       /2 x1    x2 \
  p := 2    9    16     /  |48 Pi |1 + x1 |---- - ----|
                       /   \      \       \ 3      3  /

              /   x1    5 x2    x3 \      /   x2    2 x3\\5/2\
         + x2 |- ---- + ---- - ----| + x3 |- ---- + ----||   |
              \   3      6      3  /      \   3      3  //   /

> for n from 1 to 8 do
>  
> disp([n,evalf[n](Int(Int(Int(p,x1=-infinity..1/2),x2=-infinity..1),x3=
> -infinity..3/2))]);
> od;

                            disp([1, 0.6])


                           disp([2, 0.55])


                           disp([3, 0.551])


                          disp([4, 0.5509])


                          disp([5, 0.55090])


                         disp([6, 0.550895])


                         disp([7, 0.5508953])


                        disp([8, 0.55089531])

> with(Statistics):
> X:=RandomVariable(StudentT(NU)):
> pt:=PDF(X,t);

                                          NU
                             GAMMA(1/2 + ----)
                                          2
         pt := ---------------------------------------------
                                                /       NU \
                                                |1/2 + ----|
                                                \       2  /
                                      /      2 \
                      1/2        NU   |     t  |
               (Pi NU)    GAMMA(----) |1 + ----|
                                 2    \     NU /

> simplify((pt*sqrt(2+t^2)^3),symbolic);

                                                  /         NU \
                      / NU \                      |- 1/2 - ----|
                      |----|                      \         2  /
               NU     \ 2  /       2 3/2        2                  /
  GAMMA(1/2 + ----) NU       (2 + t )    (NU + t )                /
               2                                                 /

        /  1/2        NU  \
        |Pi    GAMMA(----)|
        \             2   /

> convert(1/Beta(1/2,NU/2),GAMMA);

                                       NU
                          GAMMA(1/2 + ----)
                                       2
                          -----------------
                            1/2        NU
                          Pi    GAMMA(----)
                                       2

> coef:=1/Pi:
> for i from 1 to 101-1 do
>   if (i mod 2 = 0) and (i>2) then
>     coef := coef * (i-1) / (i-2);
>   fi;
> od;
> evalf(coef);

                             2.533404109

> (coef/Beta(1/2,101/2-1));

                                 99
                                 ---
                                   2
                                 Pi

> 
