RF(x,y,z):=INT(1/SQRT((x+t_)*(y+t_)*(z+t_)),t_,0,inf)/2

RJ(x,y,z,p):=3*INT(1/((p+t_)*SQRT((x+t_)*(y+t_)*(z+t_))),t_,0,inf)/2

RC(x,y):=INT(1/(SQRT(x+t_)*(y+t_)),t_,0,inf)/2

RD(x,y,z):=RJ(x,y,z,z)

RG(x,y,z):=z*RF(x,y,z)-(x-z)*(y-z)*RJ(x,y,z,z)/3+SQRT(x*y/z)

"To help DERIVE integrate RF and RJ I will change variables."

RF(x,y,z):=INT(t_/SQRT((t_^2+x)*(t_^2+y)*(t_^2+z)),t_,0,inf)

RJ(x,y,z,p):=3*INT(t_/((p+t_^2)*SQRT((x+t_^2)*(y+t_^2)*(z+t_^2))),t_,0,inf)

LOAD("carlson.mth")

e1:=

EF(i,n):=VECTOR(IF(i=i_),i_,1,n)

II_AUXH(h,a,b,t):=PRODUCT((a SUB j_+b SUB j_*t)^(-1/2),j_,1,h)

II_AUXN(m,n,a,b,t):=PRODUCT((a SUB j_+b SUB j_*t)^m SUB j_,j_,1,n)

II_AUX(m,h,n,a,b,t):=II_AUXH(h,a,b,t)*II_AUXN(m,n,a,b,t)

AF(m,h,n,a,b,x,y):=II_AUX(m,h,n,a,b,x)-II_AUX(m,h,n,a,b,y)

II(m,h,n,a,b,x,y):=INT(II_AUX(m,h,n,a,b,t_),t_,y,x)

MA(m,n):=SUM(m SUB j_,j_,1,n)

BA(m,n,b):=PRODUCT(b SUB j_^m SUB j_,j_,1,n)

DF(i,j,a,b):=a SUB i*b SUB j-a SUB j*b SUB i

DA(i,m,n,a,b):=PRODUCT(IF(i/=j_,DF(j_,i,a,b)^m SUB j_,1),j_,1,n)

mu_A(s,i,m,n,a,b):=-1/ABS(s)*SUM(IF(i/=j_,m SUB j_*(DF(i,j_,a,b)/b SUB j_)^s,~
0),j_,1,n)

CA1(s,i,m,n,a,b):=mu_A(SIGN(s),i,m,n,a,b)

CA2(s,i,m,n,a,b):=1/2*CA1(s,i,m,n,a,b)^2+mu_A(2*SIGN(s),i,m,n,a,b)

CA3(s,i,m,n,a,b):=1/6*CA1(s,i,m,n,a,b)^3+CA1(s,i,m,n,a,b)*mu_A(2*SIGN(s),i,m,~
n,a,b)+mu_A(3*SIGN(s),i,m,n,a,b)

CA4(s,i,m,n,a,b):=1/24*CA1(s,i,m,n,a,b)^4+1/2*CA1(s,i,m,n,a,b)^2*mu_A(2*SIGN(~
s),i,m,n,a,b)+CA1(s,i,m,n,a,b)*mu_A(3*SIGN(s),i,m,n,a,b)+1/2*mu_A(2*SIGN(s),i~
,m,n,a,b)^2+mu_A(4*SIGN(s),i,m,n,a,b)

CA(s,i,m,n,a,b):=IF(ABS(s)=1,CA1(s,i,m,n,a,b),IF(ABS(s)=2,CA2(s,i,m,n,a,b),IF~
(ABS(s)=3,CA3(s,i,m,n,a,b),IF(ABS(s)=4,CA4(s,i,m,n,a,b),1))))

IA(x):=

D219_AUX(i,m,ma_,n,a,b,e):=IF(ma_>-1,BA(m,n,b)*b SUB i^(-ma_)*SUM(CA(ma_-q_,i~
,m,n,a,b)*IA(q_*e SUB i),q_,0,ma_),0)+SUM(IF(-m SUB i_>0,DA(i_,m,n,a,b)*b SUB~
 i_^(m SUB i_-ma_)*SUM(CA(m SUB i_+q_,i_,m,n,a,b)*IA(-q_*e SUB i_),q_,1,-m SU~
B i_),0),i_,1,n)

D219(i,m,n,a,b,e):=D219_AUX(i,m,MA(m,n),n,a,b,e)

EP(p,j,s,h,a,b):=IF(p=0,1,SUM(DF(i_,j,a,b)/b SUB i_*EP(p-1,j,i_+1,h,a,b),i_,s~
,h))

sigma0(h,b):=PRODUCT(b SUB i_,i_,1,h)

åP(p,j,h,a,b):=sigma0(h,b)*EP(p,j,1,h,a,b)

AA(e):=

R35(q,j,h,a,b,e):=-b SUB j^(h-1)*AA((q+1)*e SUB j+SUM(e SUB i_,i_,1,h))+SUM((~
q+r_/2+1)*åP(h-r_,j,h,a,b)*IA((q+r_)*e SUB j),r_,0,h)

R311(q,i,j,a,b,e):=-b SUB i^q*IA(q*e SUB j)+SUM(COMB(q,r_)*b SUB j^r_*DF(j,i,~
a,b)^(q-r_)*IA(r_*e SUB i),r_,0,q)

R613(m,k,l,a,b,e):=DF(k,l,a,b)*AA(m)-b SUB l*AA(m+e SUB k)+b SUB k*AA(m+e SUB~
 l)

XF(i,a,b,x):=SQRT(a SUB i+b SUB i*x)

U12(a,b,x,y):=(SQRT(a SUB 1+y*b SUB 1)*SQRT(a SUB 2+y*b SUB 2)*SQRT(a SUB 3+x~
*b SUB 3)*SQRT(a SUB 4+x*b SUB 4)+SQRT(a SUB 1+x*b SUB 1)*SQRT(a SUB 2+x*b SU~
B 2)*SQRT(a SUB 3+y*b SUB 3)*SQRT(a SUB 4+y*b SUB 4))/(x-y)

U13(a,b,x,y):=(SQRT(a SUB 1+y*b SUB 1)*SQRT(a SUB 2+x*b SUB 2)*SQRT(a SUB 3+y~
*b SUB 3)*SQRT(a SUB 4+x*b SUB 4)+SQRT(a SUB 1+x*b SUB 1)*SQRT(a SUB 2+y*b SU~
B 2)*SQRT(a SUB 3+x*b SUB 3)*SQRT(a SUB 4+y*b SUB 4))/(x-y)

U23(a,b,x,y):=(SQRT(a SUB 1+x*b SUB 1)*SQRT(a SUB 2+y*b SUB 2)*SQRT(a SUB 3+y~
*b SUB 3)*SQRT(a SUB 4+x*b SUB 4)+SQRT(a SUB 1+y*b SUB 1)*SQRT(a SUB 2+x*b SU~
B 2)*SQRT(a SUB 3+x*b SUB 3)*SQRT(a SUB 4+y*b SUB 4))/(x-y)

I414_2(v):=[(v SUB 1+v SUB 2)*(v SUB 1+v SUB 3),(v SUB 2+v SUB 1)*(v SUB 2+v ~
SUB 3),(v SUB 3+v SUB 1)*(v SUB 3+v SUB 2)]

I414_1(v):=4*R_F(v SUB 1,v SUB 2,v SUB 3)

I414(a,b,x,y):=I414_1(I414_2([U12(a,b,x,y),U13(a,b,x,y),U23(a,b,x,y)]))

II414(a,b,x,y):=II([0,0,0,0],4,4,a,b,x,y)

U15(a,b,x,y):=SQRT(U12(a,b,x,y)^2-(a SUB 3*b SUB 1-a SUB 1*b SUB 3)*(a SUB 4*~
b SUB 1-a SUB 1*b SUB 4)*(a SUB 5*b SUB 2-a SUB 2*b SUB 5)/(a SUB 5*b SUB 1-a~
 SUB 1*b SUB 5))

Q15(a,b,x,y):=SQRT(a SUB 5+x*b SUB 5)*SQRT(a SUB 5+y*b SUB 5)*U15(a,b,x,y)/(S~
QRT(a SUB 1+x*b SUB 1)*SQRT(a SUB 1+y*b SUB 1))

S15(a,b,x,y):=(SQRT(a SUB 2+x*b SUB 2)*SQRT(a SUB 3+x*b SUB 3)*SQRT(a SUB 4+x~
*b SUB 4)*(a SUB 5+y*b SUB 5)/SQRT(a SUB 1+x*b SUB 1)+SQRT(a SUB 2+y*b SUB 2)~
*SQRT(a SUB 3+y*b SUB 3)*SQRT(a SUB 4+y*b SUB 4)*(a SUB 5+x*b SUB 5)/SQRT(a S~
UB 1+y*b SUB 1))/(x-y)

S15(a,b,x,y)^2=Q15(a,b,x,y)^2+DF(2,5,a,b)*DF(3,5,a,b)*DF(4,5,a,b)/DF(1,5,a,b)

I417_P(b,c,t):=(b*c+t)/(b+c)

I417_B(a,b,u,v,w,z):=(a SUB 5*b SUB 1-a SUB 1*b SUB 5)/((a SUB 2*b SUB 1-a SU~
B 1*b SUB 2)*(a SUB 3*b SUB 1-a SUB 1*b SUB 3)*(a SUB 4*b SUB 1-a SUB 1*b SUB~
 4))*(z+u)*(z+v)*(z+w)

I417_2(a,b,p):=4/p*R_C(1,1-(a SUB 5*b SUB 2-a SUB 2*b SUB 5)*(a SUB 5*b SUB 3~
-a SUB 3*b SUB 5)*(a SUB 5*b SUB 4-a SUB 4*b SUB 5)/(p^2*(a SUB 5*b SUB 1-a S~
UB 1*b SUB 5)))

I417_1(a,b,x,y):=I417_2(a,b,I417_P(I417_B(a,b,U12(a,b,x,y),U13(a,b,x,y),U23(a~
,b,x,y),U15(a,b,x,y)),S15(a,b,x,y)+Q15(a,b,x,y),(a SUB 5*b SUB 2-a SUB 2*b SU~
B 5)*(a SUB 5*b SUB 3-a SUB 3*b SUB 5)*(a SUB 5*b SUB 4-a SUB 4*b SUB 5)/(a S~
UB 5*b SUB 1-a SUB 1*b SUB 5)))

I417_4(v):=[(v SUB 1+v SUB 2)*(v SUB 1+v SUB 3),(v SUB 2+v SUB 1)*(v SUB 2+v ~
SUB 3),(v SUB 3+v SUB 1)*(v SUB 3+v SUB 2),v SUB 4^2+v SUB 1*v SUB 2+v SUB 1*~
v SUB 3+v SUB 2*v SUB 3]

I417_3(v):=4/3*R_J(v SUB 1,v SUB 2,v SUB 3,v SUB 4)

I417(a,b,x,y):=(a SUB 2*b SUB 1-a SUB 1*b SUB 2)*(a SUB 3*b SUB 1-a SUB 1*b S~
UB 3)*(a SUB 4*b SUB 1-a SUB 1*b SUB 4)/(a SUB 5*b SUB 1-a SUB 1*b SUB 5)*I41~
7_3(I417_4([U12(a,b,x,y),U13(a,b,x,y),U23(a,b,x,y),U15(a,b,x,y)]))+I417_1(a,b~
,x,y)

II417(a,b,x,y):=II([1,0,0,0,-1],4,5,a,b,x,y)

I427(a,b,x,y):=(b SUB 5*I417(a,b,x,y)-b SUB 1*I414(a,b,x,y))/(a SUB 1*b SUB 5~
-a SUB 5*b SUB 1)

II427(a,b,x,y):=II([0,0,0,0,-1],4,5,a,b,x,y)

I428_1(v):=2*R_D(v SUB 2,v SUB 3,v SUB 1)

I428(a,b,x,y):=(b SUB 1*2*((a SUB 2*b SUB 3-a SUB 3*b SUB 2)*(a SUB 2*b SUB 4~
-a SUB 4*b SUB 2)*I428_1(I414_2([U12(a,b,x,y),U13(a,b,x,y),U23(a,b,x,y)]))/3+~
(x-y)*(SQRT(a SUB 1+y*b SUB 1)*SQRT(a SUB 2+x*b SUB 2)+SQRT(a SUB 1+x*b SUB 1~
)*SQRT(a SUB 2+y*b SUB 2))/(SQRT(a SUB 1+x*b SUB 1)*SQRT(a SUB 1+y*b SUB 1)*(~
SQRT(a SUB 1+y*b SUB 1)*SQRT(a SUB 3+x*b SUB 3)+SQRT(a SUB 1+x*b SUB 1)*SQRT(~
a SUB 3+y*b SUB 3))*(SQRT(a SUB 1+y*b SUB 1)*SQRT(a SUB 4+x*b SUB 4)+SQRT(a S~
UB 1+x*b SUB 1)*SQRT(a SUB 4+y*b SUB 4))))-b SUB 2*I414(a,b,x,y))/(a SUB 2*b ~
SUB 1-a SUB 1*b SUB 2)

II428(a,b,x,y):=II([-1,0,0,0],4,4,a,b,x,y)

U10(a,b,x,y):=SQRT(U12(a,b,x,y)^2-b SUB 2*(a SUB 3*b SUB 1-a SUB 1*b SUB 3)*(~
a SUB 4*b SUB 1-a SUB 1*b SUB 4)/b SUB 1)

Q10(a,b,x,y):=U10(a,b,x,y)/(SQRT(a SUB 1+x*b SUB 1)*SQRT(a SUB 1+y*b SUB 1))

S10(a,b,x,y):=(SQRT(a SUB 2+x*b SUB 2)*SQRT(a SUB 3+x*b SUB 3)*SQRT(a SUB 4+x~
*b SUB 4)/SQRT(a SUB 1+x*b SUB 1)+SQRT(a SUB 2+y*b SUB 2)*SQRT(a SUB 3+y*b SU~
B 3)*SQRT(a SUB 4+y*b SUB 4)/SQRT(a SUB 1+y*b SUB 1))/(x-y)

I429_B(a,b,u,v,w,z):=-b SUB 1/((a SUB 1*b SUB 2-a SUB 2*b SUB 1)*(a SUB 3*b S~
UB 1-a SUB 1*b SUB 3)*(a SUB 4*b SUB 1-a SUB 1*b SUB 4))*(z+u)*(z+v)*(z+w)

I429_2(a,b,p):=4/p*R_C(1,1-b SUB 2*b SUB 3*b SUB 4/(p^2*b SUB 1))

I429_1(a,b,x,y):=I429_2(a,b,I417_P(I429_B(a,b,U12(a,b,x,y),U13(a,b,x,y),U23(a~
,b,x,y),U10(a,b,x,y)),S10(a,b,x,y)+Q10(a,b,x,y),b SUB 2*b SUB 3*b SUB 4/b SUB~
 1))

I429(a,b,x,y):=(a SUB 2*b SUB 1-a SUB 1*b SUB 2)*(a SUB 3*b SUB 1-a SUB 1*b S~
UB 3)*(a SUB 4*b SUB 1-a SUB 1*b SUB 4)/b SUB 1*I417_3(I417_4([U12(a,b,x,y),U~
13(a,b,x,y),U23(a,b,x,y),U10(a,b,x,y)]))+I429_1(a,b,x,y)

II429(a,b,x,y):=II([1,0,0,0],4,4,a,b,x,y)

I414A(a,b,x,y):=I414([a SUB 1,a SUB 2,a SUB 3,1],[b SUB 1,b SUB 2,b SUB 3,0],~
x,y)

II414A(a,b,x,y):=II([0,0,0],3,3,a,b,x,y)

I417A(a,b,x,y):=I417([a SUB 1,a SUB 2,a SUB 3,1,a SUB 4],[b SUB 1,b SUB 2,b S~
UB 3,0,b SUB 4],x,y)

II417A(a,b,x,y):=II([1,0,0,-1],3,4,a,b,x,y)

I427A(a,b,x,y):=I427([a SUB 1,a SUB 2,a SUB 3,1,a SUB 4],[b SUB 1,b SUB 2,b S~
UB 3,0,b SUB 4],x,y)

II427A(a,b,x,y):=II([0,0,0,-1],3,4,a,b,x,y)

I428A(a,b,x,y):=I428([a SUB 1,a SUB 2,a SUB 3,1],[b SUB 1,b SUB 2,b SUB 3,0],~
x,y)

II428A(a,b,x,y):=II([-1,0,0],3,3,a,b,x,y)

I429A(a,b,x,y):=I429([a SUB 1,a SUB 2,a SUB 3,1],[b SUB 1,b SUB 2,b SUB 3,0],~
x,y)

II429A(a,b,x,y):=II([1,0,0],3,3,a,b,x,y)

list1998:=[[IA(0),'(I414(a,b,x,y))],[IA(e1-e5),'(I417(a,b,x,y))],[IA(-e5),'(I~
427(a,b,x,y))],[IA(-e1),'(I428(a,b,x,y))],[IA(-e2),'(I428([a SUB 2,a SUB 1,a ~
SUB 3,a SUB 4],[b SUB 2,b SUB 1,b SUB 3,b SUB 4],x,y))],[IA(-e3),'(I428([a SU~
B 3,a SUB 2,a SUB 1,a SUB 4],[b SUB 3,b SUB 2,b SUB 1,b SUB 4],x,y))],[IA(-e4~
),'(I428([a SUB 4,a SUB 2,a SUB 3,a SUB 1],[b SUB 4,b SUB 2,b SUB 3,b SUB 1],~
x,y))],[IA(e1),'(I429(a,b,x,y))],[IA(e2),'(I429([a SUB 2,a SUB 1,a SUB 3,a SU~
B 4],[b SUB 2,b SUB 1,b SUB 3,b SUB 4],x,y))],[IA(e3),'(I429([a SUB 3,a SUB 2~
,a SUB 1,a SUB 4],[b SUB 3,b SUB 2,b SUB 1,b SUB 4],x,y))],[IA(e4),'(I429([a ~
SUB 4,a SUB 2,a SUB 3,a SUB 1],[b SUB 4,b SUB 2,b SUB 3,b SUB 1],x,y))]]

IAS_AUX(v):=SELECT(IF(k SUB 1=v,1,0)=1,k,list1998)

IAS(v):=VECTOR((IAS_AUX(v SUB i_)) SUB 1 SUB 2,i_,1,DIMENSION(v))

list1998a:=[[IA(0),'(I414A(a,b,x,y))],[IA(e1-e4),'(I417A(a,b,x,y))],[IA(-e4),~
'(I427A(a,b,x,y))],[IA(-e1),'(I428A(a,b,x,y))],[IA(-e2),'(I428A([a SUB 2,a SU~
B 1,a SUB 3],[b SUB 2,b SUB 1,b SUB 3],x,y))],[IA(-e3),'(I428A([a SUB 3,a SUB~
 2,a SUB 1],[b SUB 3,b SUB 2,b SUB 1],x,y))],[IA(e1),'(I429A(a,b,x,y))],[IA(e~
2),'(I429A([a SUB 2,a SUB 1,a SUB 3],[b SUB 2,b SUB 1,b SUB 3],x,y))],[IA(e3)~
,'(I429A([a SUB 3,a SUB 2,a SUB 1],[b SUB 3,b SUB 2,b SUB 1],x,y))]]

IASA_AUX(v):=SELECT(IF(k SUB 1=v,1,0)=1,k,list1998a)

IASA(v):=VECTOR((IASA_AUX(v SUB i_)) SUB 1 SUB 2,i_,1,DIMENSION(v))

U12Y(a,b,y):=SQRT(b SUB 1)*SQRT(b SUB 2)*SQRT(a SUB 3+y*b SUB 3)*SQRT(a SUB 4~
+y*b SUB 4)+SQRT(b SUB 3)*SQRT(b SUB 4)*SQRT(a SUB 1+y*b SUB 1)*SQRT(a SUB 2+~
y*b SUB 2)

U13Y(a,b,y):=SQRT(b SUB 1)*SQRT(b SUB 3)*SQRT(a SUB 2+y*b SUB 2)*SQRT(a SUB 4~
+y*b SUB 4)+SQRT(b SUB 2)*SQRT(b SUB 4)*SQRT(a SUB 1+y*b SUB 1)*SQRT(a SUB 3+~
y*b SUB 3)

U23Y(a,b,y):=SQRT(b SUB 2)*SQRT(b SUB 3)*SQRT(a SUB 1+y*b SUB 1)*SQRT(a SUB 4~
+y*b SUB 4)+SQRT(b SUB 1)*SQRT(b SUB 4)*SQRT(a SUB 2+y*b SUB 2)*SQRT(a SUB 3+~
y*b SUB 3)

I414Y(a,b,y):=I414_1(I414_2([U12Y(a,b,y),U13Y(a,b,y),U23Y(a,b,y)]))

II414Y(a,b,y):=II([0,0,0,0],4,4,a,b,inf,y)

I48(z):=I414Y(z,[1,1,1,1],0)

II48(z):=INT(1/PRODUCT(SQRT(t_+z SUB j_),j_,1,4),t_,0,inf)

I414X(a,b,x):=I414Y(a,-b,-x)

II414X(a,b,x):=II([0,0,0,0],4,4,a,b,x,-inf)

U15Y(a,b,y):=SQRT(U12Y(a,b,y)^2-(a SUB 3*b SUB 1-a SUB 1*b SUB 3)*(a SUB 4*b ~
SUB 1-a SUB 1*b SUB 4)*(a SUB 5*b SUB 2-a SUB 2*b SUB 5)/(a SUB 5*b SUB 1-a S~
UB 1*b SUB 5))

Q15Y(a,b,y):=SQRT(b SUB 5)*SQRT(a SUB 5+y*b SUB 5)/(SQRT(b SUB 1)*SQRT(a SUB ~
1+y*b SUB 1))*U15Y(a,b,y)

S15Y(a,b,y):=(a SUB 5*SQRT(b SUB 2)*SQRT(b SUB 3)*SQRT(b SUB 4)*SQRT(a SUB 1+~
y*b SUB 1)+b SUB 5*(SQRT(b SUB 1)*SQRT(a SUB 2+y*b SUB 2)*SQRT(a SUB 3+y*b SU~
B 3)*SQRT(a SUB 4+y*b SUB 4)+y*SQRT(b SUB 2)*SQRT(b SUB 3)*SQRT(b SUB 4)*SQRT~
(a SUB 1+y*b SUB 1)))/(SQRT(b SUB 1)*SQRT(a SUB 1+y*b SUB 1))

I417Y_1(a,b,y):=I417_2(a,b,I417_P(I417_B(a,b,U12Y(a,b,y),U13Y(a,b,y),U23Y(a,b~
,y),U15Y(a,b,y)),S15Y(a,b,y)+Q15Y(a,b,y),(a SUB 5*b SUB 2-a SUB 2*b SUB 5)*(a~
 SUB 5*b SUB 3-a SUB 3*b SUB 5)*(a SUB 5*b SUB 4-a SUB 4*b SUB 5)/(a SUB 5*b ~
SUB 1-a SUB 1*b SUB 5)))

I417Y(a,b,y):=(a SUB 2*b SUB 1-a SUB 1*b SUB 2)*(a SUB 3*b SUB 1-a SUB 1*b SU~
B 3)*(a SUB 4*b SUB 1-a SUB 1*b SUB 4)/(a SUB 5*b SUB 1-a SUB 1*b SUB 5)*I417~
_3(I417_4([U12Y(a,b,y),U13Y(a,b,y),U23Y(a,b,y),U15Y(a,b,y)]))+I417Y_1(a,b,y)

II417Y(a,b,y):=II([1,0,0,0,-1],4,5,a,b,inf,y)

I410(z):=I417Y(z,[1,1,1,1,1],0)

II410(z):=INT((t_+z SUB 1)/((t_+z SUB 5)*PRODUCT(SQRT(t_+z SUB j_),j_,1,4)),t~
_,0,inf)

I417X(a,b,x):=I417Y(a,-b,-x)

II417X(a,b,x):=II([1,0,0,0,-1],4,5,a,b,x,-inf)

I427Y(a,b,y):=(b SUB 5*I417Y(a,b,y)-b SUB 1*I414Y(a,b,y))/(a SUB 1*b SUB 5-a ~
SUB 5*b SUB 1)

II427Y(a,b,y):=II([0,0,0,0,-1],4,5,a,b,inf,y)

I427X(a,b,x):=I427Y(a,-b,-x)

II427X(a,b,x):=II([0,0,0,0,-1],4,5,a,b,x,-inf)

I428Y(a,b,y):=(b SUB 1*2*((a SUB 2*b SUB 3-a SUB 3*b SUB 2)*(a SUB 2*b SUB 4-~
a SUB 4*b SUB 2)*I428_1(I414_2([U12Y(a,b,y),U13Y(a,b,y),U23Y(a,b,y)]))/3+(SQR~
T(b SUB 1)*SQRT(a SUB 2+y*b SUB 2)+SQRT(b SUB 2)*SQRT(a SUB 1+y*b SUB 1))*(SQ~
RT(b SUB 1)*SQRT(a SUB 3+y*b SUB 3)-SQRT(b SUB 3)*SQRT(a SUB 1+y*b SUB 1))*(S~
QRT(b SUB 1)*SQRT(a SUB 4+y*b SUB 4)-SQRT(b SUB 4)*SQRT(a SUB 1+y*b SUB 1))/(~
SQRT(b SUB 1)*SQRT(a SUB 1+y*b SUB 1)*(a SUB 3*b SUB 1-a SUB 1*b SUB 3)*(a SU~
B 4*b SUB 1-a SUB 1*b SUB 4)))-b SUB 2*I414Y(a,b,y))/(a SUB 2*b SUB 1-a SUB 1~
*b SUB 2)

II428Y(a,b,y):=II([-1,0,0,0],4,4,a,b,inf,y)

I428X(a,b,x):=I428Y(a,-b,-x)

II428X(a,b,x):=II([-1,0,0,0],4,4,a,b,x,-inf)

list1998y:=[[IA(0),'(I414Y(a,b,y))],[IA(e1-e5),'(I417Y(a,b,y))],[IA(-e5),'(I4~
27Y(a,b,y))],[IA(-e1),'(I428Y(a,b,y))],[IA(-e2),'(I428Y([a SUB 2,a SUB 1,a SU~
B 3,a SUB 4],[b SUB 2,b SUB 1,b SUB 3,b SUB 4],y))],[IA(-e3),'(I428Y([a SUB 3~
,a SUB 2,a SUB 1,a SUB 4],[b SUB 3,b SUB 2,b SUB 1,b SUB 4],y))],[IA(-e4),'(I~
428Y([a SUB 4,a SUB 2,a SUB 3,a SUB 1],[b SUB 4,b SUB 2,b SUB 3,b SUB 1],y))]~
]

IASY_AUX(v):=SELECT(IF(k SUB 1=v,1,0)=1,k,list1998y)

IASY(v):=VECTOR((IASY_AUX(v SUB i_)) SUB 1 SUB 2,i_,1,DIMENSION(v))

list1998x:=[[IA(0),'(I414X(a,b,x))],[IA(e1-e5),'(I417X(a,b,x))],[IA(-e5),'(I4~
27X(a,b,x))],[IA(-e1),'(I428X(a,b,x))]]

IASX_AUX(v):=SELECT(IF(k SUB 1=v,1,0)=1,k,list1998x)

IASX(v):=VECTOR((IASX_AUX(v SUB i_)) SUB 1 SUB 2,i_,1,DIMENSION(v))

U12B(a,b,x,y):=(SQRT(a SUB 1+x*b SUB 1)*SQRT(a SUB 2+x*b SUB 2)+SQRT(a SUB 1+~
y*b SUB 1)*SQRT(a SUB 2+y*b SUB 2))/(x-y)

U10B(a,b,x,y):=(SQRT(a SUB 1+y*b SUB 1)*SQRT(a SUB 2+x*b SUB 2)+SQRT(a SUB 1+~
x*b SUB 1)*SQRT(a SUB 2+y*b SUB 2))/(x-y)

I414B_1(u,v):=4*R_C(1,2*v/(u+v))/(u+v)

I414B(a,b,x,y):=I414B_1(U12B(a,b,x,y),U10B(a,b,x,y))

II414B(a,b,x,y):=II([0,0],2,2,a,b,x,y)

U13C(a,b,x,y):=SQRT(a SUB 3+x*b SUB 3)*SQRT(a SUB 3+y*b SUB 3)*(SQRT(a SUB 1+~
y*b SUB 1)*SQRT(a SUB 2+x*b SUB 2)+SQRT(a SUB 1+x*b SUB 1)*SQRT(a SUB 2+y*b S~
UB 2))/(x-y)

I427B_1(x1,x2,x3,y1,y2,y3):=SQRT(x1)*SQRT(x2)*y3+SQRT(y1)*SQRT(y2)*x3

U12C(a,b,x,y):=I427B_1(a SUB 1+x*b SUB 1,a SUB 2+x*b SUB 2,a SUB 3+x*b SUB 3,~
a SUB 1+y*b SUB 1,a SUB 2+y*b SUB 2,a SUB 3+y*b SUB 3)/(x-y)

I427B(a,b,x,y):=I414B_1(U12C(a,b,x,y),U13C(a,b,x,y))

II427B(a,b,x,y):=II([0,0,-1],2,3,a,b,x,y)

I417B(a,b,x,y):=(I427B(a,b,x,y)*(a SUB 1*b SUB 3-a SUB 3*b SUB 1)+I414B(a,b,x~
,y)*b SUB 1)/b SUB 3

II417B(a,b,x,y):=II([1,0,-1],2,3,a,b,x,y)

I428B_1(x1,x2,y1,y2):=SQRT(x1)*y1*SQRT(x2)+SQRT(y1)*x1*SQRT(y2)

I428B(a,b,x,y):=2*(x-y)/I428B_1(a SUB 1+x*b SUB 1,a SUB 2+x*b SUB 2,a SUB 1+y~
*b SUB 1,a SUB 2+y*b SUB 2)

II428B(a,b,x,y):=II([-1,0],2,2,a,b,x,y)

list1998b:=[[IA(0),'(I414B(a,b,x,y))],[IA(e1-e3),'(I417B(a,b,x,y))],[IA(-e3),~
'(I427B(a,b,x,y))],[IA(-e1),'(I428B(a,b,x,y))],[IA(-e2),'(I428B([a SUB 2,a SU~
B 1],[b SUB 2,b SUB 1],x,y))]]

IASB_AUX(v):=SELECT(IF(k SUB 1=v,1,0)=1,k,list1998b)

IASB(v):=VECTOR((IASB_AUX(v SUB i_)) SUB 1 SUB 2,i_,1,DIMENSION(v))

I414C(a,b,x,y):=2*SQRT(a SUB 1+x*b SUB 1)/b SUB 1-2*SQRT(a SUB 1+y*b SUB 1)/b~
 SUB 1

II414C(a,b,x,y):=II([0],1,1,a,b,x,y)

U12D(a,b,x,y):=((a SUB 2+y*b SUB 2)*SQRT(a SUB 1+x*b SUB 1)+(a SUB 2+x*b SUB ~
2)*SQRT(a SUB 1+y*b SUB 1))/(x-y)

U10D(a,b,x,y):=SQRT(a SUB 2+x*b SUB 2)*SQRT(a SUB 2+y*b SUB 2)*(SQRT(a SUB 1+~
y*b SUB 1)+SQRT(a SUB 1+x*b SUB 1))/(x-y)

I427C(a,b,x,y):=I414B_1(U12D(a,b,x,y),U10D(a,b,x,y))

II427C(a,b,x,y):=II([0,-1],1,2,a,b,x,y)

I417C(a,b,x,y):=(I427C(a,b,x,y)*(a SUB 1*b SUB 2-a SUB 2*b SUB 1)+2*(SQRT(a S~
UB 1+x*b SUB 1)-SQRT(a SUB 1+y*b SUB 1)))/b SUB 2

II417C(a,b,x,y):=II([1,-1],1,2,a,b,x,y)

I428C(a,b,x,y):=2*(1/SQRT(a SUB 1+y*b SUB 1)-1/SQRT(a SUB 1+x*b SUB 1))/b SUB~
 1

II428C(a,b,x,y):=II([-1],1,1,a,b,x,y)

list1998c:=[[IA(0),'(I414C(a,b,x,y))],[IA(e1-e2),'(I417C(a,b,x,y))],[IA(-e2),~
'(I427C(a,b,x,y))],[IA(-e1),'(I428C(a,b,x,y))]]

IASC_AUX(v):=SELECT(IF(k SUB 1=v,1,0)=1,k,list1998c)

IASC(v):=VECTOR((IASC_AUX(v SUB i_)) SUB 1 SUB 2,i_,1,DIMENSION(v))

IS(v,h):=IF(h=4,IAS(v),IF(h=3,IASA(v),IF(h=2,IASB(v),IF(h=1,IASC(v),[]))))

I414AY(a,b,y):=I414Y([a SUB 1,a SUB 2,a SUB 3,1],[b SUB 1,b SUB 2,b SUB 3,0],~
y)

II414AY(a,b,y):=2/(SQRT(b SUB 1)*SQRT(b SUB 2)*SQRT(b SUB 3))*RF(a SUB 1/b SU~
B 1,a SUB 2/b SUB 2,a SUB 3/b SUB 3)-II414A(a,b,y,0)

I414AX(a,b,x):=I414AY(a,-b,-x)

II414AX(a,b,x):=II414AY(a,-b,-x)

I427AY(a,b,y):=I427Y([a SUB 1,a SUB 2,a SUB 3,1,a SUB 4],[b SUB 1,b SUB 2,b S~
UB 3,0,b SUB 4],y)

II427AY(a,b,y):=2/(3*SQRT(b SUB 1)*SQRT(b SUB 2)*SQRT(b SUB 3)*b SUB 4)*RJ(a ~
SUB 1/b SUB 1,a SUB 2/b SUB 2,a SUB 3/b SUB 3,a SUB 4/b SUB 4)-II427A(a,b,y,0~
)

I427AX(a,b,x):=I427AY(a,-b,-x)

II427AX(a,b,x):=II427AY(a,-b,-x)

I417AY(a,b,y):=I417Y([a SUB 1,a SUB 2,a SUB 3,1,a SUB 4],[b SUB 1,b SUB 2,b S~
UB 3,0,b SUB 4],y)

II417AY(a,b,y):=(II414AY(a,b,y)*b SUB 1+II427AY(a,b,y)*(a SUB 1*b SUB 4-a SUB~
 4*b SUB 1))/b SUB 4

I417AX(a,b,x):=I417AY(a,-b,-x)

II417AX(a,b,x):=II417AY(a,-b,-x)

I428AY(a,b,y):=I428Y([a SUB 1,a SUB 2,a SUB 3,1],[b SUB 1,b SUB 2,b SUB 3,0],~
y)

II428AY(a,b,y):=2/(3*SQRT(b SUB 2)*SQRT(b SUB 3)*b SUB 1^(3/2))*RD(a SUB 2/b ~
SUB 2,a SUB 3/b SUB 3,a SUB 1/b SUB 1)-II428A(a,b,y,0)

I428AX(a,b,x):=I428AY(a,-b,-x)

II428AX(a,b,x):=II428AY(a,-b,-x)

list1998ay:=[[IA(0),'(I414AY(a,b,y))],[IA(e1-e4),'(I417AY(a,b,y))],[IA(-e4),'~
(I427AY(a,b,y))],[IA(-e1),'(I428AY(a,b,y))],[IA(-e2),'(I428AY([a SUB 2,a SUB ~
1,a SUB 3],[b SUB 2,b SUB 1,b SUB 3],y))],[IA(-e3),'(I428AY([a SUB 3,a SUB 2,~
a SUB 1],[b SUB 3,b SUB 2,b SUB 1],y))]]

IASAY_AUX(v):=SELECT(IF(k SUB 1=v,1,0)=1,k,list1998ay)

IASAY(v):=VECTOR((IASAY_AUX(v SUB i_)) SUB 1 SUB 2,i_,1,DIMENSION(v))

ISY(v,h):=IF(h=4,IASY(v),IF(h=3,IASAY(v),[]))

list1998ax:=[[IA(0),'(I414AX(a,b,x))],[IA(e1-e4),'(I417AX(a,b,x))],[IA(-e4),'~
(I427AX(a,b,x))],[IA(-e1),'(I428AX(a,b,x))]]

IASAX_AUX(v):=SELECT(IF(k SUB 1=v,1,0)=1,k,list1998ax)

IASAX(v):=VECTOR((IASAX_AUX(v SUB i_)) SUB 1 SUB 2,i_,1,DIMENSION(v))

ISX(v,h):=IF(h=4,IASX(v),IF(h=3,IASAX(v),[]))