rat=1/(a+b1*x+b2*y+b3*z+c3*x*y+c2*x*z+c1*y*z+d1*x^2*y+d2*y^2*z+d3*z^2*x)

P3=-216*a*d1*d2*d3*x^3+144*b1*c3*d2*d3*x^3+144*b2*c1*d1*d3*x^3+144*b3*c2*d1*d2*x^3-48*c1^2*c2*d1*x^3-48*c1*c3^2*d3*x^3-48*c2^2*c3*d2*x^3+24*a*b1*c2*d2*x^2+24*a*b2*c3*d3*x^2+24*a*b3*c1*d1*x^2+24*a*c1*c2*c3*x^2-48*b1^2*b3*d2*x^2+16*b1^2*c1^2*x^2-48*b1*b2^2*d3*x^2-16*b1*b2*c1*c2*x^2-16*b1*b3*c1*c3*x^2+16*b2^2*c2^2*x^2-48*b2*b3^2*d1*x^2-16*b2*b3*c2*c3*x^2+16*b3^2*c3^2*x^2-8*a^2*b1*c1*x-8*a^2*b2*c2*x-8*a^2*b3*c3*x+24*a*b1*b2*b3*x+a^4;
P6=-5832*d1^2*d2^2*d3^2*x^6+3888*b1*c2*d1*d2^2*d3*x^5+3888*b2*c3*d1*d2*d3^2*x^5+3888*b3*c1*d1^2*d2*d3*x^5-864*c1^3*d1^2*d3*x^5-1296*c1*c2*c3*d1*d2*d3*x^5-864*c2^3*d1*d2^2*x^5-864*c3^3*d2*d3^2*x^5-1296*a*b1*c1*d1*d2*d3*x^4-1296*a*b2*c2*d1*d2*d3*x^4-1296*a*b3*c3*d1*d2*d3*x^4+864*a*c1^2*c3*d1*d3*x^4+864*a*c1*c2^2*d1*d2*x^4+864*a*c2*c3^2*d2*d3*x^4-864*b1^3*d2^2*d3*x^4+864*b1^2*c1*c3*d2*d3*x^4+216*b1^2*c2^2*d2^2*x^4-1296*b1*b2*b3*d1*d2*d3*x^4+864*b1*b2*c1^2*d1*d3*x^4-1296*b1*b2*c2*c3*d2*d3*x^4-1296*b1*b3*c1*c2*d1*d2*x^4+864*b1*b3*c3^2*d2*d3*x^4+288*b1*c1^3*c2*d1*x^4-576*b1*c1^2*c3^2*d3*x^4-144*b1*c1*c2^2*c3*d2*x^4-864*b2^3*d1*d3^2*x^4+864*b2^2*c1*c2*d1*d3*x^4+216*b2^2*c3^2*d3^2*x^4-1296*b2*b3*c1*c3*d1*d3*x^4+864*b2*b3*c2^2*d1*d2*x^4-576*b2*c1^2*c2^2*d1*x^4-144*b2*c1*c2*c3^2*d3*x^4+288*b2*c2^3*c3*d2*x^4-864*b3^3*d1^2*d2*x^4+216*b3^2*c1^2*d1^2*x^4+864*b3^2*c2*c3*d1*d2*x^4-144*b3*c1^2*c2*c3*d1*x^4+288*b3*c1*c3^3*d3*x^4-576*b3*c2^2*c3^2*d2*x^4+216*c1^2*c2^2*c3^2*x^4+540*a^3*d1*d2*d3*x^3-648*a^2*b1*c3*d2*d3*x^3-648*a^2*b2*c1*d1*d3*x^3-648*a^2*b3*c2*d1*d2*x^3-72*a^2*c1^2*c2*d1*x^3-72*a^2*c1*c3^2*d3*x^3-72*a^2*c2^2*c3*d2*x^3+864*a*b1^2*b2*d2*d3*x^3-144*a*b1^2*c1*c2*d2*x^3+720*a*b1*b2*c1*c3*d3*x^3-144*a*b1*b2*c2^2*d2*x^3+864*a*b1*b3^2*d1*d2*x^3-144*a*b1*b3*c1^2*d1*x^3+720*a*b1*b3*c2*c3*d2*x^3-144*a*b1*c1^2*c2*c3*x^3+864*a*b2^2*b3*d1*d3*x^3-144*a*b2^2*c2*c3*d3*x^3+720*a*b2*b3*c1*c2*d1*x^3-144*a*b2*b3*c3^2*d3*x^3-144*a*b2*c1*c2^2*c3*x^3-144*a*b3^2*c1*c3*d1*x^3-144*a*b3*c1*c2*c3^2*x^3+288*b1^3*b3*c1*d2*x^3-64*b1^3*c1^3*x^3-576*b1^2*b2^2*c1*d3*x^3-144*b1^2*b2*b3*c2*d2*x^3+96*b1^2*b2*c1^2*c2*x^3-576*b1^2*b3^2*c3*d2*x^3+96*b1^2*b3*c1^2*c3*x^3+288*b1*b2^3*c2*d3*x^3-144*b1*b2^2*b3*c3*d3*x^3+96*b1*b2^2*c1*c2^2*x^3-144*b1*b2*b3^2*c1*d1*x^3+48*b1*b2*b3*c1*c2*c3*x^3+96*b1*b3^2*c1*c3^2*x^3-64*b2^3*c2^3*x^3-576*b2^2*b3^2*c2*d1*x^3+96*b2^2*b3*c2^2*c3*x^3+288*b2*b3^3*c3*d1*x^3+96*b2*b3^2*c2*c3^2*x^3-64*b3^3*c3^3*x^3+36*a^3*b1*c2*d2*x^2+36*a^3*b2*c3*d3*x^2+36*a^3*b3*c1*d1*x^2+36*a^3*c1*c2*c3*x^2-72*a^2*b1^2*b3*d2*x^2+48*a^2*b1^2*c1^2*x^2-72*a^2*b1*b2^2*d3*x^2+24*a^2*b1*b2*c1*c2*x^2+24*a^2*b1*b3*c1*c3*x^2+48*a^2*b2^2*c2^2*x^2-72*a^2*b2*b3^2*d1*x^2+24*a^2*b2*b3*c2*c3*x^2+48*a^2*b3^2*c3^2*x^2-144*a*b1^2*b2*b3*c1*x^2-144*a*b1*b2^2*b3*c2*x^2-144*a*b1*b2*b3^2*c3*x^2+216*b1^2*b2^2*b3^2*x^2-12*a^4*b1*c1*x-12*a^4*b2*c2*x-12*a^4*b3*c3*x+36*a^3*b1*b2*b3*x+a^6;
subs={a->31,b1->17,b2->1,b3->11,c1->13,c2-> 1,c3->19,d1->10,d2->9,d3->7};

ratser=Expand[Normal[Series[rat/.subs,{x,0,10},{y,0,10},{z,0,10}]]];
diag=Sum[Coefficient[Coefficient[Coefficient[ratser,x,n],y,n],z,n]*x^n,{n,0,10}]

hyper=1/P3^(1/4)*Hypergeometric2F1[1/12,5/12,1,1-P6^2/P3^3]/.subs;
hser=Series[hyper,{x,0,10}]

diag-hser