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

P4=16*d^2*e^2*x^4-16*b1*c1*d*e*x^3+224*b2*c2*d*e*x^3-16*b3*c3*d*e*x^3-48*c1^2*c2*d*x^3-48*c2*c3^2*e*x^3-8*a^2*d*e*x^2+24*a*b1*c3*e*x^2+24*a*b3*c1*d*x^2+24*a*c1*c2*c3*x^2-48*b1^2*b2*e*x^2+16*b1^2*c1^2*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*d*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=-64*d^3*e^3*x^6+96*b1*c1*d^2*e^2*x^5+2112*b2*c2*d^2*e^2*x^5+96*b3*c3*d^2*e^2*x^5-576*c1^2*c2*d^2*e*x^5-576*c2*c3^2*d*e^2*x^5+48*a^2*d^2*e^2*x^4-144*a*b1*c3*d*e^2*x^4-144*a*b3*c1*d^2*e*x^4+720*a*c1*c2*c3*d*e*x^4-576*b1^2*b2*d*e^2*x^4+96*b1^2*c1^2*d*e*x^4+216*b1^2*c3^2*e^2*x^4-960*b1*b2*c1*c2*d*e*x^4+48*b1*b3*c1*c3*d*e*x^4+288*b1*c1^3*c2*d*x^4-144*b1*c1*c2*c3^2*e*x^4+2112*b2^2*c2^2*d*e*x^4-576*b2*b3^2*d^2*e*x^4-960*b2*b3*c2*c3*d*e*x^4-576*b2*c1^2*c2^2*d*x^4-576*b2*c2^2*c3^2*e*x^4+216*b3^2*c1^2*d^2*x^4+96*b3^2*c3^2*d*e*x^4-144*b3*c1^2*c2*c3*d*x^4+288*b3*c2*c3^3*e*x^4+216*c1^2*c2^2*c3^2*x^4+24*a^2*b1*c1*d*e*x^3-480*a^2*b2*c2*d*e*x^3+24*a^2*b3*c3*d*e*x^3-72*a^2*c1^2*c2*d*x^3-72*a^2*c2*c3^2*e*x^3-144*a*b1^2*c1*c3*e*x^3+720*a*b1*b2*b3*d*e*x^3+720*a*b1*b2*c2*c3*e*x^3-144*a*b1*b3*c1^2*d*x^3-144*a*b1*b3*c3^2*e*x^3-144*a*b1*c1^2*c2*c3*x^3+720*a*b2*b3*c1*c2*d*x^3-144*a*b2*c1*c2^2*c3*x^3-144*a*b3^2*c1*c3*d*x^3-144*a*b3*c1*c2*c3^2*x^3+288*b1^3*b2*c1*e*x^3-64*b1^3*c1^3*x^3-576*b1^2*b2^2*c2*e*x^3-144*b1^2*b2*b3*c3*e*x^3+96*b1^2*b2*c1^2*c2*x^3+96*b1^2*b3*c1^2*c3*x^3+96*b1*b2^2*c1*c2^2*x^3-144*b1*b2*b3^2*c1*d*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*d*x^3+96*b2^2*b3*c2^2*c3*x^3+288*b2*b3^3*c3*d*x^3+96*b2*b3^2*c2*c3^2*x^3-64*b3^3*c3^3*x^3-12*a^4*d*e*x^2+36*a^3*b1*c3*e*x^2+36*a^3*b3*c1*d*x^2+36*a^3*c1*c2*c3*x^2-72*a^2*b1^2*b2*e*x^2+48*a^2*b1^2*c1^2*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*d*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->13,b1->11,b2->17,b3->23,c1->19,c2->233,c3->33,d->211,e->215};

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/P4^(1/4)*Hypergeometric2F1[1/12,5/12,1,1-P6^2/P4^3]/.subs;
hser=Series[hyper,{x,0,10}]

diag-hser