L6iinv[arg_]:=If[Sort[Factor[{arg,1/arg}]][[1]]===Factor[arg],L6i[Factor[arg]],-L6i[Factor[1/arg]]];
L6can[expr_]:=expr/.L6i[arg_]:>L6iinv[arg]

L6piinv[arg_]:=If[Sort[Factor[{arg,1/arg}]][[1]]===Factor[arg],Lip[Factor[arg]],-Lip[Factor[1/arg]]];
L6pcan[expr_]:=expr/.Lip[arg_]:>L6piinv[arg]

li6termsAlg3Full[y_] := -1/(b*d*a*b*(b-a)*(2a + b)) * L6i[t/((1-y)^b y^a)]+
-1/(a*d*a*b*(b-a)*(a + 2b)) * L6i[t/((1-y)^a y^b)]+
-1/(b*d*a*b*(b-a)*(2a + b))*L6i[t/((y-1)^b y^d)]+
-1/(a*d*a*b*(b-a)*(a + 2b))*L6i[t/((y-1)^a y^d)]+

+((-1)/d) Lip[1-(1/p)]+((-1)/(2a)) Lip[(1/(1-p))]+(1/(2b))Lip[(1/p)]+
((-1)/(2d)) Lip[(1/((1-(1/p)) (1-y)))]-((-1)/(2d)) Lip[(((-1+(1/p)) (1-y))/y)]+
(1/(2d)) Lip[((1-(1/p))/(1-y))]-(1/(2d))Lip[((-1+y)/((1-(1/p)) y))]+
((-1)/a) Lip[((1-p)/(1-y))]-((-1)/a) Lip[-((1-y)/((1-p) y))]+
(1/b) Lip[(p/(1-y))]-(1/b)Lip[-((1-y)/(p y))]+
(1/(6(b-a))) Lip[((1-y)/((1-(1/p)) (p^2) (y^2)))]+ (1/(6(b-a)))Lip[(((-1+y) y)/((1-(1/p)) (p^2)))]+
((9a+2b)/(4a^2)) Lip[(1/((1-p) y))]+((9a+2b)/(4a^2)) Lip[(y/(1-p))]+
((2a+9b)/(4b^2)) Lip[(p/y)]+((2a+9b)/(4b^2)) Lip[p y]+
-(1/(12(2a+b))) Lip[((((1-y))^2)/((((1-(1/p)))^2) p y))]-(1/(12(a + 2b))) Lip[-((p y)/((1-(1/p)) (((1-y))^2)))];

li6term3[x_] := -I51i[1,x]-L6i[1-x]+3 L6i[(1/x)]-L6i[((-1+x)/x)]+((3 L6i[x])/2)+(1/2) L6i[1-(1/y)]+(5/2) L6i[x (1-(1/y))]+(1/2) L6i[(1/(1-y))]+(5/2) L6i[(x/(1-y))]+(1/2) L6i[1-y]-(1/2) L6i[(1/((-1+x) (-1+y)))]-(5/4) L6i[-(1/(x (-1+y)))]+(1/2) L6i[-((-1+x)/(x (-1+y)))]-(1/2) L6i[((-1+y)/(-1+x))]+(5/4) L6i[-((-1+y)/x)]+(1/2) L6i[-(((-1+x) (-1+y))/x)]+(1/18) L6i[-((-1+y)/((-1+x) x (y^2)))]-(1/36) L6i[-(((((-1+x))^2) (-1+y))/(x (y^2)))]+(1/2) L6i[(1/y)]-(1/2) L6i[-(1/((-1+x) y))]+(5/4) L6i[(1/(x y))]+(1/2) L6i[((-1+x)/(x y))]-(1/18) L6i[-((-1+x)/((x^2) (-1+y) y))]-(1/2) L6i[((-1+y)/y)]-(1/2) L6i[-((-1+y)/((-1+x) y))]-(5/4) L6i[((-1+y)/(x y))]+(1/2) L6i[(((-1+x) (-1+y))/(x y))]-(1/18) L6i[-(((-1+x) (((-1+y))^2))/((x^2) y))]+((L6i[y])/2)-(1/2) L6i[-(y/(-1+x))]-(5/4) L6i[(y/x)]+(1/2) L6i[(((-1+x) y)/x)]+(5/2) L6i[x y]+(1/18) L6i[(y/((-1+x) x (((-1+y))^2)))]-(1/36) L6i[(((((-1+x))^2) y)/(x (((-1+y))^2)))]-(1/2) L6i[-(y/((-1+x) (-1+y)))]+(5/4) L6i[(y/(x (-1+y)))]+(1/2) L6i[(((-1+x) y)/(x (-1+y)))]+(1/18) L6i[(((-1+y) y)/((-1+x) x))]-(1/36) L6i[(((((-1+x))^2) (-1+y) y)/x)]-(1/18) L6i[(((-1+x) (y^2))/((x^2) (-1+y)))];

leadingsimp[expr_] := expr//.{(1-(1/y))^exp_:>(-1)^exp(1-y)^exp/(y)^exp,(1/y)^exp_:>y^-exp,(-1/y)^exp_:>(-1)^exp y^-exp,(-1+(1/(1-y)))^exp_:>y^exp/(1-y)^exp,(1-(1/(1-y)))^exp_:>(((-1)^exp(y)^exp)/((1-y)^exp)),((1/(1-y)))^exp_:>(1-y)^-exp,(-1+y)^exp_:>(-1)^exp(1-y)^exp, (-1)^-exp_:>(-1)^exp};
Llead[expr_] := expr/.L6i[arg_]:>L6i[leadingsimp[arg/.{d->-a-b}]];

fe = ((li6termsAlg3Full[y] + li6termsAlg3Full[1-1/y] + li6termsAlg3Full[1/(1-y)]) - (((-1/a)li6term3[1-p] + (1/b)li6term3[p])/.{I51i->I51roots,L6i->Lip}))//L6can//L6pcan//Llead//Expand;

(* --------- *)

nielsenAsLip=-(-((I51roots[1,1-p])/a)+((I51roots[1,p])/b)) + ( (b+4a)/(a^2)Lip[1-p]+(4b+a)/(b^2)Lip[(1/p)]-1/(a + b)Lip[((-1+p)/p)]);

feli6 = (nielsenAsLip + fe)*a^2b^2/((a^2)+4 a b+(b^2))//Expand

coeffsimp[expr_] := Plus@@(#[[1]]#[[2]]&/@(({Factor[Plus@@#[[All,1]]],#[[1,2]]})&/@(expr//Cases[#,a_ b_L6i| a_ b_Lip:>{a,b}]&//GatherBy[#, #[[2]]&]&)))

result = feli6/.{d->-a-b}//L6can//L6pcan//coeffsimp

(* ------------------- *)

SetAttributes[L6sym, HoldAll];
Options[L6sym] = {oneMinusOneOverY->False, oneMinusP->False,sumOverP->False};
L6sym[coeff_, arg_, opts:OptionsPattern[]] := Module[{oneMinusOneOverYTF, oneMinusPTF, sumOverPTF, coeffs, args}, (
oneMinusOneOverYTF = OptionValue[oneMinusOneOverY]== True;
oneMinusPTF = OptionValue[oneMinusP]== True;
sumOverPTF = OptionValue[sumOverP]== True;

coeffs = {Hold[coeff]};
args =  {Hold[arg]};

If[oneMinusPTF,
coeffs = Join[coeffs, -coeffs/.{HoldPattern[a]->b,HoldPattern[b]->a}];
args = Join[args, args/.{p->1-p}];
];
If[oneMinusOneOverYTF,
coeffs = Join[coeffs, coeffs, coeffs];
args = Join[args, args/.{y->1-1/y}, args/.{y->1/(1-y)}];
];
If[sumOverPTF,
Return[Plus@@Release[MapThread[#1Lip[#2]&, {coeffs, args}]]];
];
Return[Plus@@Release[MapThread[#1L6i[#2]&, {coeffs, args}]]];
)];

g[p,y_] := (1-y)^b y^a;
g[1-p, y_] := (1-y)^a y^b;

result-
(  L6sym[-1/((a-b) (a+b) (2 a+b) (a+2 b)) , t/g[p,y], {oneMinusOneOverY->True, oneMinusP->True}] + 
+L6sym[a b/(36(a-b)) ,(((-1+p) p)/((-1+y) y)), {oneMinusOneOverY->True, oneMinusP->True,sumOverP->True}]+
-L6sym[a b/(18(a +2 b)) ,(((p^2) (-1+y))/((-1+p) (y^2))), {oneMinusOneOverY->True, oneMinusP->True,sumOverP->True}]+
+L6sym[a b/(2(a + b)) ,((p y)/(p-1 )), {oneMinusOneOverY->True, oneMinusP->True,sumOverP->True}]+
+ L6sym[1/2 a , p y, {oneMinusOneOverY->True, oneMinusP->True,sumOverP->True}]+
+L6sym[1/2 a ,p/y, {oneMinusOneOverY->True, oneMinusP->True,sumOverP->True}]+
+L6sym[ b/3 ,1-p, {oneMinusOneOverY->True, oneMinusP->True,sumOverP->True}]+
 +L6sym[ a b / (6(a + b)) ,((p-1 )/(p )), {oneMinusOneOverY->True, oneMinusP->True,sumOverP->True}])//L6can//L6pcan//Llead//Expand//Simplify;
Length[%]
%%

(* ----- *)

px[1] = 1/(1 - x + x^2);
px[2] = px[1]/.x->1-1/x//Factor;
px[3] = px[2]/.x->1-1/x//Factor;

tx = px[1]^1(1-px[1])^2//Factor;

LipX[arg_] := Sum[L6i[arg/.p->px[i]], {i, 1, 3}]

result/.{a->1,b->2,d->-3,t->tx,Lip->LipX}//L6can//Expand