Fast multiplication of high dimensional matrix

Better try

s = Total[a b, 3];

or

s = Flatten[a].ArrayReshape[b, {c^3, d, d, d}];

On my machine, the latter is faster. In general, rephrasing summations in terms of Dot (.) should lead to more efficient code as Dot is highly optimized.

The code can be faster if we compile:

cf = Compile[{{a, _Real, 3}, {b, _Real, 6}}, Flatten[a].Flatten[b, 2]];

test = cf[a, b]; // AbsoluteTiming

(* {0.492055, Null} *)

and even faster if compile to C and extract the LibraryFunction[…]:

cfc = 
  Compile[{{a, _Real, 3}, {b, _Real, 6}}, Flatten[a].Flatten[b, 2], 
    CompilationTarget -> C][[-1]];

testc = cfc[a, b]; // AbsoluteTiming
(* {0.234145, Null} *)

Tested on v9.0.1, with TDM-GCC-5.1.0-2 64-bit compiler, "SystemCompileOptions"->"-Ofast".