Is it possible that you are using the free version of WolframAlpha?

If you scroll down just a bit from that answer do you see a pink warning "Standard computation time exceeded" and a button to click to try again using more computation time if you have a paid pro license?

If you see all that then perhaps the explanation is that it ran out of time and used whatever partial calculation it had finished to provide an (incorrect) answer.

Cannot be certain that this is exactly what happened behind the curtain.

Using the full version of Mathematica, with fewer time limitations, this

Sum[(1/(5*n+3)+1/(5*n+2))^2*(1/(5*n+4)+1/(5*n+1))^2*(3*n+1),{n,-Infinity,Infinity}]

returns

(35*Sqrt[5*(5 - 2*Sqrt[5])]*Pi - 45*Sqrt[5*(5 + 2*Sqrt[5])]*Pi + 100*Pi^2 + 
  16*Sqrt[5]*Pi^2)/(25*(-5 + Sqrt[5])*(5 + Sqrt[5]))

and then this

N[%]

approximates that complicated result as an approximate decimal value giving

-1.09159