It looks like the size of the intermediate results won't fit in the buffer space in Wolfram|Alpha, or I haven't been able to find a trick to coax Wolfram|Alpha into doing this.
But Mathematica has different product, slightly different notation.
FullSimplify[FunctionExpand[Sum[PolyGamma[0,n+3]/(n+1/a),{n,2,m}]]]
returns
((a*(-14+25*a-14*a^2-9*a^3+6*(-1+a)^2*(2+a)*EulerGamma))/(-1+a)-
6*(1+a)*(1+a*(-2+EulerGamma)-EulerGamma)*HarmonicNumber[a^(-1)+m]+
6*(-1+a^2)*HarmonicNumber[3+m]*HarmonicNumber[a^(-1)+m]-
6*(1+a)*(EulerGamma+PolyGamma[0,a^(-1)])*(-1+2*a+(-1+a)*
PolyGamma[0,4+m])+6*(-1+a^2)*Sum[(PolyGamma[0,a^(-1)]-
PolyGamma[0,1+a^(-1)+K[1]])/(3+K[1]),{K[1],-1,m}])/(6*(-1+a^2))
but that still contains a sum that it is not able to completely reduce, perhaps because of the symbolic upper bound in your original sum.
Perhaps this might give a little more insight into what the first few terms of your sum look like
Table[FullSimplify[PolyGamma[0,n+3]/(n+1/a)],{n,2,12}]
returns
{(a*(25/12 - EulerGamma))/(1 + 2*a),
(a*(137/60 - EulerGamma))/(1 + 3*a),
(a*(49/20 - EulerGamma))/(1 + 4*a),
(a*(363/140 - EulerGamma))/(1 + 5*a),
(a*(761/280 - EulerGamma))/(1 + 6*a),
(a*(7129/2520 - EulerGamma))/(1 + 7*a),
(a*(7381/2520 - EulerGamma))/(1 + 8*a),
(a*(83711/27720 - EulerGamma))/(1 + 9*a),
(a*(86021/27720 - EulerGamma))/(1 + 10*a),
(a*(1145993/360360 - EulerGamma))/(1 + 11*a),
(a*(1171733/360360 - EulerGamma))/(1 + 12*a)}
but I don't find a nice compact form for that sum.