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Problem: InverseLaplaceTransform with unexpected result

Any ideas why Mathematica 7.0.0 does not give the correct solution

In[1]:= ClearAll["Globalâ??*"]
$Assumptions= Element[T1,Reals] && T1 >0 && Element[n,Integers] && n >0 ;
In[3]:= H[s_]:=s/((T2*s +1)^n)
In[5]:= FullSimplify[LaplaceTransform[imp[t], t, s]]
Out[5]= 1/2 s (1+s T2)^-n


Sidenote: Seems that InverseLaplaceTransform inhibits unexpexcted factor

In[6]:= InverseLaplaceTransform[H[s],s,t]
Out[6]= -((E^(-(t/T2)) (t/T2)^(1+n) (t+T2-n T2))/(2 t3 Gamma[n]))


whereas for specific n it is correct:

In[7]:= ClearAll["Globalâ??*"]
$Assumptions= Element[T1,Reals] && T1 >0 && Element[n,Integers] && n >0 ;
In[7]:= H[s_]:=s/((T2*s +1)^n) /. n-> 5
In[12]:= FullSimplify[LaplaceTransform[imp[t], t, s]]
Out[12]= s/(1+s T2)5

Thanks in advance,


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