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I don't think we're going to be able to fix this very easily at all. The problem is Maxima's framework, which there has been much discussion about and which is not going to change.

(%i5) assume(n>0);
(%o5)                               [n > 0]
(%i6) display2d:false;

(%o6) false
(%i7) integrate(sqrt(s/n)*1/sqrt(2*pi)*exp(-s*t^2/(2*n))*exp(-s/2)*s^(n/2-1)/(gamma(n/2)*2^(n/2)),s,0,inf);

Is (n+1)/2 an integer?

no
;
(%o7) gamma(n/2+1/2)*2^(-n/2-1/2)*(t^2/(2*n)+1/2)^(-n/2-1/2)
 /(sqrt(n)*gamma(n/2)*sqrt(pi))
(%i8) integrate(sqrt(s/n)*1/sqrt(2*pi)*exp(-s*t^2/(2*n))*exp(-s/2)*s^(n/2-1)/(gamma(n/2)*2^(n/2)),s,0,inf);

Is (n+1)/2 an integer?

yes;
Is n an integer?

yes;
(%o8) gamma(n/2+1/2)*2^(-n/2-1/2)*(t^2/(2*n)+1/2)^(-n/2-1/2)
 /(sqrt(n)*gamma(n/2)*sqrt(pi))

Don't even ask about why it asks if n is an integer when it knows (n+1)/2 is an integer; Maxima itself acknowledges that its assumption framework is weak, and of course every open source project has limited resources to fix big architectural issues like that. I don't think it would be trivial for them to fix that.

As for Sage, assuming n is an integer will help a little, but only so much.

(%i9) declare(n,integer);

(%o9) done
(%i10) integrate(sqrt(s/n)*1/sqrt(2*pi)*exp(-s*t^2/(2*n))*exp(-s/2)*s^(n/2-1)/(gamma(n/2)*2^(n/2)),s,0,inf);

Is (n+1)/2 an integer?

yes;
(%o10) gamma(n/2+1/2)*2^(-n/2-1/2)*(t^2/(2*n)+1/2)^(-n/2-1/2)
 /(sqrt(n)*gamma(n/2)*sqrt(pi))

(%i11) declare((n+1)/2,integer);

declare: improper argument: (1+n)/2
 -- an error. To debug this try: debugmode(true);

So we wouldn't even be able to try that, even if we wanted to; Maxima only lets "atoms" be declared to be something (i.e., variables), though we can assume positive or negative about them. And even though the answers are the same, many have pointed out that going through a decision tree of this kind and trying to check that can be arbitrarily computationally expensive, if it's even possible.

I'm sorry, but this is something at the core of how Maxima does business. On the plus side, the answer works nicely enough that one gets from Maxima, and you could use it in Sage.