In[]:=
A=Integrate[E^(-a(x^2))x^(1/2-h)BesselK[1/2+n,xb],x]/.n->2/.h->1//Expand
Out[]=
π
2

-bx-a
2
x

1+
3
2
b
2
x
+
3
bx
x
bx
x
PacletInstall["https://rulebasedintegration.org/Rubi-4.17.3.0.paclet"]
In[]:=
<<Rubi`
Rubi 4.17.3.0 running on Mathematica 14.2
In[]:=
B=E^(-a(x^2))x^(1/2-h)BesselK[1/2+n,xb]/.n->2/.h->1//Expand
Out[]=
3
-bx-a
2
x

π
2
bx
3
b
7/2
x
+
3
-bx-a
2
x

π
2
bx
2
b
5/2
x
+
-bx-a
2
x

π
2
bx
b
3/2
x
In[]:=
B1=Int[B,x]//ExpandAll
Out[]=
-
3
-bx-a
2
x

π
2
bx
4
3
b
9/2
x
-
3
-bx-a
2
x

π
2
bx
4
2
b
7/2
x
+
3a
-bx-a
2
x

π
2
bx
4
3
b
5/2
x
-
-bx-a
2
x

π
2
bx
8b
5/2
x
+
-bx-a
2
x

π
2
bx
8
3/2
x
-
5a
-bx-a
2
x

π
2
bx
4
2
b
3/2
x
+
a
-bx-a
2
x

2π
bx
2
b
3/2
x
+
a
2
b
4a

π
bx
Erf
b
2
a
+
a
x
8
2
x
-
5
3/2
a
2
b
4a

π
bx
Erf
b
2
a
+
a
x
4
2
2
b
x
+
2
3/2
a
2
b
4a

π
bx
Erf
b
2
a
+
a
x
2
b
x
+
3
2
a
π
2
bx
Int
-bx-a
2
x

x
,x
2
3
b
x
+
a
π
2
bx
Int
-bx-a
2
x

x
,x
2b
x
+
b
π
2
bx
Int
-bx-a
2
x

x
,x
8
x
Int
-bx-a
2
x

x
,x(*Can't*)
Out[]=
Int
-bx-a
2
x

x
,x
Integrate
-bx-a
2
x

x
,x(*Can't*)
Out[]=

-bx-a
2
x

x
x
In[]:=
Int
-bx

x
(-a
2
x
)^n
n!
,x//PowerExpand
Out[]=
-
n
(-1)
n
a
-2n
b
Gamma[2n,bx]
n!
In[]:=
InactivateIntegrate
-bx-a
2
x

x
,x,Integrate==InactivateSum-
n
(-1)
n
a
-2n
b
Gamma[2n,bx]
n!
,{n,0,Infinity}+C,Sum
Out[]=

-bx-a
2
x

x
xC+
∞
∑
n=0
-
n
(-1)
n
a
-2n
b
Gamma[2n,bx]
n!