ξp[r_] = Exp[-αp r^2]

ξq[r_] = Exp[-αq r^2]

Integrate[ 4 π ξp[r] ξq[r] r^2, {r,0,Infinity}]

Integrate[ 4 π ξp[r] 1/r ξq[r] r^2, {r,0,Infinity}]

D[ξp[r],r]

D[r^2 (-2 (E^(-(r^2) αp)) r αp),r]

1/r^2 (-6 (E^(-(r^2) αp)) (r^2) αp+4 (E^(-(r^2) αp)) (r^4) ((αp)^2))

-1/2 Integrate[ 4 π ξq[r]  (((-6 (E^(-(r^2) αp)) (r^2) αp+4 (E^(-(r^2) αp)) (r^4) ((αp)^2))/(r^2))) r^2, {r,0,Infinity}]