CrN = Series[BesselJ[1 - ν, x*β], {β, 0, 4}]

CrD = Series[BesselJ[ν - 1, x*β], {β, 0, 4}]

Cra = FullSimplify[CrN/CrD] /. x -> a

Crb = FullSimplify[CrN/CrD] /. x -> b

Jnβx = 
      Series[BesselJ[-ν, x*β], {β, 0, 4}]

Jβx = Series[BesselJ[ν, x*β], {β, 0, 4}]

DJβx = D[Jβx, x]/β

DJnβx = D[Jnβx, x]/β

JsumβxSq = FullSimplify[(Jnβx + Cra*Jβx)^2]

DJsumβxSq = 
      FullSimplify[(DJnβx + Cra*DJβx)^2]

JsumβbSq = JsumβxSq /. x -> b

JsumβaSq = JsumβxSq /. x -> a

DJsumβbSq = DJsumβxSq /. x -> b

DJsumβaSq = DJsumβxSq /. x -> a

Normβb = 
      FullSimplify[
        (b^2)/2*
          (DJsumβbSq + 
              (1 - (ν/(b*β))^2)*JsumβbSq)]

Normβa = 
      FullSimplify[
        (a^2)/2*
          (DJsumβaSq + 
              (1 - (ν/(a*β))^2)*JsumβaSq)]

Cintegrand1 = 
      FullSimplify[(x^ν)*(Jnβx + Crb*Jβx)]

Cintgnd1norm = 
      FullSimplify[Cintegrand1/(Normβb - Normβa)]

Cintgnd1norm /. β -> 0

TermN0 = FullSimplify[Cintgnd1norm*(x^α)*Jnβx]

Term00 = Cintgnd1norm*Crb*(x^α)*Jβx

Term0 = FullSimplify[Term00]

TermT = FullSimplify[Term0 + TermN0]

FullSimplify[TermT /. β -> 0]

TermT1 = 
      FullSimplify[ 
        Gamma[2 - ν]/( Gamma[2 - ν]  + ν*Gamma[1 - ν])]

TermT1 = 
      FullSimplify[ 
        1/( 1 + (ν*Gamma[1 - ν])/Gamma[2 - ν])]

TermT2 = 
      FullSimplify[
        ((a^(2 ν)) (b^(2 ν)))/
          ((a^(2 ν)) (b^2) - (a^2) (b^(2 ν)))]

TermT = TermT1*TermT2*2*x^(α - ν)