Clear[P, Q, A, IAQP1,  IAQP2,  IAQP3, Ib, 
          M];
Ib = (1/(2 Q))*(3 - (ra/Q)^2) rb;
M=(3/(2 A))*(1/P)^3;
M*((∫_(A - P)^Q) 
                ra*((∫_(A - ra)^P) 
                        Ib \[DifferentialD]rb ) \[DifferentialD]ra + 
(∫_Q^A) ((∫_(A - ra)^P) 
                        rb \[DifferentialD]rb) \[DifferentialD]ra  + 
(∫_Q^(A + P)) ((∫_(ra - A)^P) 
                        rb \[DifferentialD]rb) \[DifferentialD]ra);
IAQP1 = %;
Expand[IAQP1]

M*((∫_(P - A)^Q) 
                  ra*((∫_(A - ra)^P) 
                          Ib \[DifferentialD]rb ) \[DifferentialD]ra + 
(∫_0^(P - A)) 
                  ra*((∫_(A - ra)^(A + ra)) 
                        Ib \[DifferentialD]rb ) \[DifferentialD]ra + 
(∫_Q^A) ((∫_(A - ra)^P) 
                          rb \[DifferentialD]rb) \[DifferentialD]ra + 
(∫_A^(A + P)) ((∫_(ra - A)^P) 
                          rb \[DifferentialD]rb) \[DifferentialD]ra);
IAQP2 = %;
Expand[IAQP2]

M*((∫_A^Q) 
                  ra*((∫_(ra - A)^P) 
                          Ib \[DifferentialD]rb ) \[DifferentialD]ra + 
(∫_(P - A)^A) 
                  ra*((∫_(A - ra)^P) 
                        Ib \[DifferentialD]rb ) \[DifferentialD]ra + 
(∫_0^(P - A)) 
                  ra*((∫_(A - ra)^(A + ra))Ib \[DifferentialD]rb ) \[DifferentialD]ra  + 
(∫_Q^(A + P)) ((∫_(ra - A)^P) 
                          rb \[DifferentialD]rb) \[DifferentialD]ra);

IAQP3 = %;
Expand[IAQP3]

Q = 
        P;
IAP = Expand[IAQP3]

A = 
        0;
Expand[IAP]

Clear[P, Q, 
          A];
Ic = (1/(2 P))*(3 - (rb/P)^2) rb;
M1=(3/(2 A))*(1/Q^3);
M1*((∫_(A - Q)^A) 
                ra*((∫_(A - rb)^Q) 
                        Ic \[DifferentialD]rb ) \[DifferentialD]ra + 
(∫_A^(A + Q)) 
                ra*((∫_(rb - A)^Q) 
                      Ic \[DifferentialD]rb ) \[DifferentialD]ra);
Expand[%]

Clear[P, Q, 
          A];
Ic = (1/(2 P))*(3 - (rb/P)^2) rb;
M1=(3/(2 A))*(1/Q^3);
M1*((∫_(-Q)^0) 
                ra*((∫_(A - ra)^Q) 
                        Ic \[DifferentialD]rb ) \[DifferentialD]ra + 
(∫_0^Q) 
                ra*((∫_(ra - A)^Q) 
                      Ic \[DifferentialD]rb ) \[DifferentialD]ra);
Expand[%]

Clear[P, Q, 
          A];
Ic = (1/(2 P))*(3 - (rb/P)^2) rb;
M1=(3/(2 A))*(1/Q^3);

M1*((∫_A^Q) 
                ra*((∫_(ra - A)^(ra + A))Ic \[DifferentialD]rb ) \[DifferentialD]ra + 
(∫_0^A) 
                ra*((∫_(A - ra)^(ra + A)) 
                      Ic \[DifferentialD]rb ) \[DifferentialD]ra);

Expand[%]