\[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b x^6+a \left (q+p x^3\right )^6\right )}{x^9} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {p^{2} x^{6}+2 p q \,x^{3}-2 p q \,x^{2}+q^{2}}\, \left (6 a \,p^{7} x^{21}+42 a \,p^{6} q \,x^{18}-2 a \,p^{6} q \,x^{17}+126 a \,p^{5} q^{2} x^{15}-10 a \,p^{5} q^{2} x^{14}-5 a \,p^{5} q^{2} x^{13}+210 a \,p^{4} q^{3} x^{12}-20 a \,p^{4} q^{3} x^{11}-15 a \,p^{4} q^{3} x^{10}-15 a \,p^{4} q^{3} x^{9}+210 a \,p^{3} q^{4} x^{9}-20 a \,p^{3} q^{4} x^{8}-15 a \,p^{3} q^{4} x^{7}-15 a \,p^{3} q^{4} x^{6}+126 a \,p^{2} q^{5} x^{6}-10 a \,p^{2} q^{5} x^{5}-5 a \,p^{2} q^{5} x^{4}+42 a p \,q^{6} x^{3}+24 b p \,x^{9}-2 a p \,q^{6} x^{2}+6 a \,q^{7}+24 b q \,x^{6}\right )}{48 x^{8}}+\frac {\left (5 a \,p^{4} q^{4}+8 b p q \right ) \ln \left (x \right )}{8}+\frac {\left (-5 a \,p^{4} q^{4}-8 b p q \right ) \ln \left (q +p \,x^{3}+\sqrt {p^{2} x^{6}+2 p q \,x^{3}-2 p q \,x^{2}+q^{2}}\right )}{8} \]
command
Integrate[((-q + 2*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(b*x^6 + a*(q + p*x^3)^6))/x^9,x]
Mathematica 13.1 output
\[ \frac {1}{48} \left (\frac {\left (q+p x^3\right ) \sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6} \left (24 b x^6+a \left (6 q^6+6 p^6 x^{18}+2 p q^5 x^2 (-1+18 x)+2 p^5 q x^{14} (-1+18 x)+p^2 q^4 x^4 \left (-5-8 x+90 x^2\right )+p^4 q^2 x^{10} \left (-5-8 x+90 x^2\right )+p^3 q^3 x^6 \left (-15-10 x-12 x^2+120 x^3\right )\right )\right )}{x^8}-6 p q \left (8 b+5 a p^3 q^3\right ) \tanh ^{-1}\left (\frac {\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}}{q+p x^3}\right )\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b x^6+a \left (q+p x^3\right )^6\right )}{x^9} \, dx \]________________________________________________________________________________________