\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^9} \, dx \]
Optimal antiderivative \[ -\frac {A \left (b \,x^{3}+a \right )^{\frac {3}{2}}}{8 a \,x^{8}}+\frac {\left (7 A b -16 B a \right ) \sqrt {b \,x^{3}+a}}{80 a \,x^{5}}+\frac {3 b \left (7 A b -16 B a \right ) \sqrt {b \,x^{3}+a}}{320 a^{2} x^{2}}+\frac {3^{\frac {3}{4}} b^{\frac {5}{3}} \left (7 A b -16 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{320 a^{2} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((B*x^3+A)*(b*x^3+a)^(1/2)/x^9,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {3 \, {\left (16 \, B a b - 7 \, A b^{2}\right )} \sqrt {b} x^{8} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (3 \, {\left (16 \, B a b - 7 \, A b^{2}\right )} x^{6} + 4 \, {\left (16 \, B a^{2} + 3 \, A a b\right )} x^{3} + 40 \, A a^{2}\right )} \sqrt {b x^{3} + a}}{320 \, a^{2} x^{8}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{x^{9}}, x\right ) \]