\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{x^{11}} \, dx \]
Optimal antiderivative \[ \frac {\left (A b -4 B a \right ) \left (b \,x^{3}+a \right )^{\frac {3}{2}}}{28 a \,x^{7}}-\frac {A \left (b \,x^{3}+a \right )^{\frac {5}{2}}}{10 a \,x^{10}}+\frac {9 b \left (A b -4 B a \right ) \sqrt {b \,x^{3}+a}}{224 a \,x^{4}}+\frac {27 b^{2} \left (A b -4 B a \right ) \sqrt {b \,x^{3}+a}}{448 a^{2} x}-\frac {27 b^{\frac {7}{3}} \left (A b -4 B a \right ) \sqrt {b \,x^{3}+a}}{448 a^{2} \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}-\frac {9 \,3^{\frac {3}{4}} b^{\frac {7}{3}} \left (A b -4 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, \sqrt {2}}{448 a^{\frac {5}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}+\frac {27 \,3^{\frac {1}{4}} b^{\frac {7}{3}} \left (A b -4 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticE \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{896 a^{\frac {5}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((b*x^3+a)^(3/2)*(B*x^3+A)/x^11,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {135 \, {\left (4 \, B a b^{2} - A b^{3}\right )} \sqrt {b} x^{10} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (135 \, {\left (4 \, B a b^{2} - A b^{3}\right )} x^{9} + 2 \, {\left (340 \, B a^{2} b + 27 \, A a b^{2}\right )} x^{6} + 224 \, A a^{3} + 16 \, {\left (20 \, B a^{3} + 23 \, A a^{2} b\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{2240 \, a^{2} x^{10}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B b x^{6} + {\left (B a + A b\right )} x^{3} + A a\right )} \sqrt {b x^{3} + a}}{x^{11}}, x\right ) \]