\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^{11}} \, dx \]
Optimal antiderivative \[ -\frac {A \left (b \,x^{3}+a \right )^{\frac {3}{2}}}{10 a \,x^{10}}+\frac {\left (11 A b -20 B a \right ) \sqrt {b \,x^{3}+a}}{140 a \,x^{7}}+\frac {3 b \left (11 A b -20 B a \right ) \sqrt {b \,x^{3}+a}}{1120 a^{2} x^{4}}-\frac {3 b^{2} \left (11 A b -20 B a \right ) \sqrt {b \,x^{3}+a}}{448 a^{3} x}+\frac {3 b^{\frac {7}{3}} \left (11 A b -20 B a \right ) \sqrt {b \,x^{3}+a}}{448 a^{3} \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}+\frac {3^{\frac {3}{4}} b^{\frac {7}{3}} \left (11 A b -20 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, \sqrt {2}}{448 a^{\frac {8}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}-\frac {3 \,3^{\frac {1}{4}} b^{\frac {7}{3}} \left (11 A b -20 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticE \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{896 a^{\frac {8}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((B*x^3+A)*(b*x^3+a)^(1/2)/x^11,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {15 \, {\left (20 \, B a b^{2} - 11 \, A b^{3}\right )} \sqrt {b} x^{10} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (15 \, {\left (20 \, B a b^{2} - 11 \, A b^{3}\right )} x^{9} - 6 \, {\left (20 \, B a^{2} b - 11 \, A a b^{2}\right )} x^{6} - 224 \, A a^{3} - 16 \, {\left (20 \, B a^{3} + 3 \, A a^{2} b\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{2240 \, a^{3} x^{10}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{x^{11}}, x\right ) \]