\[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^8} \, dx \]
Optimal antiderivative \[ -\frac {A \left (b \,x^{3}+a \right )^{\frac {3}{2}}}{7 a \,x^{7}}+\frac {\left (5 A b -14 B a \right ) \sqrt {b \,x^{3}+a}}{56 a \,x^{4}}+\frac {3 b \left (5 A b -14 B a \right ) \sqrt {b \,x^{3}+a}}{112 a^{2} x}-\frac {3 b^{\frac {4}{3}} \left (5 A b -14 B a \right ) \sqrt {b \,x^{3}+a}}{112 a^{2} \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}-\frac {3^{\frac {3}{4}} b^{\frac {4}{3}} \left (5 A b -14 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, \sqrt {2}}{112 a^{\frac {5}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}+\frac {3 \,3^{\frac {1}{4}} b^{\frac {4}{3}} \left (5 A b -14 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticE \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{224 a^{\frac {5}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((B*x^3+A)*(b*x^3+a)^(1/2)/x^8,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {3 \, {\left (14 \, B a b - 5 \, A b^{2}\right )} \sqrt {b} x^{7} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (3 \, {\left (14 \, B a b - 5 \, A b^{2}\right )} x^{6} + 2 \, {\left (14 \, B a^{2} + 3 \, A a b\right )} x^{3} + 16 \, A a^{2}\right )} \sqrt {b x^{3} + a}}{112 \, a^{2} x^{7}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{x^{8}}, x\right ) \]