\[ \int \frac {A+B x^3}{x^8 \left (a+b x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {A}{7 a \,x^{7} \sqrt {b \,x^{3}+a}}+\frac {-17 A b +14 B a}{21 a^{2} x^{4} \sqrt {b \,x^{3}+a}}+\frac {11 \left (17 A b -14 B a \right ) \sqrt {b \,x^{3}+a}}{168 a^{3} x^{4}}-\frac {55 b \left (17 A b -14 B a \right ) \sqrt {b \,x^{3}+a}}{336 a^{4} x}+\frac {55 b^{\frac {4}{3}} \left (17 A b -14 B a \right ) \sqrt {b \,x^{3}+a}}{336 a^{4} \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}+\frac {55 b^{\frac {4}{3}} \left (17 A b -14 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}} \sqrt {2}}{1008 a^{\frac {11}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}-\frac {55 b^{\frac {4}{3}} \left (17 A b -14 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticE \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {1}{4}}}{672 a^{\frac {11}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((B*x^3+A)/x^8/(b*x^3+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {55 \, {\left ({\left (14 \, B a b^{2} - 17 \, A b^{3}\right )} x^{10} + {\left (14 \, B a^{2} b - 17 \, A a b^{2}\right )} x^{7}\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left (55 \, {\left (14 \, B a b^{2} - 17 \, A b^{3}\right )} x^{9} + 33 \, {\left (14 \, B a^{2} b - 17 \, A a b^{2}\right )} x^{6} - 48 \, A a^{3} - 6 \, {\left (14 \, B a^{3} - 17 \, A a^{2} b\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{336 \, {\left (a^{4} b x^{10} + a^{5} x^{7}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{b^{2} x^{14} + 2 \, a b x^{11} + a^{2} x^{8}}, x\right ) \]