\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{x^6} \, dx \]
Optimal antiderivative \[ -\frac {\left (A b +2 B a \right ) \left (b \,x^{3}+a \right )^{\frac {3}{2}}}{4 a \,x^{2}}-\frac {A \left (b \,x^{3}+a \right )^{\frac {5}{2}}}{5 a \,x^{5}}+\frac {9 b \left (A b +2 B a \right ) x \sqrt {b \,x^{3}+a}}{20 a}+\frac {9 \,3^{\frac {3}{4}} b^{\frac {2}{3}} \left (A b +2 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{20 \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((b*x^3+a)^(3/2)*(B*x^3+A)/x^6,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {27 \, {\left (2 \, B a + A b\right )} \sqrt {b} x^{5} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (8 \, B b x^{6} - {\left (10 \, B a + 13 \, A b\right )} x^{3} - 4 \, A a\right )} \sqrt {b x^{3} + a}}{20 \, x^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B b x^{6} + {\left (B a + A b\right )} x^{3} + A a\right )} \sqrt {b x^{3} + a}}{x^{6}}, x\right ) \]