\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{x^3} \, dx \]
Optimal antiderivative \[ \frac {\left (11 A b +4 B a \right ) x \left (b \,x^{3}+a \right )^{\frac {3}{2}}}{22 a}-\frac {A \left (b \,x^{3}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {9 \left (11 A b +4 B a \right ) x \sqrt {b \,x^{3}+a}}{110}+\frac {9 \,3^{\frac {3}{4}} a \left (11 A b +4 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{110 b^{\frac {1}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((b*x^3+a)^(3/2)*(B*x^3+A)/x^3,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {27 \, {\left (4 \, B a^{2} + 11 \, A a b\right )} \sqrt {b} x^{2} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (20 \, B b^{2} x^{6} + 4 \, {\left (14 \, B a b + 11 \, A b^{2}\right )} x^{3} - 55 \, A a b\right )} \sqrt {b x^{3} + a}}{110 \, b x^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B b x^{6} + {\left (B a + A b\right )} x^{3} + A a\right )} \sqrt {b x^{3} + a}}{x^{3}}, x\right ) \]