\[ \int \frac {\left (a+b x^3\right )^{3/2} \left (A+B x^3\right )}{x^5} \, dx \]
Optimal antiderivative \[ -\frac {\left (7 A b +8 B a \right ) \left (b \,x^{3}+a \right )^{\frac {3}{2}}}{8 a x}-\frac {A \left (b \,x^{3}+a \right )^{\frac {5}{2}}}{4 a \,x^{4}}+\frac {9 b \left (7 A b +8 B a \right ) x^{2} \sqrt {b \,x^{3}+a}}{56 a}+\frac {27 b^{\frac {1}{3}} \left (7 A b +8 B a \right ) \sqrt {b \,x^{3}+a}}{56 \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}+\frac {9 \,3^{\frac {3}{4}} a^{\frac {1}{3}} b^{\frac {1}{3}} \left (7 A b +8 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, \sqrt {2}}{56 \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}-\frac {27 \,3^{\frac {1}{4}} a^{\frac {1}{3}} b^{\frac {1}{3}} \left (7 A b +8 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticE \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{112 \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((b*x^3+a)^(3/2)*(B*x^3+A)/x^5,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {27 \, {\left (8 \, B a + 7 \, A b\right )} \sqrt {b} x^{4} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (16 \, B b x^{6} - 7 \, {\left (8 \, B a + 11 \, A b\right )} x^{3} - 14 \, A a\right )} \sqrt {b x^{3} + a}}{56 \, x^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B b x^{6} + {\left (B a + A b\right )} x^{3} + A a\right )} \sqrt {b x^{3} + a}}{x^{5}}, x\right ) \]