\[ \int x^4 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx \]
Optimal antiderivative \[ \frac {2 \left (5 A b -2 B a \right ) x^{5} \left (b \,x^{3}+a \right )^{\frac {3}{2}}}{95 b}+\frac {2 B \,x^{5} \left (b \,x^{3}+a \right )^{\frac {5}{2}}}{25 b}+\frac {54 a^{2} \left (5 A b -2 B a \right ) x^{2} \sqrt {b \,x^{3}+a}}{8645 b^{2}}+\frac {18 a \left (5 A b -2 B a \right ) x^{5} \sqrt {b \,x^{3}+a}}{1235 b}-\frac {216 a^{3} \left (5 A b -2 B a \right ) \sqrt {b \,x^{3}+a}}{8645 b^{\frac {8}{3}} \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}-\frac {72 \,3^{\frac {3}{4}} a^{\frac {10}{3}} \left (5 A b -2 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \sqrt {2}\, \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{8645 b^{\frac {8}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}+\frac {108 \,3^{\frac {1}{4}} a^{\frac {10}{3}} \left (5 A b -2 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticE \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{8645 b^{\frac {8}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate(x^4*(b*x^3+a)^(3/2)*(B*x^3+A),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (540 \, {\left (2 \, B a^{4} - 5 \, A a^{3} b\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (1729 \, B b^{4} x^{11} + 91 \, {\left (28 \, B a b^{3} + 25 \, A b^{4}\right )} x^{8} + 7 \, {\left (27 \, B a^{2} b^{2} + 550 \, A a b^{3}\right )} x^{5} - 135 \, {\left (2 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{3} + a}\right )}}{43225 \, b^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B b x^{10} + {\left (B a + A b\right )} x^{7} + A a x^{4}\right )} \sqrt {b x^{3} + a}, x\right ) \]