\[ \int x^4 \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx \]
Optimal antiderivative \[ \frac {2 B \,x^{5} \left (b \,x^{3}+a \right )^{\frac {3}{2}}}{19 b}+\frac {6 a \left (19 A b -10 B a \right ) x^{2} \sqrt {b \,x^{3}+a}}{1729 b^{2}}+\frac {2 \left (19 A b -10 B a \right ) x^{5} \sqrt {b \,x^{3}+a}}{247 b}-\frac {24 a^{2} \left (19 A b -10 B a \right ) \sqrt {b \,x^{3}+a}}{1729 b^{\frac {8}{3}} \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}-\frac {8 \,3^{\frac {3}{4}} a^{\frac {7}{3}} \left (19 A b -10 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \sqrt {2}\, \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{1729 b^{\frac {8}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}+\frac {12 \,3^{\frac {1}{4}} a^{\frac {7}{3}} \left (19 A b -10 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticE \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{1729 b^{\frac {8}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate(x^4*(B*x^3+A)*(b*x^3+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (12 \, {\left (10 \, B a^{3} - 19 \, A a^{2} b\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (91 \, B b^{3} x^{8} + 7 \, {\left (3 \, B a b^{2} + 19 \, A b^{3}\right )} x^{5} - 3 \, {\left (10 \, B a^{2} b - 19 \, A a b^{2}\right )} x^{2}\right )} \sqrt {b x^{3} + a}\right )}}{1729 \, b^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B x^{7} + A x^{4}\right )} \sqrt {b x^{3} + a}, x\right ) \]