\[ \int x^3 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx \]
Optimal antiderivative \[ \frac {2 \left (23 A b -8 B a \right ) x^{4} \left (b \,x^{3}+a \right )^{\frac {3}{2}}}{391 b}+\frac {2 B \,x^{4} \left (b \,x^{3}+a \right )^{\frac {5}{2}}}{23 b}+\frac {54 a^{2} \left (23 A b -8 B a \right ) x \sqrt {b \,x^{3}+a}}{21505 b^{2}}+\frac {18 a \left (23 A b -8 B a \right ) x^{4} \sqrt {b \,x^{3}+a}}{4301 b}-\frac {36 \,3^{\frac {3}{4}} a^{3} \left (23 A b -8 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{21505 b^{\frac {7}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate(x^3*(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
\[ \frac {2 \, {\left (54 \, {\left (8 \, B a^{4} - 23 \, A a^{3} b\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (935 \, B b^{4} x^{10} + 55 \, {\left (26 \, B a b^{3} + 23 \, A b^{4}\right )} x^{7} + 5 \, {\left (27 \, B a^{2} b^{2} + 460 \, A a b^{3}\right )} x^{4} - 27 \, {\left (8 \, B a^{3} b - 23 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b x^{3} + a}\right )}}{21505 \, b^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B b x^{9} + {\left (B a + A b\right )} x^{6} + A a x^{3}\right )} \sqrt {b x^{3} + a}, x\right ) \]