\[ \int \frac {x^6 \left (A+B x^3\right )}{\left (a+b x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (11 A b -14 B a \right ) x^{4}}{33 b^{2} \sqrt {b \,x^{3}+a}}+\frac {2 B \,x^{7}}{11 b \sqrt {b \,x^{3}+a}}+\frac {16 \left (11 A b -14 B a \right ) x \sqrt {b \,x^{3}+a}}{165 b^{3}}-\frac {32 a \left (11 A b -14 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{495 b^{\frac {10}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate(x^6*(B*x^3+A)/(b*x^3+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (16 \, {\left (14 \, B a^{3} - 11 \, A a^{2} b + {\left (14 \, B a^{2} b - 11 \, A a b^{2}\right )} x^{3}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (15 \, B b^{3} x^{7} - 3 \, {\left (14 \, B a b^{2} - 11 \, A b^{3}\right )} x^{4} - 8 \, {\left (14 \, B a^{2} b - 11 \, A a b^{2}\right )} x\right )} \sqrt {b x^{3} + a}\right )}}{165 \, {\left (b^{5} x^{3} + a b^{4}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B x^{9} + A x^{6}\right )} \sqrt {b x^{3} + a}}{b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}, x\right ) \]