\[ \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {A}{5 a \,x^{5} \left (b \,x^{3}+a \right )^{\frac {3}{2}}}+\frac {-19 A b +10 B a}{45 a^{2} x^{2} \left (b \,x^{3}+a \right )^{\frac {3}{2}}}-\frac {13 \left (19 A b -10 B a \right )}{135 a^{3} x^{2} \sqrt {b \,x^{3}+a}}+\frac {91 \left (19 A b -10 B a \right ) \sqrt {b \,x^{3}+a}}{540 a^{4} x^{2}}+\frac {91 b^{\frac {2}{3}} \left (19 A b -10 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{1620 a^{4} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((B*x^3+A)/x^6/(b*x^3+a)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {91 \, {\left ({\left (10 \, B a b^{2} - 19 \, A b^{3}\right )} x^{11} + 2 \, {\left (10 \, B a^{2} b - 19 \, A a b^{2}\right )} x^{8} + {\left (10 \, B a^{3} - 19 \, A a^{2} b\right )} x^{5}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (91 \, {\left (10 \, B a b^{2} - 19 \, A b^{3}\right )} x^{9} + 130 \, {\left (10 \, B a^{2} b - 19 \, A a b^{2}\right )} x^{6} + 108 \, A a^{3} + 27 \, {\left (10 \, B a^{3} - 19 \, A a^{2} b\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{540 \, {\left (a^{4} b^{2} x^{11} + 2 \, a^{5} b x^{8} + a^{6} x^{5}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B x^{3} + A\right )} \sqrt {b x^{3} + a}}{b^{3} x^{15} + 3 \, a b^{2} x^{12} + 3 \, a^{2} b x^{9} + a^{3} x^{6}}, x\right ) \]