\[ \int \frac {x \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (A b -B a \right ) x^{2}}{9 a b \left (b \,x^{3}+a \right )^{\frac {3}{2}}}+\frac {2 \left (5 A b +4 B a \right ) x^{2}}{27 a^{2} b \sqrt {b \,x^{3}+a}}-\frac {2 \left (5 A b +4 B a \right ) \sqrt {b \,x^{3}+a}}{27 a^{2} b^{\frac {5}{3}} \left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}-\frac {2 \left (5 A b +4 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \sqrt {2}\, \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{81 a^{\frac {5}{3}} b^{\frac {5}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}+\frac {\left (5 A b +4 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticE \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {1}{4}}}{27 a^{\frac {5}{3}} b^{\frac {5}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate(x*(B*x^3+A)/(b*x^3+a)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left ({\left (4 \, B a b^{2} + 5 \, A b^{3}\right )} x^{6} + 4 \, B a^{3} + 5 \, A a^{2} b + 2 \, {\left (4 \, B a^{2} b + 5 \, A a b^{2}\right )} x^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + {\left ({\left (4 \, B a b^{2} + 5 \, A b^{3}\right )} x^{5} + {\left (B a^{2} b + 8 \, A a b^{2}\right )} x^{2}\right )} \sqrt {b x^{3} + a}\right )}}{27 \, {\left (a^{2} b^{4} x^{6} + 2 \, a^{3} b^{3} x^{3} + a^{4} b^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B x^{4} + A x\right )} \sqrt {b x^{3} + a}}{b^{3} x^{9} + 3 \, a b^{2} x^{6} + 3 \, a^{2} b x^{3} + a^{3}}, x\right ) \]