\[ \int \frac {A+B x^3}{x^3 \left (a+b x^3\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {A}{2 a \,x^{2} \left (b \,x^{3}+a \right )^{\frac {3}{2}}}-\frac {\left (13 A b -4 B a \right ) x}{18 a^{2} \left (b \,x^{3}+a \right )^{\frac {3}{2}}}-\frac {7 \left (13 A b -4 B a \right ) x}{54 a^{3} \sqrt {b \,x^{3}+a}}-\frac {7 \left (13 A b -4 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{162 a^{3} b^{\frac {1}{3}} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {a^{\frac {1}{3}} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (b^{\frac {1}{3}} x +a^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((B*x**3+A)/x**3/(b*x**3+a)**(5/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {A \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {5}{2} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {5}{2}} x^{2} \Gamma \left (\frac {1}{3}\right )} + \frac {B x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {5}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {5}{2}} \Gamma \left (\frac {4}{3}\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________