\[ \int \frac {A+B x^3}{(e x)^{7/2} \left (a+b x^3\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 A}{5 a e \left (e x \right )^{\frac {5}{2}} \sqrt {b \,x^{3}+a}}-\frac {2 \left (8 A b -5 B a \right ) \sqrt {e x}}{15 a^{2} e^{4} \sqrt {b \,x^{3}+a}}-\frac {2 \left (8 A b -5 B a \right ) \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \sqrt {\frac {\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1-\sqrt {3}\right )\right )^{2}}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}\, \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right ) \EllipticF \left (\sqrt {1-\frac {\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1-\sqrt {3}\right )\right )^{2}}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}, \frac {\sqrt {6}}{4}+\frac {\sqrt {2}}{4}\right ) \sqrt {e x}\, \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}}}{45 \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1-\sqrt {3}\right )\right ) a^{\frac {7}{3}} e^{4} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {b^{\frac {1}{3}} x \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((B*x**3+A)/(e*x)**(7/2)/(b*x**3+a)**(3/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {A \Gamma \left (- \frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{6}, \frac {3}{2} \\ \frac {1}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} e^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {1}{6}\right )} + \frac {B \sqrt {x} \Gamma \left (\frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{6}, \frac {3}{2} \\ \frac {7}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} e^{\frac {7}{2}} \Gamma \left (\frac {7}{6}\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________