\[ \int (e x)^{5/2} \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx \]
Optimal antiderivative \[ \frac {15 a \left (4 A b -B a \right ) \left (e x \right )^{\frac {7}{2}} \left (b \,x^{3}+a \right )^{\frac {3}{2}}}{704 b e}+\frac {\left (4 A b -B a \right ) \left (e x \right )^{\frac {7}{2}} \left (b \,x^{3}+a \right )^{\frac {5}{2}}}{44 b e}+\frac {B \left (e x \right )^{\frac {7}{2}} \left (b \,x^{3}+a \right )^{\frac {7}{2}}}{14 b e}+\frac {27 a^{2} \left (4 A b -B a \right ) \left (e x \right )^{\frac {7}{2}} \sqrt {b \,x^{3}+a}}{1408 b e}+\frac {81 a^{3} \left (4 A b -B a \right ) e^{2} \sqrt {e x}\, \sqrt {b \,x^{3}+a}}{5632 b^{2}}-\frac {27 \,3^{\frac {3}{4}} a^{\frac {11}{3}} \left (4 A b -B a \right ) e^{2} \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \sqrt {\frac {\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1-\sqrt {3}\right )\right )^{2}}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}\, \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right ) \EllipticF \left (\sqrt {1-\frac {\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1-\sqrt {3}\right )\right )^{2}}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}, \frac {\sqrt {6}}{4}+\frac {\sqrt {2}}{4}\right ) \sqrt {e x}\, \sqrt {\frac {a^{\frac {2}{3}}-a^{\frac {1}{3}} b^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}}{11264 \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1-\sqrt {3}\right )\right ) b^{2} \sqrt {b \,x^{3}+a}\, \sqrt {\frac {b^{\frac {1}{3}} x \left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate((e*x)**(5/2)*(b*x**3+a)**(5/2)*(B*x**3+A),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {A a^{\frac {5}{2}} e^{\frac {5}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{6} \\ \frac {13}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {13}{6}\right )} + \frac {2 A a^{\frac {3}{2}} b e^{\frac {5}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{6} \\ \frac {19}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {19}{6}\right )} + \frac {A \sqrt {a} b^{2} e^{\frac {5}{2}} x^{\frac {19}{2}} \Gamma \left (\frac {19}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {19}{6} \\ \frac {25}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {25}{6}\right )} + \frac {B a^{\frac {5}{2}} e^{\frac {5}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{6} \\ \frac {19}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {19}{6}\right )} + \frac {2 B a^{\frac {3}{2}} b e^{\frac {5}{2}} x^{\frac {19}{2}} \Gamma \left (\frac {19}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {19}{6} \\ \frac {25}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {25}{6}\right )} + \frac {B \sqrt {a} b^{2} e^{\frac {5}{2}} x^{\frac {25}{2}} \Gamma \left (\frac {25}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {25}{6} \\ \frac {31}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {31}{6}\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________