\[ \int \frac {(e x)^{7/2} (A+B x)}{\left (a+c x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {e \left (e x \right )^{\frac {5}{2}} \left (B x +A \right )}{c \sqrt {c \,x^{2}+a}}+\frac {7 B \,e^{2} \left (e x \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+a}}{5 c^{2}}-\frac {21 a B \,e^{4} x \sqrt {c \,x^{2}+a}}{5 c^{\frac {5}{2}} \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {e x}}+\frac {5 A \,e^{3} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}{3 c^{2}}+\frac {21 a^{\frac {5}{4}} B \,e^{4} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{5 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {11}{4}} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}-\frac {a^{\frac {3}{4}} e^{4} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (63 B \sqrt {a}+25 A \sqrt {c}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{30 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {11}{4}} \sqrt {e x}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((e*x)**(7/2)*(B*x+A)/(c*x**2+a)**(3/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {A e^{\frac {7}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {13}{4}\right )} + \frac {B e^{\frac {7}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} \Gamma \left (\frac {15}{4}\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________