\[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {B x +A}{3 a e \left (c \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {e x}}+\frac {5 B x +7 A}{6 a^{2} e \sqrt {e x}\, \sqrt {c \,x^{2}+a}}-\frac {7 A \sqrt {c \,x^{2}+a}}{2 a^{3} e \sqrt {e x}}+\frac {7 A x \sqrt {c}\, \sqrt {c \,x^{2}+a}}{2 a^{3} e \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {e x}}-\frac {7 A \,c^{\frac {1}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{2 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {11}{4}} e \sqrt {e x}\, \sqrt {c \,x^{2}+a}}+\frac {\sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (5 B \sqrt {a}+21 A \sqrt {c}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{12 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {11}{4}} c^{\frac {1}{4}} e \sqrt {e x}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((B*x+A)/(e*x)**(3/2)/(c*x**2+a)**(5/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {A \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} e^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________