\[ \int \frac {A+B x}{(e x)^{5/2} \sqrt {a+c x^2}} \, dx \]
Optimal antiderivative \[ -\frac {2 A \sqrt {c \,x^{2}+a}}{3 a e \left (e x \right )^{\frac {3}{2}}}-\frac {2 B \sqrt {c \,x^{2}+a}}{a \,e^{2} \sqrt {e x}}+\frac {2 B x \sqrt {c}\, \sqrt {c \,x^{2}+a}}{a \,e^{2} \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {e x}}-\frac {2 B \,c^{\frac {1}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{\cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {3}{4}} e^{2} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}+\frac {c^{\frac {1}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (3 B \sqrt {a}-A \sqrt {c}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{3 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {5}{4}} e^{2} \sqrt {e x}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((B*x+A)/(e*x)^(5/2)/(c*x^2+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (A \sqrt {c} x^{2} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) + 3 \, B \sqrt {c} x^{2} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + \sqrt {c x^{2} + a} {\left (3 \, B x + A\right )} \sqrt {x}\right )} e^{\left (-\frac {5}{2}\right )}}{3 \, a x^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x}}{c e^{3} x^{5} + a e^{3} x^{3}}, x\right ) \]