\[ \int \frac {(A+B x) \sqrt {a+c x^2}}{(e x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-B x +3 A \right ) \sqrt {c \,x^{2}+a}}{3 e \sqrt {e x}}+\frac {4 A x \sqrt {c}\, \sqrt {c \,x^{2}+a}}{e \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {e x}}-\frac {4 a^{\frac {1}{4}} A \,c^{\frac {1}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{\cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) e \sqrt {e x}\, \sqrt {c \,x^{2}+a}}+\frac {2 a^{\frac {1}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (B \sqrt {a}+3 A \sqrt {c}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{3 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {1}{4}} e \sqrt {e x}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((B*x+A)*(c*x^2+a)^(1/2)/(e*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (2 \, B a \sqrt {c} x {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 6 \, A c^{\frac {3}{2}} x {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (B c x - 3 \, A c\right )} \sqrt {c x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {3}{2}\right )}}{3 \, c x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x}}{e^{2} x^{2}}, x\right ) \]