\[ \int \frac {A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {B x +A}{a e \sqrt {e x}\, \sqrt {c \,x^{2}+a}}-\frac {3 A \sqrt {c \,x^{2}+a}}{a^{2} e \sqrt {e x}}+\frac {3 A x \sqrt {c}\, \sqrt {c \,x^{2}+a}}{a^{2} e \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {e x}}-\frac {3 A \,c^{\frac {1}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{\cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {7}{4}} e \sqrt {e x}\, \sqrt {c \,x^{2}+a}}+\frac {\sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (B \sqrt {a}+3 A \sqrt {c}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{2 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {7}{4}} c^{\frac {1}{4}} e \sqrt {e x}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((B*x+A)/(e*x)^(3/2)/(c*x^2+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left ({\left (B a c x^{3} + B a^{2} x\right )} \sqrt {c} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 3 \, {\left (A c^{2} x^{3} + A a c x\right )} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (3 \, A c^{2} x^{2} - B a c x + 2 \, A a c\right )} \sqrt {c x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {3}{2}\right )}}{a^{2} c^{2} x^{3} + a^{3} c x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x}}{c^{2} e^{2} x^{6} + 2 \, a c e^{2} x^{4} + a^{2} e^{2} x^{2}}, x\right ) \]