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