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