\[ \int \frac {(A+B x) \sqrt {a+c x^2}}{\sqrt {e x}} \, dx \]
Optimal antiderivative \[ \frac {4 a B x \sqrt {c \,x^{2}+a}}{5 \sqrt {c}\, \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {e x}}+\frac {2 \left (3 B x +5 A \right ) \sqrt {e x}\, \sqrt {c \,x^{2}+a}}{15 e}-\frac {4 a^{\frac {5}{4}} B \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{5 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {3}{4}} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}+\frac {2 a^{\frac {3}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (3 B \sqrt {a}+5 A \sqrt {c}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {3}{4}} \sqrt {e x}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((B*x+A)*(c*x^2+a)^(1/2)/(e*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (10 \, A a \sqrt {c} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 6 \, B a \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (3 \, B c x + 5 \, A c\right )} \sqrt {c x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {1}{2}\right )}}{15 \, c} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x}}{e x}, x\right ) \]