\[ \int \frac {(A+B x) \left (a+c x^2\right )^{3/2}}{(e x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-3 B x +5 A \right ) \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{15 e \left (e x \right )^{\frac {3}{2}}}-\frac {4 \left (-5 A c x +9 B a \right ) \sqrt {c \,x^{2}+a}}{15 e^{2} \sqrt {e x}}+\frac {24 a B x \sqrt {c}\, \sqrt {c \,x^{2}+a}}{5 e^{2} \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {e x}}-\frac {24 a^{\frac {5}{4}} B \,c^{\frac {1}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{5 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) e^{2} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}+\frac {4 a^{\frac {3}{4}} c^{\frac {1}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (9 B \sqrt {a}+5 A \sqrt {c}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) e^{2} \sqrt {e x}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((B*x+A)*(c*x^2+a)^(3/2)/(e*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (20 \, A a \sqrt {c} x^{2} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 36 \, B a \sqrt {c} x^{2} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (3 \, B c x^{3} + 5 \, A c x^{2} - 15 \, B a x - 5 \, A a\right )} \sqrt {c x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {5}{2}\right )}}{15 \, x^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B c x^{3} + A c x^{2} + B a x + A a\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{e^{3} x^{3}}, x\right ) \]