\[ \int \frac {A+B x}{(e x)^{7/2} \left (a+c x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {B x +A}{3 a e \left (e x \right )^{\frac {5}{2}} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {9 B x +11 A}{6 a^{2} e \left (e x \right )^{\frac {5}{2}} \sqrt {c \,x^{2}+a}}-\frac {77 A \sqrt {c \,x^{2}+a}}{30 a^{3} e \left (e x \right )^{\frac {5}{2}}}-\frac {5 B \sqrt {c \,x^{2}+a}}{2 a^{3} e^{2} \left (e x \right )^{\frac {3}{2}}}+\frac {77 A c \sqrt {c \,x^{2}+a}}{10 a^{4} e^{3} \sqrt {e x}}-\frac {77 A \,c^{\frac {3}{2}} x \sqrt {c \,x^{2}+a}}{10 a^{4} e^{3} \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {e x}}+\frac {77 A \,c^{\frac {5}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{10 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {15}{4}} e^{3} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}-\frac {c^{\frac {3}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (25 B \sqrt {a}+77 A \sqrt {c}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{20 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {15}{4}} e^{3} \sqrt {e x}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((B*x+A)/(e*x)^(7/2)/(c*x^2+a)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {{\left (75 \, {\left (B a c^{2} x^{7} + 2 \, B a^{2} c x^{5} + B a^{3} x^{3}\right )} \sqrt {c} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 231 \, {\left (A c^{3} x^{7} + 2 \, A a c^{2} x^{5} + A a^{2} c x^{3}\right )} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (231 \, A c^{3} x^{6} - 75 \, B a c^{2} x^{5} + 385 \, A a c^{2} x^{4} - 105 \, B a^{2} c x^{3} + 132 \, A a^{2} c x^{2} - 20 \, B a^{3} x - 12 \, A a^{3}\right )} \sqrt {c x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {7}{2}\right )}}{30 \, {\left (a^{4} c^{2} x^{7} + 2 \, a^{5} c x^{5} + a^{6} x^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} {\left (B x + A\right )} \sqrt {e x}}{c^{3} e^{4} x^{10} + 3 \, a c^{2} e^{4} x^{8} + 3 \, a^{2} c e^{4} x^{6} + a^{3} e^{4} x^{4}}, x\right ) \]