\[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {4 \left (-21 A c x +25 B a \right ) \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{105 e^{2} \left (e x \right )^{\frac {3}{2}}}-\frac {2 \left (-5 B x +7 A \right ) \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{35 e \left (e x \right )^{\frac {5}{2}}}-\frac {8 a c \left (-25 B x +63 A \right ) \sqrt {c \,x^{2}+a}}{105 e^{3} \sqrt {e x}}+\frac {48 a A \,c^{\frac {3}{2}} x \sqrt {c \,x^{2}+a}}{5 e^{3} \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {e x}}-\frac {48 a^{\frac {5}{4}} A \,c^{\frac {5}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{5 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) e^{3} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}+\frac {8 a^{\frac {5}{4}} c^{\frac {3}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (25 B \sqrt {a}+63 A \sqrt {c}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{105 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) e^{3} \sqrt {e x}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((B*x+A)*(c*x^2+a)^(5/2)/(e*x)^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (200 \, B a^{2} \sqrt {c} x^{3} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 504 \, A a c^{\frac {3}{2}} x^{3} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (15 \, B c^{2} x^{5} + 21 \, A c^{2} x^{4} + 80 \, B a c x^{3} - 252 \, A a c x^{2} - 35 \, B a^{2} x - 21 \, A a^{2}\right )} \sqrt {c x^{2} + a} \sqrt {x}\right )} e^{\left (-\frac {7}{2}\right )}}{105 \, x^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (B c^{2} x^{5} + A c^{2} x^{4} + 2 \, B a c x^{3} + 2 \, A a c x^{2} + B a^{2} x + A a^{2}\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{e^{4} x^{4}}, x\right ) \]