\[ \int \sqrt {e x} (A+B x) \left (a+c x^2\right )^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {4 a \left (-385 A c x +39 B a \right ) \left (c \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {e x}}{9009 c}-\frac {2 \left (-165 A c x +13 B a \right ) \left (c \,x^{2}+a \right )^{\frac {5}{2}} \sqrt {e x}}{2145 c}+\frac {2 B \left (c \,x^{2}+a \right )^{\frac {7}{2}} \sqrt {e x}}{15 c}+\frac {16 a^{3} A e x \sqrt {c \,x^{2}+a}}{39 \sqrt {c}\, \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {e x}}-\frac {8 a^{2} \left (-77 A c x +13 B a \right ) \sqrt {e x}\, \sqrt {c \,x^{2}+a}}{3003 c}-\frac {16 a^{\frac {13}{4}} A e \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{39 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {3}{4}} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}-\frac {8 a^{\frac {13}{4}} e \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (13 B \sqrt {a}-77 A \sqrt {c}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{3003 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {5}{4}} \sqrt {e x}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((e*x)^(1/2)*(B*x+A)*(c*x^2+a)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (1560 \, B a^{4} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) + 9240 \, A a^{3} c^{\frac {3}{2}} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (3003 \, B c^{4} x^{6} + 3465 \, A c^{4} x^{5} + 8736 \, B a c^{3} x^{4} + 10780 \, A a c^{3} x^{3} + 8073 \, B a^{2} c^{2} x^{2} + 11935 \, A a^{2} c^{2} x + 1560 \, B a^{3} c\right )} \sqrt {c x^{2} + a} \sqrt {x} e^{\frac {1}{2}}\right )}}{45045 \, c^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B c^{2} x^{5} + A c^{2} x^{4} + 2 \, B a c x^{3} + 2 \, A a c x^{2} + B a^{2} x + A a^{2}\right )} \sqrt {c x^{2} + a} \sqrt {e x}, x\right ) \]