\[ \int (e x)^{3/2} (A+B x) \left (a+c x^2\right )^{5/2} \, dx \]
Optimal antiderivative \[ \frac {2 B \left (e x \right )^{\frac {3}{2}} \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{17 c}-\frac {4 a^{2} e \left (385 B x +221 A \right ) \left (c \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {e x}}{51051 c}-\frac {2 a e \left (495 B x +221 A \right ) \left (c \,x^{2}+a \right )^{\frac {5}{2}} \sqrt {e x}}{36465 c}+\frac {2 A e \left (c \,x^{2}+a \right )^{\frac {7}{2}} \sqrt {e x}}{15 c}-\frac {16 a^{4} B \,e^{2} x \sqrt {c \,x^{2}+a}}{221 c^{\frac {3}{2}} \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {e x}}-\frac {8 a^{3} e \left (231 B x +221 A \right ) \sqrt {e x}\, \sqrt {c \,x^{2}+a}}{51051 c}+\frac {16 a^{\frac {17}{4}} B \,e^{2} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{221 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {7}{4}} \sqrt {e x}\, \sqrt {c \,x^{2}+a}}-\frac {8 a^{\frac {15}{4}} e^{2} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (231 B \sqrt {a}+221 A \sqrt {c}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{51051 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {7}{4}} \sqrt {e x}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((e*x)^(3/2)*(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
\[ -\frac {2 \, {\left (8840 \, A a^{4} \sqrt {c} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 9240 \, B a^{4} \sqrt {c} e^{\frac {3}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (15015 \, B c^{4} x^{7} + 17017 \, A c^{4} x^{6} + 41580 \, B a c^{3} x^{5} + 49504 \, A a c^{3} x^{4} + 34265 \, B a^{2} c^{2} x^{3} + 45747 \, A a^{2} c^{2} x^{2} + 3080 \, B a^{3} c x + 8840 \, A a^{3} c\right )} \sqrt {c x^{2} + a} \sqrt {x} e^{\frac {3}{2}}\right )}}{255255 \, c^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B c^{2} e x^{6} + A c^{2} e x^{5} + 2 \, B a c e x^{4} + 2 \, A a c e x^{3} + B a^{2} e x^{2} + A a^{2} e x\right )} \sqrt {c x^{2} + a} \sqrt {e x}, x\right ) \]