\[ \int \sqrt {e x} \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx \]
Optimal antiderivative \[ \frac {2 B \left (e x \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{9 b e}+\frac {2 \left (3 A b -B a \right ) \left (e x \right )^{\frac {3}{2}} \sqrt {b \,x^{2}+a}}{15 b e}+\frac {4 a \left (3 A b -B a \right ) \sqrt {e x}\, \sqrt {b \,x^{2}+a}}{15 b^{\frac {3}{2}} \left (\sqrt {a}+x \sqrt {b}\right )}-\frac {4 a^{\frac {5}{4}} \left (3 A b -B a \right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {e}\, \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ) b^{\frac {7}{4}} \sqrt {b \,x^{2}+a}}+\frac {2 a^{\frac {5}{4}} \left (3 A b -B a \right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {a}+x \sqrt {b}\right ) \sqrt {e}\, \sqrt {\frac {b \,x^{2}+a}{\left (\sqrt {a}+x \sqrt {b}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{a^{\frac {1}{4}} \sqrt {e}}\right )\right ) b^{\frac {7}{4}} \sqrt {b \,x^{2}+a}} \]
command
integrate((B*x^2+A)*(e*x)^(1/2)*(b*x^2+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (6 \, {\left (B a^{2} - 3 \, A a b\right )} \sqrt {b} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (5 \, B b^{2} x^{3} + {\left (2 \, B a b + 9 \, A b^{2}\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {x} e^{\frac {1}{2}}\right )}}{45 \, b^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (B x^{2} + A\right )} \sqrt {b x^{2} + a} \sqrt {e x}, x\right ) \]