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