\[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\sqrt {c+d x^2}} \, dx \]
Optimal antiderivative \[ -\frac {2 b \left (-18 a d +7 b c \right ) \left (e x \right )^{\frac {3}{2}} \sqrt {d \,x^{2}+c}}{45 d^{2} e}+\frac {2 b^{2} \left (e x \right )^{\frac {7}{2}} \sqrt {d \,x^{2}+c}}{9 d \,e^{3}}+\frac {2 \left (15 a^{2} d^{2}+b c \left (-18 a d +7 b c \right )\right ) \sqrt {e x}\, \sqrt {d \,x^{2}+c}}{15 d^{\frac {5}{2}} \left (\sqrt {c}+x \sqrt {d}\right )}-\frac {2 c^{\frac {1}{4}} \left (15 a^{2} d^{2}+b c \left (-18 a d +7 b c \right )\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {c}+x \sqrt {d}\right ) \sqrt {e}\, \sqrt {\frac {d \,x^{2}+c}{\left (\sqrt {c}+x \sqrt {d}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ) d^{\frac {11}{4}} \sqrt {d \,x^{2}+c}}+\frac {c^{\frac {1}{4}} \left (15 a^{2} d^{2}+b c \left (-18 a d +7 b c \right )\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {c}+x \sqrt {d}\right ) \sqrt {e}\, \sqrt {\frac {d \,x^{2}+c}{\left (\sqrt {c}+x \sqrt {d}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ) d^{\frac {11}{4}} \sqrt {d \,x^{2}+c}} \]
command
integrate((b*x^2+a)^2*(e*x)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (3 \, {\left (7 \, b^{2} c^{2} - 18 \, a b c d + 15 \, a^{2} d^{2}\right )} \sqrt {d} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) - {\left (5 \, b^{2} d^{2} x^{3} - {\left (7 \, b^{2} c d - 18 \, a b d^{2}\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {x} e^{\frac {1}{2}}\right )}}{45 \, d^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {e x}}{\sqrt {d x^{2} + c}}, x\right ) \]