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