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