\[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{(e x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 a^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{3 c e \left (e x \right )^{\frac {3}{2}}}+\frac {2 b^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {e x}}{7 d \,e^{3}}-\frac {2 \left (b^{2} c^{2}-7 a d \left (a d +2 b c \right )\right ) \sqrt {e x}\, \sqrt {d \,x^{2}+c}}{21 c d \,e^{3}}-\frac {2 \left (b^{2} c^{2}-7 a d \left (a d +2 b c \right )\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {c}+x \sqrt {d}\right ) \sqrt {\frac {d \,x^{2}+c}{\left (\sqrt {c}+x \sqrt {d}\right )^{2}}}}{21 \cos \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ) c^{\frac {1}{4}} d^{\frac {5}{4}} e^{\frac {5}{2}} \sqrt {d \,x^{2}+c}} \]
command
integrate((b*x^2+a)^2*(d*x^2+c)^(1/2)/(e*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (2 \, {\left (b^{2} c^{2} - 14 \, a b c d - 7 \, a^{2} d^{2}\right )} \sqrt {d} x^{2} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (3 \, b^{2} d^{2} x^{4} - 7 \, a^{2} d^{2} + 2 \, {\left (b^{2} c d + 7 \, a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x}\right )} e^{\left (-\frac {5}{2}\right )}}{21 \, d^{2} x^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{e^{3} x^{3}}, x\right ) \]