\[ \int \frac {(e x)^{3/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (-a d +b c \right )^{2} \left (e x \right )^{\frac {5}{2}}}{c \,d^{2} e \sqrt {d \,x^{2}+c}}+\frac {2 b^{2} \left (e x \right )^{\frac {5}{2}} \sqrt {d \,x^{2}+c}}{7 d^{2} e}-\frac {\left (21 a^{2} d^{2}-70 a b c d +45 b^{2} c^{2}\right ) e \sqrt {e x}\, \sqrt {d \,x^{2}+c}}{21 c \,d^{3}}+\frac {\left (21 a^{2} d^{2}-70 a b c d +45 b^{2} c^{2}\right ) e^{\frac {3}{2}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {c}+x \sqrt {d}\right ) \sqrt {\frac {d \,x^{2}+c}{\left (\sqrt {c}+x \sqrt {d}\right )^{2}}}}{42 \cos \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ) c^{\frac {1}{4}} d^{\frac {13}{4}} \sqrt {d \,x^{2}+c}} \]
command
integrate((e*x)^(3/2)*(b*x^2+a)^2/(d*x^2+c)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (45 \, b^{2} c^{3} - 70 \, a b c^{2} d + 21 \, a^{2} c d^{2} + {\left (45 \, b^{2} c^{2} d - 70 \, a b c d^{2} + 21 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d} e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) + {\left (6 \, b^{2} d^{3} x^{4} - 45 \, b^{2} c^{2} d + 70 \, a b c d^{2} - 21 \, a^{2} d^{3} - 2 \, {\left (9 \, b^{2} c d^{2} - 14 \, a b d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x} e^{\frac {3}{2}}}{21 \, {\left (d^{5} x^{2} + c d^{4}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b^{2} e x^{5} + 2 \, a b e x^{3} + a^{2} e x\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}, x\right ) \]