\[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{\sqrt {e x}} \, dx \]
Optimal antiderivative \[ \frac {2 b^{2} \left (e x \right )^{\frac {5}{2}} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{15 d \,e^{3}}+\frac {2 \left (33 a^{2} d^{2}+b c \left (-6 a d +b c \right )\right ) \left (d \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {e x}}{231 d^{2} e}-\frac {2 b \left (-6 a d +b c \right ) \left (d \,x^{2}+c \right )^{\frac {5}{2}} \sqrt {e x}}{33 d^{2} e}+\frac {4 c \left (33 a^{2} d^{2}+b c \left (-6 a d +b c \right )\right ) \sqrt {e x}\, \sqrt {d \,x^{2}+c}}{231 d^{2} e}+\frac {4 c^{\frac {7}{4}} \left (33 a^{2} d^{2}+b c \left (-6 a d +b c \right )\right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ), \frac {\sqrt {2}}{2}\right ) \left (\sqrt {c}+x \sqrt {d}\right ) \sqrt {\frac {d \,x^{2}+c}{\left (\sqrt {c}+x \sqrt {d}\right )^{2}}}}{231 \cos \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ) d^{\frac {9}{4}} \sqrt {e}\, \sqrt {d \,x^{2}+c}} \]
command
integrate((b*x^2+a)^2*(d*x^2+c)^(3/2)/(e*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (20 \, {\left (b^{2} c^{4} - 6 \, a b c^{3} d + 33 \, a^{2} c^{2} d^{2}\right )} \sqrt {d} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) + {\left (77 \, b^{2} d^{4} x^{6} - 20 \, b^{2} c^{3} d + 120 \, a b c^{2} d^{2} + 495 \, a^{2} c d^{3} + 7 \, {\left (17 \, b^{2} c d^{3} + 30 \, a b d^{4}\right )} x^{4} + 3 \, {\left (4 \, b^{2} c^{2} d^{2} + 130 \, a b c d^{3} + 55 \, a^{2} d^{4}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x}\right )} e^{\left (-\frac {1}{2}\right )}}{1155 \, d^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b^{2} d x^{6} + {\left (b^{2} c + 2 \, a b d\right )} x^{4} + a^{2} c + {\left (2 \, a b c + a^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{e x}, x\right ) \]