\[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (b^{2} c^{2}+9 a d \left (a d +2 b c \right )\right ) \left (e x \right )^{\frac {3}{2}} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{9 c^{2} e^{5}}-\frac {2 a^{2} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{5 c e \left (e x \right )^{\frac {5}{2}}}-\frac {2 a \left (a d +2 b c \right ) \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{c^{2} e^{3} \sqrt {e x}}+\frac {4 \left (b^{2} c^{2}+9 a d \left (a d +2 b c \right )\right ) \left (e x \right )^{\frac {3}{2}} \sqrt {d \,x^{2}+c}}{15 c \,e^{5}}+\frac {8 \left (b^{2} c^{2}+9 a d \left (a d +2 b c \right )\right ) \sqrt {e x}\, \sqrt {d \,x^{2}+c}}{15 e^{4} \sqrt {d}\, \left (\sqrt {c}+x \sqrt {d}\right )}-\frac {8 c^{\frac {1}{4}} \left (b^{2} c^{2}+9 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}}\, \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 {3}{4}} e^{\frac {7}{2}} \sqrt {d \,x^{2}+c}}+\frac {4 c^{\frac {1}{4}} \left (b^{2} c^{2}+9 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}}}}{15 \cos \left (2 \arctan \left (\frac {d^{\frac {1}{4}} \sqrt {e x}}{c^{\frac {1}{4}} \sqrt {e}}\right )\right ) d^{\frac {3}{4}} e^{\frac {7}{2}} \sqrt {d \,x^{2}+c}} \]
command
integrate((b*x^2+a)^2*(d*x^2+c)^(3/2)/(e*x)^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (12 \, {\left (b^{2} c^{2} + 18 \, a b c d + 9 \, a^{2} d^{2}\right )} \sqrt {d} x^{3} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) - {\left (5 \, b^{2} d^{2} x^{6} + {\left (11 \, b^{2} c d + 18 \, a b d^{2}\right )} x^{4} - 9 \, a^{2} c d - 9 \, {\left (10 \, a b c d + 7 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {x}\right )} e^{\left (-\frac {7}{2}\right )}}{45 \, d x^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b^{2} d x^{6} + {\left (b^{2} c + 2 \, a b d\right )} x^{4} + a^{2} c + {\left (2 \, a b c + a^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{e^{4} x^{4}}, x\right ) \]