\[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{7 e \left (e x +d \right )^{\frac {7}{2}}}-\frac {4 c \left (2 d \left (a \,e^{2}+2 c \,d^{2}\right )+e \left (5 a \,e^{2}+7 c \,d^{2}\right ) x \right ) \sqrt {c \,x^{2}+a}}{35 e^{3} \left (a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}+\frac {32 c^{2} d \left (2 a \,e^{2}+c \,d^{2}\right ) \sqrt {c \,x^{2}+a}}{35 e^{3} \left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {e x +d}}+\frac {32 c^{\frac {5}{2}} d \left (2 a \,e^{2}+c \,d^{2}\right ) \EllipticE \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {e x +d}\, \sqrt {1+\frac {c \,x^{2}}{a}}}{35 e^{4} \left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {c \,x^{2}+a}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}-\frac {8 c^{\frac {3}{2}} \left (5 a \,e^{2}+4 c \,d^{2}\right ) \EllipticF \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}{35 e^{4} \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((c*x^2+a)^(3/2)/(e*x+d)^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (4 \, {\left (16 \, c^{3} d^{7} x e + 4 \, c^{3} d^{8} + 15 \, a^{2} c x^{4} e^{8} + 60 \, a^{2} c d x^{3} e^{7} + {\left (11 \, a c^{2} d^{2} x^{4} + 90 \, a^{2} c d^{2} x^{2}\right )} e^{6} + 4 \, {\left (11 \, a c^{2} d^{3} x^{3} + 15 \, a^{2} c d^{3} x\right )} e^{5} + {\left (4 \, c^{3} d^{4} x^{4} + 66 \, a c^{2} d^{4} x^{2} + 15 \, a^{2} c d^{4}\right )} e^{4} + 4 \, {\left (4 \, c^{3} d^{5} x^{3} + 11 \, a c^{2} d^{5} x\right )} e^{3} + {\left (24 \, c^{3} d^{6} x^{2} + 11 \, a c^{2} d^{6}\right )} e^{2}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right ) + 48 \, {\left (4 \, c^{3} d^{6} x e^{2} + c^{3} d^{7} e + 2 \, a c^{2} d x^{4} e^{7} + 8 \, a c^{2} d^{2} x^{3} e^{6} + {\left (c^{3} d^{3} x^{4} + 12 \, a c^{2} d^{3} x^{2}\right )} e^{5} + 4 \, {\left (c^{3} d^{4} x^{3} + 2 \, a c^{2} d^{4} x\right )} e^{4} + 2 \, {\left (3 \, c^{3} d^{5} x^{2} + a c^{2} d^{5}\right )} e^{3}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right )\right ) + 3 \, {\left (26 \, c^{3} d^{5} x e^{3} + 8 \, c^{3} d^{6} e^{2} - 5 \, {\left (3 \, a^{2} c x^{2} + a^{3}\right )} e^{8} + 2 \, {\left (16 \, a c^{2} d x^{3} - 7 \, a^{2} c d x\right )} e^{7} + 2 \, {\left (31 \, a c^{2} d^{2} x^{2} - 7 \, a^{2} c d^{2}\right )} e^{6} + 4 \, {\left (4 \, c^{3} d^{3} x^{3} + 15 \, a c^{2} d^{3} x\right )} e^{5} + {\left (29 \, c^{3} d^{4} x^{2} + 15 \, a c^{2} d^{4}\right )} e^{4}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )}}{105 \, {\left (4 \, c^{2} d^{7} x e^{6} + c^{2} d^{8} e^{5} + a^{2} x^{4} e^{13} + 4 \, a^{2} d x^{3} e^{12} + 2 \, {\left (a c d^{2} x^{4} + 3 \, a^{2} d^{2} x^{2}\right )} e^{11} + 4 \, {\left (2 \, a c d^{3} x^{3} + a^{2} d^{3} x\right )} e^{10} + {\left (c^{2} d^{4} x^{4} + 12 \, a c d^{4} x^{2} + a^{2} d^{4}\right )} e^{9} + 4 \, {\left (c^{2} d^{5} x^{3} + 2 \, a c d^{5} x\right )} e^{8} + 2 \, {\left (3 \, c^{2} d^{6} x^{2} + a c d^{6}\right )} e^{7}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} \sqrt {e x + d}}{e^{5} x^{5} + 5 \, d e^{4} x^{4} + 10 \, d^{2} e^{3} x^{3} + 10 \, d^{3} e^{2} x^{2} + 5 \, d^{4} e x + d^{5}}, x\right ) \]