\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx \]
Optimal antiderivative \[ -\frac {4 c \left (2 d \left (a \,e^{2}+4 c \,d^{2}\right )+e \left (7 a \,e^{2}+13 c \,d^{2}\right ) x \right ) \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{63 e^{3} \left (a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}-\frac {2 \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{9 e \left (e x +d \right )^{\frac {9}{2}}}-\frac {8 c^{2} \left (d \left (9 a^{2} e^{4}+49 a c \,d^{2} e^{2}+32 c^{2} d^{4}\right )+e \left (21 a^{2} e^{4}+69 a c \,d^{2} e^{2}+40 c^{2} d^{4}\right ) x \right ) \sqrt {c \,x^{2}+a}}{63 e^{5} \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {16 c^{\frac {5}{2}} \left (21 a^{2} e^{4}+57 a c \,d^{2} e^{2}+32 c^{2} d^{4}\right ) \EllipticE \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {e x +d}\, \sqrt {1+\frac {c \,x^{2}}{a}}}{63 e^{6} \left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {c \,x^{2}+a}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}+\frac {16 c^{\frac {5}{2}} d \left (33 a \,e^{2}+32 c \,d^{2}\right ) \EllipticF \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}{63 e^{6} \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((c*x^2+a)^(5/2)/(e*x+d)^(11/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (8 \, {\left (160 \, c^{4} d^{9} x e + 32 \, c^{4} d^{10} + 57 \, a^{2} c^{2} d x^{5} e^{9} + 285 \, a^{2} c^{2} d^{2} x^{4} e^{8} + 3 \, {\left (27 \, a c^{3} d^{3} x^{5} + 190 \, a^{2} c^{2} d^{3} x^{3}\right )} e^{7} + 15 \, {\left (27 \, a c^{3} d^{4} x^{4} + 38 \, a^{2} c^{2} d^{4} x^{2}\right )} e^{6} + {\left (32 \, c^{4} d^{5} x^{5} + 810 \, a c^{3} d^{5} x^{3} + 285 \, a^{2} c^{2} d^{5} x\right )} e^{5} + {\left (160 \, c^{4} d^{6} x^{4} + 810 \, a c^{3} d^{6} x^{2} + 57 \, a^{2} c^{2} d^{6}\right )} e^{4} + 5 \, {\left (64 \, c^{4} d^{7} x^{3} + 81 \, a c^{3} d^{7} x\right )} e^{3} + {\left (320 \, c^{4} d^{8} x^{2} + 81 \, a c^{3} d^{8}\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 ) + 24 \, {\left (160 \, c^{4} d^{8} x e^{2} + 32 \, c^{4} d^{9} e + 21 \, a^{2} c^{2} x^{5} e^{10} + 105 \, a^{2} c^{2} d x^{4} e^{9} + 3 \, {\left (19 \, a c^{3} d^{2} x^{5} + 70 \, a^{2} c^{2} d^{2} x^{3}\right )} e^{8} + 15 \, {\left (19 \, a c^{3} d^{3} x^{4} + 14 \, a^{2} c^{2} d^{3} x^{2}\right )} e^{7} + {\left (32 \, c^{4} d^{4} x^{5} + 570 \, a c^{3} d^{4} x^{3} + 105 \, a^{2} c^{2} d^{4} x\right )} e^{6} + {\left (160 \, c^{4} d^{5} x^{4} + 570 \, a c^{3} d^{5} x^{2} + 21 \, a^{2} c^{2} d^{5}\right )} e^{5} + 5 \, {\left (64 \, c^{4} d^{6} x^{3} + 57 \, a c^{3} d^{6} x\right )} e^{4} + {\left (320 \, c^{4} d^{7} x^{2} + 57 \, a c^{3} d^{7}\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 (544 \, c^{4} d^{7} x e^{3} + 128 \, c^{4} d^{8} e^{2} + 7 \, {\left (15 \, a^{2} c^{2} x^{4} + 4 \, a^{3} c x^{2} + a^{4}\right )} e^{10} + 18 \, {\left (17 \, a^{2} c^{2} d x^{3} + a^{3} c d x\right )} e^{9} + 6 \, {\left (55 \, a c^{3} d^{2} x^{4} + 72 \, a^{2} c^{2} d^{2} x^{2} + 3 \, a^{3} c d^{2}\right )} e^{8} + 4 \, {\left (271 \, a c^{3} d^{3} x^{3} + 63 \, a^{2} c^{2} d^{3} x\right )} e^{7} + {\left (193 \, c^{4} d^{4} x^{4} + 1476 \, a c^{3} d^{4} x^{2} + 63 \, a^{2} c^{2} d^{4}\right )} e^{6} + 2 \, {\left (325 \, c^{4} d^{5} x^{3} + 453 \, a c^{3} d^{5} x\right )} e^{5} + 4 \, {\left (220 \, c^{4} d^{6} x^{2} + 53 \, a c^{3} d^{6}\right )} e^{4}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )}}{189 \, {\left (5 \, c^{2} d^{8} x e^{8} + c^{2} d^{9} e^{7} + a^{2} x^{5} e^{16} + 5 \, a^{2} d x^{4} e^{15} + 2 \, {\left (a c d^{2} x^{5} + 5 \, a^{2} d^{2} x^{3}\right )} e^{14} + 10 \, {\left (a c d^{3} x^{4} + a^{2} d^{3} x^{2}\right )} e^{13} + {\left (c^{2} d^{4} x^{5} + 20 \, a c d^{4} x^{3} + 5 \, a^{2} d^{4} x\right )} e^{12} + {\left (5 \, c^{2} d^{5} x^{4} + 20 \, a c d^{5} x^{2} + a^{2} d^{5}\right )} e^{11} + 10 \, {\left (c^{2} d^{6} x^{3} + a c d^{6} x\right )} e^{10} + 2 \, {\left (5 \, c^{2} d^{7} x^{2} + a c d^{7}\right )} e^{9}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}}{e^{6} x^{6} + 6 \, d e^{5} x^{5} + 15 \, d^{2} e^{4} x^{4} + 20 \, d^{3} e^{3} x^{3} + 15 \, d^{4} e^{2} x^{2} + 6 \, d^{5} e x + d^{6}}, x\right ) \]