\[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+c x^2}} \, dx \]
Optimal antiderivative \[ -\frac {2 e \sqrt {c \,x^{2}+a}}{5 \left (a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}-\frac {16 c d e \sqrt {c \,x^{2}+a}}{15 \left (a \,e^{2}+c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 c e \left (-9 a \,e^{2}+23 c \,d^{2}\right ) \sqrt {c \,x^{2}+a}}{15 \left (a \,e^{2}+c \,d^{2}\right )^{3} \sqrt {e x +d}}-\frac {2 c^{\frac {3}{2}} \left (-9 a \,e^{2}+23 c \,d^{2}\right ) \EllipticE \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {e x +d}\, \sqrt {1+\frac {c \,x^{2}}{a}}}{15 \left (a \,e^{2}+c \,d^{2}\right )^{3} \sqrt {c \,x^{2}+a}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}+\frac {16 c^{\frac {3}{2}} d \EllipticF \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}{15 \left (a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}} \]
command
integrate(1/(e*x+d)^(7/2)/(c*x^2+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (2 \, {\left (33 \, c^{2} d^{5} x e + 11 \, c^{2} d^{6} - 21 \, a c d x^{3} e^{5} - 63 \, a c d^{2} x^{2} e^{4} + {\left (11 \, c^{2} d^{3} x^{3} - 63 \, a c d^{3} x\right )} e^{3} + 3 \, {\left (11 \, c^{2} d^{4} x^{2} - 7 \, a c d^{4}\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 ) - 3 \, {\left (69 \, c^{2} d^{4} x e^{2} + 23 \, c^{2} d^{5} e - 9 \, a c x^{3} e^{6} - 27 \, a c d x^{2} e^{5} + {\left (23 \, c^{2} d^{2} x^{3} - 27 \, a c d^{2} x\right )} e^{4} + 3 \, {\left (23 \, c^{2} d^{3} x^{2} - 3 \, a c d^{3}\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 (54 \, c^{2} d^{3} x e^{3} + 34 \, c^{2} d^{4} e^{2} - 10 \, a c d x e^{5} - 3 \, {\left (3 \, a c x^{2} - a^{2}\right )} e^{6} + {\left (23 \, c^{2} d^{2} x^{2} + 5 \, a c d^{2}\right )} e^{4}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )}}{45 \, {\left (3 \, c^{3} d^{8} x e^{2} + c^{3} d^{9} e + a^{3} x^{3} e^{10} + 3 \, a^{3} d x^{2} e^{9} + 3 \, {\left (a^{2} c d^{2} x^{3} + a^{3} d^{2} x\right )} e^{8} + {\left (9 \, a^{2} c d^{3} x^{2} + a^{3} d^{3}\right )} e^{7} + 3 \, {\left (a c^{2} d^{4} x^{3} + 3 \, a^{2} c d^{4} x\right )} e^{6} + 3 \, {\left (3 \, a c^{2} d^{5} x^{2} + a^{2} c d^{5}\right )} e^{5} + {\left (c^{3} d^{6} x^{3} + 9 \, a c^{2} d^{6} x\right )} e^{4} + 3 \, {\left (c^{3} d^{7} x^{2} + a c^{2} d^{7}\right )} e^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} \sqrt {e x + d}}{c e^{4} x^{6} + 4 \, c d e^{3} x^{5} + 4 \, a d^{3} e x + a d^{4} + {\left (6 \, c d^{2} e^{2} + a e^{4}\right )} x^{4} + 4 \, {\left (c d^{3} e + a d e^{3}\right )} x^{3} + {\left (c d^{4} + 6 \, a d^{2} e^{2}\right )} x^{2}}, x\right ) \]