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