\[ \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {x \sqrt {e x +d}}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {\left (a d e +\left (3 a \,e^{2}+4 c \,d^{2}\right ) x \right ) \sqrt {e x +d}}{6 a^{2} \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {c \,x^{2}+a}}+\frac {\left (3 a \,e^{2}+4 c \,d^{2}\right ) \EllipticE \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {e x +d}\, \sqrt {1+\frac {c \,x^{2}}{a}}}{6 \left (-a \right )^{\frac {3}{2}} \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {c}\, \sqrt {c \,x^{2}+a}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}-\frac {2 d \EllipticF \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {1+\frac {c \,x^{2}}{a}}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}{3 \left (-a \right )^{\frac {3}{2}} \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((e*x+d)^(1/2)/(c*x^2+a)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (2 \, c^{3} d^{3} x^{4} + 4 \, a c^{2} d^{3} x^{2} + 2 \, a^{2} c d^{3} + 3 \, {\left (a c^{2} d x^{4} + 2 \, a^{2} c d x^{2} + a^{3} d\right )} e^{2}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right ) + 3 \, {\left (3 \, {\left (a c^{2} x^{4} + 2 \, a^{2} c x^{2} + a^{3}\right )} e^{3} + 4 \, {\left (c^{3} d^{2} x^{4} + 2 \, a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right )\right ) + 3 \, \sqrt {c x^{2} + a} {\left ({\left (3 \, a c^{2} x^{3} + 5 \, a^{2} c x\right )} e^{3} + {\left (a c^{2} d x^{2} + a^{2} c d\right )} e^{2} + 2 \, {\left (2 \, c^{3} d^{2} x^{3} + 3 \, a c^{2} d^{2} x\right )} e\right )} \sqrt {x e + d}}{18 \, {\left ({\left (a^{3} c^{3} x^{4} + 2 \, a^{4} c^{2} x^{2} + a^{5} c\right )} e^{3} + {\left (a^{2} c^{4} d^{2} x^{4} + 2 \, a^{3} c^{3} d^{2} x^{2} + a^{4} c^{2} d^{2}\right )} e\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} \sqrt {e x + d}}{c^{3} x^{6} + 3 \, a c^{2} x^{4} + 3 \, a^{2} c x^{2} + a^{3}}, x\right ) \]