\[ \int \frac {\left (a+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx \]
Optimal antiderivative \[ \frac {20 \left (-7 c d e x +9 a \,e^{2}+8 c \,d^{2}\right ) \left (c \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {e x +d}}{693 e^{3}}+\frac {2 \left (c \,x^{2}+a \right )^{\frac {5}{2}} \sqrt {e x +d}}{11 e}+\frac {8 \left (32 c^{2} d^{4}+69 a c \,d^{2} e^{2}+45 a^{2} e^{4}-24 c d e \left (2 a \,e^{2}+c \,d^{2}\right ) x \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}{693 e^{5}}+\frac {16 d \left (93 a^{2} e^{4}+93 a c \,d^{2} e^{2}+32 c^{2} d^{4}\right ) \EllipticE \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {1+\frac {c \,x^{2}}{a}}}{693 e^{6} \sqrt {c \,x^{2}+a}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}-\frac {16 \left (a \,e^{2}+c \,d^{2}\right ) \left (45 a^{2} e^{4}+69 a c \,d^{2} e^{2}+32 c^{2} d^{4}\right ) \EllipticF \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}{693 e^{6} \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((c*x^2+a)^(5/2)/(e*x+d)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (8 \, {\left (32 \, c^{3} d^{6} + 117 \, a c^{2} d^{4} e^{2} + 156 \, a^{2} c d^{2} e^{4} + 135 \, a^{3} e^{6}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right ) + 24 \, {\left (32 \, c^{3} d^{5} e + 93 \, a c^{2} d^{3} e^{3} + 93 \, a^{2} c d e^{5}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right )\right ) - 3 \, {\left (96 \, c^{3} d^{3} x e^{3} - 128 \, c^{3} d^{4} e^{2} - 9 \, {\left (7 \, c^{3} x^{4} + 24 \, a c^{2} x^{2} + 37 \, a^{2} c\right )} e^{6} + 2 \, {\left (35 \, c^{3} d x^{3} + 131 \, a c^{2} d x\right )} e^{5} - 4 \, {\left (20 \, c^{3} d^{2} x^{2} + 89 \, a c^{2} d^{2}\right )} e^{4}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )} e^{\left (-7\right )}}{2079 \, c} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\right )} \sqrt {c x^{2} + a}}{\sqrt {e x + d}}, x\right ) \]