\[ \int \sqrt {d+e x} \left (a+c x^2\right )^{5/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{13 e}+\frac {20 \left (4 d \left (5 a \,e^{2}+2 c \,d^{2}\right )-7 e \left (-11 a \,e^{2}+c \,d^{2}\right ) x \right ) \left (c \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {e x +d}}{9009 e^{3}}-\frac {20 d \left (c \,x^{2}+a \right )^{\frac {5}{2}} \sqrt {e x +d}}{143 e}+\frac {8 \left (d \left (177 a^{2} e^{4}+113 a c \,d^{2} e^{2}+32 c^{2} d^{4}\right )-3 e \left (-77 a^{2} e^{4}+27 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right ) x \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}{9009 e^{5}}+\frac {16 \left (-231 a^{3} e^{6}+258 a^{2} c \,d^{2} e^{4}+137 a \,c^{2} d^{4} e^{2}+32 c^{3} d^{6}\right ) \EllipticE \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {e x +d}\, \sqrt {1+\frac {c \,x^{2}}{a}}}{9009 e^{6} \sqrt {c}\, \sqrt {c \,x^{2}+a}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}-\frac {16 d \left (a \,e^{2}+c \,d^{2}\right ) \left (177 a^{2} e^{4}+113 a c \,d^{2} e^{2}+32 c^{2} d^{4}\right ) \EllipticF \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}{9009 e^{6} \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}} \]
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
\[ \frac {2 \, {\left (8 \, {\left (32 \, c^{3} d^{7} + 161 \, a c^{2} d^{5} e^{2} + 354 \, a^{2} c d^{3} e^{4} + 993 \, a^{3} d e^{6}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right ) + 24 \, {\left (32 \, c^{3} d^{6} e + 137 \, a c^{2} d^{4} e^{3} + 258 \, a^{2} c d^{2} e^{5} - 231 \, a^{3} e^{7}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right )\right ) - 3 \, {\left (96 \, c^{3} d^{4} x e^{3} - 128 \, c^{3} d^{5} e^{2} - 77 \, {\left (9 \, c^{3} x^{5} + 28 \, a c^{2} x^{3} + 31 \, a^{2} c x\right )} e^{7} - {\left (63 \, c^{3} d x^{4} + 326 \, a c^{2} d x^{2} + 971 \, a^{2} c d\right )} e^{6} + 2 \, {\left (35 \, c^{3} d^{2} x^{3} + 197 \, a c^{2} d^{2} x\right )} e^{5} - 4 \, {\left (20 \, c^{3} d^{3} x^{2} + 133 \, a c^{2} d^{3}\right )} e^{4}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )} e^{\left (-7\right )}}{27027 \, c} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}, x\right ) \]