\[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (28 c d e x -3 a \,e^{2}+c \,d^{2}\right ) \left (c \,x^{2}+a \right )^{\frac {3}{2}} \sqrt {e x +d}}{231 c e}+\frac {2 e \left (c \,x^{2}+a \right )^{\frac {5}{2}} \sqrt {e x +d}}{11 c}+\frac {4 \left (4 c^{2} d^{4}+21 a c \,d^{2} e^{2}-15 a^{2} e^{4}-3 c d e \left (-31 a \,e^{2}+c \,d^{2}\right ) x \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}{1155 c \,e^{3}}+\frac {32 d \left (-3 a \,e^{2}+c \,d^{2}\right ) \left (9 a \,e^{2}+c \,d^{2}\right ) \EllipticE \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {e x +d}\, \sqrt {1+\frac {c \,x^{2}}{a}}}{1155 e^{4} \sqrt {c}\, \sqrt {c \,x^{2}+a}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}-\frac {8 \left (a \,e^{2}+c \,d^{2}\right ) \left (-15 a^{2} e^{4}+21 a c \,d^{2} e^{2}+4 c^{2} d^{4}\right ) \EllipticF \left (\frac {\sqrt {1-\frac {x \sqrt {c}}{\sqrt {-a}}}\, \sqrt {2}}{2}, \sqrt {-\frac {2 a e}{-a e +d \sqrt {-a}\, \sqrt {c}}}\right ) \sqrt {-a}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \sqrt {\frac {\left (e x +d \right ) \sqrt {c}}{e \sqrt {-a}+d \sqrt {c}}}}{1155 c^{\frac {3}{2}} e^{4} \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}} \]
command
integrate((e*x+d)^(3/2)*(c*x^2+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (4 \, {\left (4 \, c^{3} d^{6} + 27 \, a c^{2} d^{4} e^{2} + 234 \, a^{2} c d^{2} e^{4} - 45 \, a^{3} e^{6}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right ) + 48 \, {\left (c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} - 27 \, a^{2} c d e^{5}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right )\right ) - 3 \, {\left (6 \, c^{3} d^{3} x e^{3} - 8 \, c^{3} d^{4} e^{2} - 15 \, {\left (7 \, c^{3} x^{4} + 13 \, a c^{2} x^{2} + 4 \, a^{2} c\right )} e^{6} - 2 \, {\left (70 \, c^{3} d x^{3} + 163 \, a c^{2} d x\right )} e^{5} - {\left (5 \, c^{3} d^{2} x^{2} + 47 \, a c^{2} d^{2}\right )} e^{4}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}\right )} e^{\left (-5\right )}}{3465 \, c^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (c e x^{3} + c d x^{2} + a e x + a d\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}, x\right ) \]