\[ \int \sqrt {d+e x} \left (b x+c x^2\right )^{5/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{13 e}+\frac {10 \left (16 c^{3} d^{3}-31 b \,c^{2} d^{2} e +9 b^{2} c d \,e^{2}-18 b^{3} e^{3}-14 c e \left (3 b^{2} e^{2}-b c d e +c^{2} d^{2}\right ) x \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {e x +d}}{9009 c^{2} e^{3}}-\frac {10 \left (-b e +2 c d \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}} \sqrt {e x +d}}{143 c e}-\frac {4 \left (24 b^{6} e^{6}-20 b^{5} c d \,e^{5}-21 b^{4} c^{2} d^{2} e^{4}-46 b^{3} c^{3} d^{3} e^{3}+343 b^{2} c^{4} d^{4} e^{2}-384 b \,c^{5} d^{5} e +128 c^{6} d^{6}\right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{9009 c^{\frac {7}{2}} e^{6} \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}+\frac {2 d \left (-b e +c d \right ) \left (-b e +2 c d \right ) \left (24 b^{4} e^{4}+49 b^{3} c d \,e^{3}+79 b^{2} c^{2} d^{2} e^{2}-256 b \,c^{3} d^{3} e +128 c^{4} d^{4}\right ) \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{9009 c^{\frac {7}{2}} e^{6} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}+\frac {2 \left (128 c^{5} d^{5}-368 b \,c^{4} d^{4} e +303 b^{2} c^{3} d^{3} e^{2}-22 b^{3} c^{2} d^{2} e^{3}-17 b^{4} c d \,e^{4}+24 b^{5} e^{5}-3 c e \left (-24 b^{4} e^{4}+11 b^{3} c d \,e^{3}+21 b^{2} c^{2} d^{2} e^{2}-64 b \,c^{3} d^{3} e +32 c^{4} d^{4}\right ) x \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}{9009 c^{3} e^{5}} \]
command
integrate((c*x^2+b*x)^(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 ({\left (256 \, c^{7} d^{7} - 896 \, b c^{6} d^{6} e + 1022 \, b^{2} c^{5} d^{5} e^{2} - 315 \, b^{3} c^{4} d^{4} e^{3} - 68 \, b^{4} c^{3} d^{3} e^{4} - 31 \, b^{5} c^{2} d^{2} e^{5} - 64 \, b^{6} c d e^{6} + 48 \, b^{7} e^{7}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 6 \, {\left (128 \, c^{7} d^{6} e - 384 \, b c^{6} d^{5} e^{2} + 343 \, b^{2} c^{5} d^{4} e^{3} - 46 \, b^{3} c^{4} d^{3} e^{4} - 21 \, b^{4} c^{3} d^{2} e^{5} - 20 \, b^{5} c^{2} d e^{6} + 24 \, b^{6} c e^{7}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) + 3 \, {\left (128 \, c^{7} d^{5} e^{2} + 3 \, {\left (231 \, c^{7} x^{5} + 567 \, b c^{6} x^{4} + 371 \, b^{2} c^{5} x^{3} + 5 \, b^{3} c^{4} x^{2} - 6 \, b^{4} c^{3} x + 8 \, b^{5} c^{2}\right )} e^{7} + {\left (63 \, c^{7} d x^{4} + 196 \, b c^{6} d x^{3} + 178 \, b^{2} c^{5} d x^{2} + 12 \, b^{3} c^{4} d x - 17 \, b^{4} c^{3} d\right )} e^{6} - {\left (70 \, c^{7} d^{2} x^{3} + 225 \, b c^{6} d^{2} x^{2} + 218 \, b^{2} c^{5} d^{2} x + 22 \, b^{3} c^{4} d^{2}\right )} e^{5} + {\left (80 \, c^{7} d^{3} x^{2} + 272 \, b c^{6} d^{3} x + 303 \, b^{2} c^{5} d^{3}\right )} e^{4} - 16 \, {\left (6 \, c^{7} d^{4} x + 23 \, b c^{6} d^{4}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )} e^{\left (-7\right )}}{27027 \, c^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}, x\right ) \]