\[ \int (d+e x)^{3/2} \left (b x+c x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (c^{2} d^{2}+13 b c d e -6 b^{2} e^{2}+14 c e \left (-b e +2 c d \right ) x \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {e x +d}}{231 c^{2} e}+\frac {2 e \left (c \,x^{2}+b x \right )^{\frac {5}{2}} \sqrt {e x +d}}{11 c}-\frac {16 \left (-2 b e +c d \right ) \left (-b e +2 c d \right ) \left (b e +c d \right ) \left (b^{2} e^{2}-b c d e +c^{2} d^{2}\right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{1155 c^{\frac {7}{2}} e^{4} \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}+\frac {2 d \left (-b e +c d \right ) \left (-8 b^{4} e^{4}+13 b^{3} c d \,e^{3}+3 b^{2} c^{2} d^{2} e^{2}-32 b \,c^{3} d^{3} e +16 c^{4} d^{4}\right ) \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{1155 c^{\frac {7}{2}} e^{4} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}+\frac {2 \left (8 c^{4} d^{4}-19 b \,c^{3} d^{3} e +6 b^{2} c^{2} d^{2} e^{2}-19 b^{3} c d \,e^{3}+8 b^{4} e^{4}-3 c e \left (-b e +2 c d \right ) \left (8 b^{2} e^{2}-b c d e +c^{2} d^{2}\right ) x \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}{1155 c^{3} e^{3}} \]
command
integrate((e*x+d)^(3/2)*(c*x^2+b*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (16 \, c^{6} d^{6} - 48 \, b c^{5} d^{5} e + 33 \, b^{2} c^{4} d^{4} e^{2} + 14 \, b^{3} c^{3} d^{3} e^{3} + 33 \, b^{4} c^{2} d^{2} e^{4} - 48 \, b^{5} c d e^{5} + 16 \, b^{6} e^{6}\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 ) + 24 \, {\left (2 \, c^{6} d^{5} e - 5 \, b c^{5} d^{4} e^{2} + 2 \, b^{2} c^{4} d^{3} e^{3} + 2 \, b^{3} c^{3} d^{2} e^{4} - 5 \, b^{4} c^{2} d e^{5} + 2 \, b^{5} c e^{6}\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 (8 \, c^{6} d^{4} e^{2} + {\left (105 \, c^{6} x^{4} + 140 \, b c^{5} x^{3} + 5 \, b^{2} c^{4} x^{2} - 6 \, b^{3} c^{3} x + 8 \, b^{4} c^{2}\right )} e^{6} + {\left (140 \, c^{6} d x^{3} + 205 \, b c^{5} d x^{2} + 14 \, b^{2} c^{4} d x - 19 \, b^{3} c^{3} d\right )} e^{5} + {\left (5 \, c^{6} d^{2} x^{2} + 14 \, b c^{5} d^{2} x + 6 \, b^{2} c^{4} d^{2}\right )} e^{4} - {\left (6 \, c^{6} d^{3} x + 19 \, b c^{5} d^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )} e^{\left (-5\right )}}{3465 \, c^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (c e x^{3} + b d x + {\left (c d + b e\right )} x^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}, x\right ) \]