\[ \int (d+e x)^{3/2} \sqrt {b x+c x^2} \, dx \]
Optimal antiderivative \[ \frac {2 e \left (c \,x^{2}+b x \right )^{\frac {3}{2}} \sqrt {e x +d}}{7 c}-\frac {2 \left (-b e +2 c d \right ) \left (8 b^{2} e^{2}-3 b c d e +3 c^{2} d^{2}\right ) \EllipticE \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {e x +d}}{105 c^{\frac {5}{2}} e^{2} \sqrt {1+\frac {e x}{d}}\, \sqrt {c \,x^{2}+b x}}+\frac {4 d \left (-b e +c d \right ) \left (2 b^{2} e^{2}-3 b c d e +3 c^{2} d^{2}\right ) \EllipticF \left (\frac {\sqrt {c}\, \sqrt {x}}{\sqrt {-b}}, \sqrt {\frac {b e}{c d}}\right ) \sqrt {-b}\, \sqrt {x}\, \sqrt {1+\frac {c x}{b}}\, \sqrt {1+\frac {e x}{d}}}{105 c^{\frac {5}{2}} e^{2} \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}+\frac {2 \left (3 c^{2} d^{2}+9 b c d e -4 b^{2} e^{2}+12 c e \left (-b e +2 c d \right ) x \right ) \sqrt {e x +d}\, \sqrt {c \,x^{2}+b x}}{105 c^{2} e} \]
command
integrate((e*x+d)^(3/2)*(c*x^2+b*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left ({\left (6 \, c^{4} d^{4} - 12 \, b c^{3} d^{3} e - 17 \, b^{2} c^{2} d^{2} e^{2} + 23 \, b^{3} c d e^{3} - 8 \, b^{4} e^{4}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 3 \, {\left (6 \, c^{4} d^{3} e - 9 \, b c^{3} d^{2} e^{2} + 19 \, b^{2} c^{2} d e^{3} - 8 \, b^{3} c e^{4}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) + 3 \, {\left (3 \, c^{4} d^{2} e^{2} + {\left (15 \, c^{4} x^{2} + 3 \, b c^{3} x - 4 \, b^{2} c^{2}\right )} e^{4} + 3 \, {\left (8 \, c^{4} d x + 3 \, b c^{3} d\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x e + d}\right )} e^{\left (-3\right )}}{315 \, c^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {c x^{2} + b x} {\left (e x + d\right )}^{\frac {3}{2}}, x\right ) \]