\[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {\left (2 c d x +b d \right )^{\frac {3}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{9 c d}-\frac {\left (-4 a c +b^{2}\right ) \left (2 c d x +b d \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}}{30 c^{2} d}+\frac {\left (-4 a c +b^{2}\right )^{\frac {11}{4}} \EllipticE \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{30 c^{3} \sqrt {c \,x^{2}+b x +a}}-\frac {\left (-4 a c +b^{2}\right )^{\frac {11}{4}} \EllipticF \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {d}\, \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{30 c^{3} \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {3 \, \sqrt {2} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {c^{2} d} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) - {\left (20 \, c^{4} x^{3} + 30 \, b c^{3} x^{2} - 3 \, b^{3} c + 22 \, a b c^{2} + 4 \, {\left (b^{2} c^{2} + 11 \, a c^{3}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{90 \, c^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {2 \, c d x + b d} {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}, x\right ) \]