\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\sqrt {b d+2 c d x}} \, dx \]
Optimal antiderivative \[ \frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \sqrt {2 c d x +b d}}{7 c d}-\frac {\left (-4 a c +b^{2}\right ) \sqrt {2 c d x +b d}\, \sqrt {c \,x^{2}+b x +a}}{14 c^{2} d}+\frac {\left (-4 a c +b^{2}\right )^{\frac {9}{4}} \EllipticF \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{14 c^{3} \sqrt {d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + 2 \, {\left (2 \, c^{4} x^{2} + 2 \, b c^{3} x - b^{2} c^{2} + 6 \, a c^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{28 \, c^{4} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{\sqrt {2 \, c d x + b d}}, x\right ) \]