\[ \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right ) d \sqrt {c \,x^{2}+b x +a}}-\frac {4 \EllipticF \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{\left (-4 a c +b^{2}\right )^{\frac {3}{4}} \sqrt {d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate(1/(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
\[ -\frac {2 \, {\left (\sqrt {2} \sqrt {c^{2} d} {\left (c x^{2} + b x + a\right )} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a} c\right )}}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d x^{2} + {\left (b^{3} c - 4 \, a b c^{2}\right )} d x + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{2 \, c^{3} d x^{5} + 5 \, b c^{2} d x^{4} + 4 \, {\left (b^{2} c + a c^{2}\right )} d x^{3} + a^{2} b d + {\left (b^{3} + 6 \, a b c\right )} d x^{2} + 2 \, {\left (a b^{2} + a^{2} c\right )} d x}, x\right ) \]