\[ \int \frac {1}{(b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2}} \, dx \]
Optimal antiderivative \[ \frac {4 \sqrt {c \,x^{2}+b x +a}}{3 \left (-4 a c +b^{2}\right ) d \left (2 c d x +b d \right )^{\frac {3}{2}}}+\frac {2 \EllipticF \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{3 c \left (-4 a c +b^{2}\right )^{\frac {3}{4}} d^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}} \]
command
integrate(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {4 \, \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a} c^{2} + \sqrt {2} {\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )}{3 \, {\left (4 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{3} x^{2} + 4 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d^{3} x + {\left (b^{4} c^{2} - 4 \, a b^{2} c^{3}\right )} d^{3}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{8 \, c^{4} d^{3} x^{5} + 20 \, b c^{3} d^{3} x^{4} + a b^{3} d^{3} + 2 \, {\left (9 \, b^{2} c^{2} + 4 \, a c^{3}\right )} d^{3} x^{3} + {\left (7 \, b^{3} c + 12 \, a b c^{2}\right )} d^{3} x^{2} + {\left (b^{4} + 6 \, a b^{2} c\right )} d^{3} x}, x\right ) \]