\[ \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \sqrt {2 c d x +b d}}{3 \left (-4 a c +b^{2}\right ) d \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {20 c \sqrt {2 c d x +b d}}{3 \left (-4 a c +b^{2}\right )^{2} d \sqrt {c \,x^{2}+b x +a}}+\frac {40 c \EllipticF \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{3 \left (-4 a c +b^{2}\right )^{\frac {7}{4}} \sqrt {d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate(1/(2*c*d*x+b*d)^(1/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (10 \, \sqrt {2} {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + {\left (10 \, c^{2} x^{2} + 10 \, b c x - b^{2} + 14 \, a c\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} d x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} d x + {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )} d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{2 \, c^{4} d x^{7} + 7 \, b c^{3} d x^{6} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d x^{5} + 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d x^{4} + a^{3} b d + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d x^{3} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d x^{2} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d x}, x\right ) \]