\[ \int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {c \,x^{2}+b x +a}}{5 c d \left (2 c d x +b d \right )^{\frac {5}{2}}}+\frac {2 \sqrt {c \,x^{2}+b x +a}}{5 c \left (-4 a c +b^{2}\right ) d^{3} \sqrt {2 c d x +b d}}-\frac {\EllipticE \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{5 c^{2} \left (-4 a c +b^{2}\right )^{\frac {1}{4}} d^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}}+\frac {\EllipticF \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{5 c^{2} \left (-4 a c +b^{2}\right )^{\frac {1}{4}} d^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (8 \, c^{3} x^{3} + 12 \, b c^{2} x^{2} + 6 \, b^{2} c x + b^{3}\right )} \sqrt {c^{2} d} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + {\left (8 \, c^{3} x^{2} + 8 \, b c^{2} x + b^{2} c + 4 \, a c^{2}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{5 \, {\left (8 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} d^{4} x^{3} + 12 \, {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} d^{4} x^{2} + 6 \, {\left (b^{4} c^{3} - 4 \, a b^{2} c^{4}\right )} d^{4} x + {\left (b^{5} c^{2} - 4 \, a b^{3} c^{3}\right )} d^{4}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{16 \, c^{4} d^{4} x^{4} + 32 \, b c^{3} d^{4} x^{3} + 24 \, b^{2} c^{2} d^{4} x^{2} + 8 \, b^{3} c d^{4} x + b^{4} d^{4}}, x\right ) \]