\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{11/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{18 c^{2} d^{3} \left (2 c d x +b d \right )^{\frac {5}{2}}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{9 c d \left (2 c d x +b d \right )^{\frac {9}{2}}}-\frac {\sqrt {c \,x^{2}+b x +a}}{12 c^{3} d^{5} \sqrt {2 c d x +b d}}+\frac {\left (-4 a c +b^{2}\right )^{\frac {3}{4}} \EllipticE \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{12 c^{4} d^{\frac {11}{2}} \sqrt {c \,x^{2}+b x +a}}-\frac {\left (-4 a c +b^{2}\right )^{\frac {3}{4}} \EllipticF \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{12 c^{4} d^{\frac {11}{2}} \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(11/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {3 \, \sqrt {2} {\left (32 \, c^{5} x^{5} + 80 \, b c^{4} x^{4} + 80 \, b^{2} c^{3} x^{3} + 40 \, b^{3} c^{2} x^{2} + 10 \, b^{4} c x + b^{5}\right )} \sqrt {c^{2} d} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + {\left (60 \, c^{5} x^{4} + 120 \, b c^{4} x^{3} + 3 \, b^{4} c + 2 \, a b^{2} c^{2} + 4 \, a^{2} c^{3} + 2 \, {\left (43 \, b^{2} c^{3} + 8 \, a c^{4}\right )} x^{2} + 2 \, {\left (13 \, b^{3} c^{2} + 8 \, a b c^{3}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{36 \, {\left (32 \, c^{9} d^{6} x^{5} + 80 \, b c^{8} d^{6} x^{4} + 80 \, b^{2} c^{7} d^{6} x^{3} + 40 \, b^{3} c^{6} d^{6} x^{2} + 10 \, b^{4} c^{5} d^{6} x + b^{5} c^{4} d^{6}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{64 \, c^{6} d^{6} x^{6} + 192 \, b c^{5} d^{6} x^{5} + 240 \, b^{2} c^{4} d^{6} x^{4} + 160 \, b^{3} c^{3} d^{6} x^{3} + 60 \, b^{4} c^{2} d^{6} x^{2} + 12 \, b^{5} c d^{6} x + b^{6} d^{6}}, x\right ) \]