\[ \int \frac {(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 d \left (2 c d x +b d \right )^{\frac {9}{2}}}{\sqrt {c \,x^{2}+b x +a}}+\frac {72 c \,d^{3} \left (2 c d x +b d \right )^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}}{7}+\frac {120 c \left (-4 a c +b^{2}\right ) d^{5} \sqrt {2 c d x +b d}\, \sqrt {c \,x^{2}+b x +a}}{7}+\frac {60 \left (-4 a c +b^{2}\right )^{\frac {9}{4}} d^{\frac {11}{2}} \EllipticF \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{7 \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((2*c*d*x+b*d)^(11/2)/(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
\[ \frac {2 \, {\left (15 \, \sqrt {2} {\left ({\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{5} x^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d^{5} x + {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} d^{5}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + {\left (32 \, c^{5} d^{5} x^{4} + 64 \, b c^{4} d^{5} x^{3} + 24 \, {\left (3 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d^{5} x^{2} + 8 \, {\left (5 \, b^{3} c^{2} - 12 \, a b c^{3}\right )} d^{5} x - {\left (7 \, b^{4} c - 96 \, a b^{2} c^{2} + 240 \, a^{2} c^{3}\right )} d^{5}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{7 \, {\left (c^{2} x^{2} + b c x + a c\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (32 \, c^{5} d^{5} x^{5} + 80 \, b c^{4} d^{5} x^{4} + 80 \, b^{2} c^{3} d^{5} x^{3} + 40 \, b^{3} c^{2} d^{5} x^{2} + 10 \, b^{4} c d^{5} x + b^{5} d^{5}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}}, x\right ) \]