\[ \int (b d+2 c d x)^{7/2} \sqrt {a+b x+c x^2} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-4 a c +b^{2}\right ) d \left (2 c d x +b d \right )^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}}{77 c}+\frac {\left (2 c d x +b d \right )^{\frac {9}{2}} \sqrt {c \,x^{2}+b x +a}}{11 c d}-\frac {10 \left (-4 a c +b^{2}\right )^{2} d^{3} \sqrt {2 c d x +b d}\, \sqrt {c \,x^{2}+b x +a}}{231 c}-\frac {5 \left (-4 a c +b^{2}\right )^{\frac {13}{4}} d^{\frac {7}{2}} \EllipticF \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{231 c^{2} \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((2*c*d*x+b*d)^(7/2)*(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
\[ -\frac {5 \, \sqrt {2} {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c^{2} d} d^{3} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) - 2 \, {\left (336 \, c^{6} d^{3} x^{4} + 672 \, b c^{5} d^{3} x^{3} + 96 \, {\left (5 \, b^{2} c^{4} + a c^{5}\right )} d^{3} x^{2} + 48 \, {\left (3 \, b^{3} c^{3} + 2 \, a b c^{4}\right )} d^{3} x + {\left (5 \, b^{4} c^{2} + 104 \, a b^{2} c^{3} - 160 \, a^{2} c^{4}\right )} d^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{462 \, c^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (8 \, c^{3} d^{3} x^{3} + 12 \, b c^{2} d^{3} x^{2} + 6 \, b^{2} c d^{3} x + b^{3} d^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}, x\right ) \]