\[ \int (b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {\left (-4 a c +b^{2}\right ) \left (2 c d x +b d \right )^{\frac {9}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{114 c^{2} d}+\frac {\left (2 c d x +b d \right )^{\frac {9}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{19 c d}-\frac {\left (-4 a c +b^{2}\right )^{3} d \left (2 c d x +b d \right )^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}}{2926 c^{3}}+\frac {\left (-4 a c +b^{2}\right )^{2} \left (2 c d x +b d \right )^{\frac {9}{2}} \sqrt {c \,x^{2}+b x +a}}{836 c^{3} d}-\frac {5 \left (-4 a c +b^{2}\right )^{4} d^{3} \sqrt {2 c d x +b d}\, \sqrt {c \,x^{2}+b x +a}}{8778 c^{3}}-\frac {5 \left (-4 a c +b^{2}\right )^{\frac {21}{4}} d^{\frac {7}{2}} \EllipticF \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{17556 c^{4} \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((2*c*d*x+b*d)^(7/2)*(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
\[ -\frac {5 \, \sqrt {2} {\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\right )} \sqrt {c^{2} d} d^{3} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) - 2 \, {\left (14784 \, c^{10} d^{3} x^{8} + 59136 \, b c^{9} d^{3} x^{7} + 4928 \, {\left (19 \, b^{2} c^{8} + 8 \, a c^{9}\right )} d^{3} x^{6} + 14784 \, {\left (5 \, b^{3} c^{7} + 8 \, a b c^{8}\right )} d^{3} x^{5} + 28 \, {\left (1057 \, b^{4} c^{6} + 4744 \, a b^{2} c^{7} + 1072 \, a^{2} c^{8}\right )} d^{3} x^{4} + 56 \, {\left (89 \, b^{5} c^{5} + 1224 \, a b^{3} c^{6} + 1072 \, a^{2} b c^{7}\right )} d^{3} x^{3} + 6 \, {\left (3 \, b^{6} c^{4} + 2456 \, a b^{4} c^{5} + 7312 \, a^{2} b^{2} c^{6} + 256 \, a^{3} c^{7}\right )} d^{3} x^{2} - 2 \, {\left (5 \, b^{7} c^{3} - 88 \, a b^{5} c^{4} - 6928 \, a^{2} b^{3} c^{5} - 768 \, a^{3} b c^{6}\right )} d^{3} x + {\left (5 \, b^{8} c^{2} - 90 \, a b^{6} c^{3} + 628 \, a^{2} b^{4} c^{4} + 2944 \, a^{3} b^{2} c^{5} - 2560 \, a^{4} c^{6}\right )} d^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{35112 \, c^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (8 \, c^{5} d^{3} x^{7} + 28 \, b c^{4} d^{3} x^{6} + 2 \, {\left (19 \, b^{2} c^{3} + 8 \, a c^{4}\right )} d^{3} x^{5} + a^{2} b^{3} d^{3} + 5 \, {\left (5 \, b^{3} c^{2} + 8 \, a b c^{3}\right )} d^{3} x^{4} + 4 \, {\left (2 \, b^{4} c + 9 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} d^{3} x^{3} + {\left (b^{5} + 14 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} d^{3} x^{2} + 2 \, {\left (a b^{4} + 3 \, a^{2} b^{2} c\right )} d^{3} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}, x\right ) \]