\[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {5 \left (-4 a c +b^{2}\right ) \left (2 c d x +b d \right )^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{442 c^{2} d}+\frac {\left (2 c d x +b d \right )^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{17 c d}-\frac {\left (-4 a c +b^{2}\right )^{3} d \left (2 c d x +b d \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}}{1326 c^{3}}+\frac {5 \left (-4 a c +b^{2}\right )^{2} \left (2 c d x +b d \right )^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}}{2652 c^{3} d}-\frac {\left (-4 a c +b^{2}\right )^{\frac {19}{4}} d^{\frac {5}{2}} \EllipticE \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{884 c^{4} \sqrt {c \,x^{2}+b x +a}}+\frac {\left (-4 a c +b^{2}\right )^{\frac {19}{4}} d^{\frac {5}{2}} \EllipticF \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{884 c^{4} \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((2*c*d*x+b*d)^(5/2)*(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
\[ \frac {3 \, \sqrt {2} {\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt {c^{2} d} d^{2} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + {\left (1248 \, c^{8} d^{2} x^{7} + 4368 \, b c^{7} d^{2} x^{6} + 72 \, {\left (79 \, b^{2} c^{6} + 48 \, a c^{7}\right )} d^{2} x^{5} + 60 \, {\left (55 \, b^{3} c^{5} + 144 \, a b c^{6}\right )} d^{2} x^{4} + 4 \, {\left (187 \, b^{4} c^{4} + 1804 \, a b^{2} c^{5} + 712 \, a^{2} c^{6}\right )} d^{2} x^{3} + 6 \, {\left (b^{5} c^{3} + 364 \, a b^{3} c^{4} + 712 \, a^{2} b c^{5}\right )} d^{2} x^{2} - 4 \, {\left (b^{6} c^{2} - 15 \, a b^{4} c^{3} - 486 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d^{2} x + {\left (3 \, b^{7} c - 46 \, a b^{5} c^{2} + 260 \, a^{2} b^{3} c^{3} + 128 \, a^{3} b c^{4}\right )} d^{2}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{2652 \, c^{4}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (4 \, c^{4} d^{2} x^{6} + 12 \, b c^{3} d^{2} x^{5} + {\left (13 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} x^{4} + a^{2} b^{2} d^{2} + 2 \, {\left (3 \, b^{3} c + 8 \, a b c^{2}\right )} d^{2} x^{3} + {\left (b^{4} + 10 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{2} x^{2} + 2 \, {\left (a b^{3} + 2 \, a^{2} b c\right )} d^{2} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}, x\right ) \]