\[ \int (b d+2 c d x)^{5/2} \sqrt {a+b x+c x^2} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-4 a c +b^{2}\right ) d \left (2 c d x +b d \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}}{45 c}+\frac {\left (2 c d x +b d \right )^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}}{9 c d}-\frac {\left (-4 a c +b^{2}\right )^{\frac {11}{4}} d^{\frac {5}{2}} \EllipticE \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{15 c^{2} \sqrt {c \,x^{2}+b x +a}}+\frac {\left (-4 a c +b^{2}\right )^{\frac {11}{4}} d^{\frac {5}{2}} \EllipticF \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{15 c^{2} \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((2*c*d*x+b*d)^(5/2)*(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
\[ \frac {3 \, \sqrt {2} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {c^{2} d} d^{2} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) + {\left (40 \, c^{4} d^{2} x^{3} + 60 \, b c^{3} d^{2} x^{2} + 2 \, {\left (13 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} x + {\left (3 \, b^{3} c + 8 \, a b c^{2}\right )} d^{2}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{45 \, c^{2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (4 \, c^{2} d^{2} x^{2} + 4 \, b c d^{2} x + b^{2} d^{2}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}, x\right ) \]