\[ \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {\left (2 c d x +b d \right )^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{13 c d}+\frac {\left (-4 a c +b^{2}\right )^{2} d \left (2 c d x +b d \right )^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}}{195 c^{2}}-\frac {\left (-4 a c +b^{2}\right ) \left (2 c d x +b d \right )^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}}{78 c^{2} d}+\frac {\left (-4 a c +b^{2}\right )^{\frac {15}{4}} d^{\frac {5}{2}} \EllipticE \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{130 c^{3} \sqrt {c \,x^{2}+b x +a}}-\frac {\left (-4 a c +b^{2}\right )^{\frac {15}{4}} d^{\frac {5}{2}} \EllipticF \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{130 c^{3} \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((2*c*d*x+b*d)^(5/2)*(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
\[ -\frac {3 \, \sqrt {2} {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c^{2} d} d^{2} {\rm weierstrassZeta}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )\right ) - {\left (240 \, c^{6} d^{2} x^{5} + 600 \, b c^{5} d^{2} x^{4} + 100 \, {\left (5 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{2} x^{3} + 150 \, {\left (b^{3} c^{3} + 4 \, a b c^{4}\right )} d^{2} x^{2} + 4 \, {\left (b^{4} c^{2} + 67 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{2} x - {\left (3 \, b^{5} c - 34 \, a b^{3} c^{2} - 32 \, a^{2} b c^{3}\right )} d^{2}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{390 \, c^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (4 \, c^{3} d^{2} x^{4} + 8 \, b c^{2} d^{2} x^{3} + a b^{2} d^{2} + {\left (5 \, b^{2} c + 4 \, a c^{2}\right )} d^{2} x^{2} + {\left (b^{3} + 4 \, a b c\right )} d^{2} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}, x\right ) \]