\[ \int (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{5/2} \, dx \]
Optimal antiderivative \[ -\frac {\left (-4 a c +b^{2}\right ) \left (2 c d x +b d \right )^{\frac {5}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{66 c^{2} d}+\frac {\left (2 c d x +b d \right )^{\frac {5}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{15 c d}+\frac {\left (-4 a c +b^{2}\right )^{2} \left (2 c d x +b d \right )^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}}{308 c^{3} d}-\frac {\left (-4 a c +b^{2}\right )^{3} d \sqrt {2 c d x +b d}\, \sqrt {c \,x^{2}+b x +a}}{462 c^{3}}-\frac {\left (-4 a c +b^{2}\right )^{\frac {17}{4}} d^{\frac {3}{2}} \EllipticF \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{924 c^{4} \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((2*c*d*x+b*d)^(3/2)*(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
\[ -\frac {5 \, \sqrt {2} {\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt {c^{2} d} d {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) - 2 \, {\left (1232 \, c^{8} d x^{6} + 3696 \, b c^{7} d x^{5} + 28 \, {\left (133 \, b^{2} c^{6} + 128 \, a c^{7}\right )} d x^{4} + 56 \, {\left (23 \, b^{3} c^{5} + 128 \, a b c^{6}\right )} d x^{3} + 6 \, {\left (3 \, b^{4} c^{4} + 620 \, a b^{2} c^{5} + 552 \, a^{2} c^{6}\right )} d x^{2} - 2 \, {\left (5 \, b^{5} c^{3} - 68 \, a b^{3} c^{4} - 1656 \, a^{2} b c^{5}\right )} d x + {\left (5 \, b^{6} c^{2} - 70 \, a b^{4} c^{3} + 348 \, a^{2} b^{2} c^{4} + 640 \, a^{3} c^{5}\right )} d\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{9240 \, c^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (2 \, c^{3} d x^{5} + 5 \, b c^{2} d x^{4} + 4 \, {\left (b^{2} c + a c^{2}\right )} d x^{3} + a^{2} b d + {\left (b^{3} + 6 \, a b c\right )} d x^{2} + 2 \, {\left (a b^{2} + a^{2} c\right )} d x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}, x\right ) \]