\[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 d \left (2 c d x +b d \right )^{\frac {5}{2}}}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {20 c \,d^{3} \sqrt {2 c d x +b d}}{3 \sqrt {c \,x^{2}+b x +a}}+\frac {40 c \left (-4 a c +b^{2}\right )^{\frac {1}{4}} d^{\frac {7}{2}} \EllipticF \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{3 \sqrt {c \,x^{2}+b x +a}} \]
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
\[ \frac {2 \, {\left (10 \, \sqrt {2} {\left (c^{2} d^{3} x^{4} + 2 \, b c d^{3} x^{3} + 2 \, a b d^{3} x + {\left (b^{2} + 2 \, a c\right )} d^{3} x^{2} + a^{2} d^{3}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) - {\left (14 \, c^{2} d^{3} x^{2} + 14 \, b c d^{3} x + {\left (b^{2} + 10 \, a c\right )} d^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (8 \, c^{3} d^{3} x^{3} + 12 \, b c^{2} d^{3} x^{2} + 6 \, b^{2} c d^{3} x + b^{3} d^{3}\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x + {\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} x^{2}}, x\right ) \]