\[ \int (d x)^{5/2} \sqrt {a+b x+c x^2} \, dx \]
Optimal antiderivative \[ \frac {2 d \left (d x \right )^{\frac {3}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{9 c}-\frac {4 b \,d^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \sqrt {d x}}{21 c^{2}}-\frac {4 \left (21 a^{2} c^{2}-36 a \,b^{2} c +8 b^{4}\right ) d^{3} x \sqrt {c \,x^{2}+b x +a}}{315 c^{\frac {7}{2}} \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {d x}}+\frac {2 d^{2} \left (b \left (3 a c +8 b^{2}\right )+3 c \left (-7 a c +8 b^{2}\right ) x \right ) \sqrt {d x}\, \sqrt {c \,x^{2}+b x +a}}{315 c^{3}}+\frac {4 a^{\frac {1}{4}} \left (21 a^{2} c^{2}-36 a \,b^{2} c +8 b^{4}\right ) d^{3} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2-\frac {b}{\sqrt {a}\, \sqrt {c}}}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+b x +a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{315 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {15}{4}} \sqrt {d x}\, \sqrt {c \,x^{2}+b x +a}}-\frac {a^{\frac {1}{4}} d^{3} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2-\frac {b}{\sqrt {a}\, \sqrt {c}}}}{2}\right ) \left (\sqrt {a}+x \sqrt {c}\right ) \left (16 b^{4}-72 a \,b^{2} c +42 a^{2} c^{2}+b \left (-27 a c +8 b^{2}\right ) \sqrt {a}\, \sqrt {c}\right ) \sqrt {x}\, \sqrt {\frac {c \,x^{2}+b x +a}{\left (\sqrt {a}+x \sqrt {c}\right )^{2}}}}{315 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} \sqrt {x}}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {15}{4}} \sqrt {d x}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((d*x)^(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
\[ \frac {2 \, {\left ({\left (16 \, b^{5} - 96 \, a b^{3} c + 123 \, a^{2} b c^{2}\right )} \sqrt {c d} d^{2} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 6 \, {\left (8 \, b^{4} c - 36 \, a b^{2} c^{2} + 21 \, a^{2} c^{3}\right )} \sqrt {c d} d^{2} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) + 3 \, {\left (35 \, c^{5} d^{2} x^{3} + 5 \, b c^{4} d^{2} x^{2} - 2 \, {\left (3 \, b^{2} c^{3} - 7 \, a c^{4}\right )} d^{2} x + {\left (8 \, b^{3} c^{2} - 27 \, a b c^{3}\right )} d^{2}\right )} \sqrt {c x^{2} + b x + a} \sqrt {d x}\right )}}{945 \, c^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {c x^{2} + b x + a} \sqrt {d x} d^{2} x^{2}, x\right ) \]