9.6 Problem number 64

\[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2}} \, dx \]

Optimal antiderivative \[ -\frac {2 \left (A \,b^{2}-a \left (b B -a C \right )\right ) \left (d x +c \right )^{\frac {3}{2}} \left (f x +e \right )^{\frac {3}{2}}}{3 b \left (-a d +b c \right ) \left (-a f +b e \right ) \left (b x +a \right )^{\frac {3}{2}}}-\frac {2 \left (b B -2 a C \right ) \left (f x +e \right )^{\frac {3}{2}} \sqrt {d x +c}}{b^{2} \left (-a f +b e \right ) \sqrt {b x +a}}+\frac {2 \left (8 a^{2} C d f +b^{2} \left (A d f +3 B c f +c C e \right )-a b \left (4 B d f +7 c C f +C d e \right )\right ) \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}}{3 b^{3} \left (-a d +b c \right ) \left (-a f +b e \right )}+\frac {2 \left (16 a^{3} C \,d^{2} f^{2}-8 a^{2} b d f \left (B d f +2 C \left (c f +d e \right )\right )-b^{3} \left (c^{2} C e f +A \,d^{2} e f +c d \left (A \,f^{2}+6 B e f +C \,e^{2}\right )\right )+a \,b^{2} \left (d f \left (2 A d f +7 B c f +7 B d e \right )+C \left (c^{2} f^{2}+16 c d e f +d^{2} e^{2}\right )\right )\right ) \EllipticE \left (\frac {\sqrt {d}\, \sqrt {b x +a}}{\sqrt {a d -b c}}, \sqrt {\frac {\left (-a d +b c \right ) f}{d \left (-a f +b e \right )}}\right ) \sqrt {\frac {b \left (d x +c \right )}{-a d +b c}}\, \sqrt {f x +e}}{3 b^{4} f \left (-a f +b e \right ) \sqrt {d}\, \sqrt {a d -b c}\, \sqrt {d x +c}\, \sqrt {\frac {b \left (f x +e \right )}{-a f +b e}}}+\frac {2 \left (-c f +d e \right ) \left (8 a^{2} C d f +b^{2} \left (A d f +3 B c f +c C e \right )-a b \left (4 B d f +7 c C f +C d e \right )\right ) \EllipticF \left (\frac {\sqrt {d}\, \sqrt {b x +a}}{\sqrt {a d -b c}}, \sqrt {\frac {\left (-a d +b c \right ) f}{d \left (-a f +b e \right )}}\right ) \sqrt {\frac {b \left (d x +c \right )}{-a d +b c}}\, \sqrt {\frac {b \left (f x +e \right )}{-a f +b e}}}{3 b^{4} f \sqrt {d}\, \sqrt {a d -b c}\, \sqrt {d x +c}\, \sqrt {f x +e}} \]

command

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(b*x+a)^(5/2),x, algorithm="fricas")

Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output

\[ \text {output too large to display} \]

Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}, x\right ) \]