14.221 Problem number 2010

\[ \int \frac {(d+e x)^{11/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx \]

Optimal antiderivative \[ \frac {7 e \left (-a \,e^{2}+c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 c^{3} d^{3}}+\frac {7 e \left (e x +d \right )^{\frac {5}{2}}}{5 c^{2} d^{2}}-\frac {\left (e x +d \right )^{\frac {7}{2}}}{c d \left (c d x +a e \right )}-\frac {7 e \left (-a \,e^{2}+c \,d^{2}\right )^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {c}\, \sqrt {d}\, \sqrt {e x +d}}{\sqrt {-a \,e^{2}+c \,d^{2}}}\right )}{c^{\frac {9}{2}} d^{\frac {9}{2}}}+\frac {7 e \left (-a \,e^{2}+c \,d^{2}\right )^{2} \sqrt {e x +d}}{c^{4} d^{4}} \]

command

integrate((e*x+d)^(11/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ \frac {7 \, {\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} \arctan \left (\frac {\sqrt {x e + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}} c^{4} d^{4}} - \frac {\sqrt {x e + d} c^{3} d^{6} e - 3 \, \sqrt {x e + d} a c^{2} d^{4} e^{3} + 3 \, \sqrt {x e + d} a^{2} c d^{2} e^{5} - \sqrt {x e + d} a^{3} e^{7}}{{\left ({\left (x e + d\right )} c d - c d^{2} + a e^{2}\right )} c^{4} d^{4}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{8} d^{8} e + 10 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{8} d^{9} e + 45 \, \sqrt {x e + d} c^{8} d^{10} e - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{7} d^{7} e^{3} - 90 \, \sqrt {x e + d} a c^{7} d^{8} e^{3} + 45 \, \sqrt {x e + d} a^{2} c^{6} d^{6} e^{5}\right )}}{15 \, c^{10} d^{10}} \]

Giac 1.7.0 via sagemath 9.3 output

\[ \text {Timed out} \]________________________________________________________________________________________