14.205 Problem number 1937

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^2} \, dx \]

Optimal antiderivative \[ \frac {5 \left (a -\frac {c \,d^{2}}{e^{2}}\right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{24}+\frac {\left (c d x +a e \right ) \left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )^{\frac {3}{2}}}{4 e}-\frac {5 \left (-a \,e^{2}+c \,d^{2}\right )^{4} \arctanh \left (\frac {2 c d e x +a \,e^{2}+c \,d^{2}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}\right )}{128 c^{\frac {3}{2}} d^{\frac {3}{2}} e^{\frac {7}{2}}}+\frac {5 \left (-a \,e^{2}+c \,d^{2}\right )^{2} \left (2 c d e x +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}}}{64 c d \,e^{3}} \]

command

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^2,x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

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

Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________