4.18 Problem number 1127

\[ \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx \]

Optimal antiderivative \[ \frac {2 \left (-a d +b c \right ) \left (e x \right )^{\frac {5}{2}}}{5 a b e \left (b \,x^{2}+a \right )^{\frac {5}{4}}}+\frac {d \,e^{\frac {3}{2}} \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} \sqrt {e}}\right )}{b^{\frac {9}{4}}}+\frac {d \,e^{\frac {3}{2}} \arctanh \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} \sqrt {e}}\right )}{b^{\frac {9}{4}}}-\frac {2 d e \sqrt {e x}}{b^{2} \left (b \,x^{2}+a \right )^{\frac {1}{4}}} \]

command

integrate((e*x)^(3/2)*(d*x^2+c)/(b*x^2+a)^(9/4),x, algorithm="maxima")

Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output

\[ -\frac {1}{10} \, {\left (d {\left (\frac {4 \, {\left (b + \frac {5 \, {\left (b x^{2} + a\right )}}{x^{2}}\right )} x^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} b^{2}} + \frac {5 \, {\left (\frac {2 \, \arctan \left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} \sqrt {x}}\right )}{b^{\frac {1}{4}}} + \frac {\log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}{b^{\frac {1}{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}\right )}{b^{\frac {1}{4}}}\right )}}{b^{2}}\right )} - \frac {4 \, c x^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} a}\right )} e^{\frac {3}{2}} \]

Maxima 5.44 via sagemath 9.3 output

\[ \int \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {9}{4}}}\,{d x} \]________________________________________________________________________________________