\[ \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{5/4}} \, dx \]
Optimal antiderivative \[ \frac {d \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} \sqrt {e}}\right )}{b^{\frac {5}{4}} \sqrt {e}}+\frac {d \arctanh \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} \sqrt {e}}\right )}{b^{\frac {5}{4}} \sqrt {e}}+\frac {2 \left (-a d +b c \right ) \sqrt {e x}}{a b e \left (b \,x^{2}+a \right )^{\frac {1}{4}}} \]
command
integrate((d*x^2+c)/(e*x)^(1/2)/(b*x^2+a)^(5/4),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ -\frac {1}{2} \, {\left (d {\left (\frac {\frac {2 \, \arctan \left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} \sqrt {x}}\right )}{b^{\frac {1}{4}}} + \frac {\log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}{b^{\frac {1}{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}\right )}{b^{\frac {1}{4}}}}{b} + \frac {4 \, \sqrt {x}}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} b}\right )} - \frac {4 \, c \sqrt {x}}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} a}\right )} e^{\left (-\frac {1}{2}\right )} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} \sqrt {e x}}\,{d x} \]________________________________________________________________________________________