\[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx \]
Optimal antiderivative \[ -\frac {d \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} \sqrt {e}}\right )}{b^{\frac {3}{4}} e^{\frac {3}{2}}}+\frac {d \arctanh \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} \sqrt {e}}\right )}{b^{\frac {3}{4}} e^{\frac {3}{2}}}-\frac {2 c \left (b \,x^{2}+a \right )^{\frac {1}{4}}}{a e \sqrt {e x}} \]
command
integrate((d*x^2+c)/(e*x)^(3/2)/(b*x^2+a)^(3/4),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {1}{2} \, {\left (d {\left (\frac {2 \, \arctan \left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} \sqrt {x}}\right )}{b^{\frac {3}{4}}} - \frac {\log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}{b^{\frac {1}{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}\right )}{b^{\frac {3}{4}}}\right )} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}} c}{a \sqrt {x}}\right )} e^{\left (-\frac {3}{2}\right )} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {3}{2}}}\,{d x} \]________________________________________________________________________________________