13.4 Problem number 20

\[ \int \frac {1}{\left (3 i x+4 x^2\right )^{7/2}} \, dx \]

Optimal antiderivative \[ \frac {\frac {2 i}{15}+\frac {16 x}{45}}{\left (4 x^{2}+3 i x \right )^{\frac {5}{2}}}+\frac {\frac {128 i}{405}+\frac {1024 x}{1215}}{\left (4 x^{2}+3 i x \right )^{\frac {3}{2}}}+\frac {\frac {4096 i}{3645}+\frac {32768 x}{10935}}{\sqrt {4 x^{2}+3 i x}} \]

command

integrate(1/(3*I*x+4*x^2)^(7/2),x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ \frac {{\left (8 \, {\left (32 \, {\left (8 \, {\left (16 \, {\left (8 \, x + 15 i\right )} x - 135\right )} x - 135 i\right )} x - 405\right )} x + 729 i\right )} \sqrt {8 \, x^{2} + 2 \, \sqrt {16 \, x^{2} + 9} x} {\left (\frac {3 i \, x}{4 \, x^{2} + \sqrt {16 \, x^{4} + 9 \, x^{2}}} + 1\right )}}{10935 \, {\left (4 \, x^{2} + 3 i \, x\right )}^{3}} \]

Giac 1.7.0 via sagemath 9.3 output

\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________