\[ \int \frac {(d x)^{5/2}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx \]
Optimal antiderivative \[ \frac {7 \left (d x \right )^{\frac {7}{2}} \expIntegral \left (\frac {\frac {7 a}{2}+\frac {7 b \ln \left (c \,x^{n}\right )}{2}}{b n}\right ) {\mathrm e}^{-\frac {7 a}{2 b n}} \left (c \,x^{n}\right )^{-\frac {7}{2 n}}}{2 b^{2} d \,n^{2}}-\frac {\left (d x \right )^{\frac {7}{2}}}{b d n \left (a +b \ln \left (c \,x^{n}\right )\right )} \]
command
int((d*x)^(5/2)/(a+b*ln(c*x^n))^2,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(-\frac {2 x^{4} d^{3}}{b n \sqrt {d x}\, \left (2 a +2 b \ln \left (c \right )+2 b \ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )+i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+i b \pi \,\mathrm {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-i b \pi \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}\right )}-\frac {7 \,{\mathrm e}^{\frac {7 i \left (b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )-b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-b \pi \,\mathrm {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+b \pi \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}+2 i b n \left (\ln \left (x \right )-\ln \left (d x \right )\right )+2 i b \ln \left (c \right )+2 i b \left (\ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-n \ln \left (x \right )\right )+2 i a \right )}{4 b n}} \expIntegral \left (1, -\frac {7 \ln \left (d x \right )}{2}+\frac {7 i \left (b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )-b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-b \pi \,\mathrm {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+b \pi \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}+2 i b n \left (\ln \left (x \right )-\ln \left (d x \right )\right )+2 i b \ln \left (c \right )+2 i b \left (\ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-n \ln \left (x \right )\right )+2 i a \right )}{4 b n}\right )}{2 d \,b^{2} n^{2}}\) | \(432\) |
Maple 2021.1 output
\[ \int \frac {\left (d x \right )^{\frac {5}{2}}}{\left (b \ln \left (c \,x^{n}\right )+a \right )^{2}}\, dx \]________________________________________________________________________________________