Integrand size = 14, antiderivative size = 93 \[ \int \frac {1}{x^3 \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\frac {8 \sqrt {2 \pi } \left (a x^n\right )^{2/n} \text {erf}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2} x^2}-\frac {2}{3 n x^2 \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {8}{3 n^2 x^2 \sqrt {\log \left (a x^n\right )}} \]
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Time = 0.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2343, 2347, 2211, 2236} \[ \int \frac {1}{x^3 \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\frac {8 \sqrt {2 \pi } \left (a x^n\right )^{2/n} \text {erf}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2} x^2}+\frac {8}{3 n^2 x^2 \sqrt {\log \left (a x^n\right )}}-\frac {2}{3 n x^2 \log ^{\frac {3}{2}}\left (a x^n\right )} \]
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Rule 2211
Rule 2236
Rule 2343
Rule 2347
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3 n x^2 \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 \int \frac {1}{x^3 \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx}{3 n} \\ & = -\frac {2}{3 n x^2 \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {8}{3 n^2 x^2 \sqrt {\log \left (a x^n\right )}}+\frac {16 \int \frac {1}{x^3 \sqrt {\log \left (a x^n\right )}} \, dx}{3 n^2} \\ & = -\frac {2}{3 n x^2 \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {8}{3 n^2 x^2 \sqrt {\log \left (a x^n\right )}}+\frac {\left (16 \left (a x^n\right )^{2/n}\right ) \text {Subst}\left (\int \frac {e^{-\frac {2 x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right )}{3 n^3 x^2} \\ & = -\frac {2}{3 n x^2 \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {8}{3 n^2 x^2 \sqrt {\log \left (a x^n\right )}}+\frac {\left (32 \left (a x^n\right )^{2/n}\right ) \text {Subst}\left (\int e^{-\frac {2 x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right )}{3 n^3 x^2} \\ & = \frac {8 \sqrt {2 \pi } \left (a x^n\right )^{2/n} \text {erf}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2} x^2}-\frac {2}{3 n x^2 \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {8}{3 n^2 x^2 \sqrt {\log \left (a x^n\right )}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^3 \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=-\frac {2 \left (n-4 \log \left (a x^n\right )+4 \sqrt {2} n \left (a x^n\right )^{2/n} \Gamma \left (\frac {1}{2},\frac {2 \log \left (a x^n\right )}{n}\right ) \left (\frac {\log \left (a x^n\right )}{n}\right )^{3/2}\right )}{3 n^2 x^2 \log ^{\frac {3}{2}}\left (a x^n\right )} \]
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\[\int \frac {1}{x^{3} \ln \left (a \,x^{n}\right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{x^3 \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{x^3 \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int \frac {1}{x^{3} \log {\left (a x^{n} \right )}^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{x^3 \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int { \frac {1}{x^{3} \log \left (a x^{n}\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{x^3 \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int { \frac {1}{x^{3} \log \left (a x^{n}\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^3 \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int \frac {1}{x^3\,{\ln \left (a\,x^n\right )}^{5/2}} \,d x \]
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