Integrand size = 14, antiderivative size = 84 \[ \int \frac {1}{x^2 \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\frac {4 \sqrt {\pi } \left (a x^n\right )^{\frac {1}{n}} \text {erf}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2} x}-\frac {2}{3 n x \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {4}{3 n^2 x \sqrt {\log \left (a x^n\right )}} \]
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Time = 0.05 (sec) , antiderivative size = 84, 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} \[ \int \frac {1}{x^2 \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\frac {4 \sqrt {\pi } \left (a x^n\right )^{\frac {1}{n}} \text {erf}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2} x}+\frac {4}{3 n^2 x \sqrt {\log \left (a x^n\right )}}-\frac {2}{3 n x \log ^{\frac {3}{2}}\left (a x^n\right )} \]
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Rule 2211
Rule 2236
Rule 2343
Rule 2347
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3 n x \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {2 \int \frac {1}{x^2 \log ^{\frac {3}{2}}\left (a x^n\right )} \, dx}{3 n} \\ & = -\frac {2}{3 n x \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {4}{3 n^2 x \sqrt {\log \left (a x^n\right )}}+\frac {4 \int \frac {1}{x^2 \sqrt {\log \left (a x^n\right )}} \, dx}{3 n^2} \\ & = -\frac {2}{3 n x \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {4}{3 n^2 x \sqrt {\log \left (a x^n\right )}}+\frac {\left (4 \left (a x^n\right )^{\frac {1}{n}}\right ) \text {Subst}\left (\int \frac {e^{-\frac {x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right )}{3 n^3 x} \\ & = -\frac {2}{3 n x \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {4}{3 n^2 x \sqrt {\log \left (a x^n\right )}}+\frac {\left (8 \left (a x^n\right )^{\frac {1}{n}}\right ) \text {Subst}\left (\int e^{-\frac {x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right )}{3 n^3 x} \\ & = \frac {4 \sqrt {\pi } \left (a x^n\right )^{\frac {1}{n}} \text {erf}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{3 n^{5/2} x}-\frac {2}{3 n x \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {4}{3 n^2 x \sqrt {\log \left (a x^n\right )}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^2 \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=-\frac {2 \left (n-2 \log \left (a x^n\right )+2 n \left (a x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{2},\frac {\log \left (a x^n\right )}{n}\right ) \left (\frac {\log \left (a x^n\right )}{n}\right )^{3/2}\right )}{3 n^2 x \log ^{\frac {3}{2}}\left (a x^n\right )} \]
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\[\int \frac {1}{x^{2} \ln \left (a \,x^{n}\right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{x^2 \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{x^2 \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int \frac {1}{x^{2} \log {\left (a x^{n} \right )}^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{x^2 \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int { \frac {1}{x^{2} \log \left (a x^{n}\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{x^2 \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int { \frac {1}{x^{2} \log \left (a x^{n}\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^2 \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int \frac {1}{x^2\,{\ln \left (a\,x^n\right )}^{5/2}} \,d x \]
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