Integrand size = 14, antiderivative size = 90 \[ \int \frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{x^3} \, dx=\frac {3 n^{3/2} \sqrt {\frac {\pi }{2}} \left (a x^n\right )^{2/n} \text {erf}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{16 x^2}-\frac {3 n \sqrt {\log \left (a x^n\right )}}{8 x^2}-\frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{2 x^2} \]
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Time = 0.05 (sec) , antiderivative size = 90, 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 = {2342, 2347, 2211, 2236} \[ \int \frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{x^3} \, dx=\frac {3 \sqrt {\frac {\pi }{2}} n^{3/2} \left (a x^n\right )^{2/n} \text {erf}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{16 x^2}-\frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{2 x^2}-\frac {3 n \sqrt {\log \left (a x^n\right )}}{8 x^2} \]
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Rule 2211
Rule 2236
Rule 2342
Rule 2347
Rubi steps \begin{align*} \text {integral}& = -\frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{2 x^2}+\frac {1}{4} (3 n) \int \frac {\sqrt {\log \left (a x^n\right )}}{x^3} \, dx \\ & = -\frac {3 n \sqrt {\log \left (a x^n\right )}}{8 x^2}-\frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{2 x^2}+\frac {1}{16} \left (3 n^2\right ) \int \frac {1}{x^3 \sqrt {\log \left (a x^n\right )}} \, dx \\ & = -\frac {3 n \sqrt {\log \left (a x^n\right )}}{8 x^2}-\frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{2 x^2}+\frac {\left (3 n \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 )}{16 x^2} \\ & = -\frac {3 n \sqrt {\log \left (a x^n\right )}}{8 x^2}-\frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{2 x^2}+\frac {\left (3 n \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 )}{8 x^2} \\ & = \frac {3 n^{3/2} \sqrt {\frac {\pi }{2}} \left (a x^n\right )^{2/n} \text {erf}\left (\frac {\sqrt {2} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{16 x^2}-\frac {3 n \sqrt {\log \left (a x^n\right )}}{8 x^2}-\frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{2 x^2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.98 \[ \int \frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{x^3} \, dx=-\frac {3 \sqrt {2} n^2 \left (a x^n\right )^{2/n} \Gamma \left (\frac {1}{2},\frac {2 \log \left (a x^n\right )}{n}\right ) \sqrt {\frac {\log \left (a x^n\right )}{n}}+4 \log \left (a x^n\right ) \left (3 n+4 \log \left (a x^n\right )\right )}{32 x^2 \sqrt {\log \left (a x^n\right )}} \]
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\[\int \frac {\ln \left (a \,x^{n}\right )^{\frac {3}{2}}}{x^{3}}d x\]
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Exception generated. \[ \int \frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{x^3} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{x^3} \, dx=\int \frac {\log {\left (a x^{n} \right )}^{\frac {3}{2}}}{x^{3}}\, dx \]
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\[ \int \frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{x^3} \, dx=\int { \frac {\log \left (a x^{n}\right )^{\frac {3}{2}}}{x^{3}} \,d x } \]
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\[ \int \frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{x^3} \, dx=\int { \frac {\log \left (a x^{n}\right )^{\frac {3}{2}}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{x^3} \, dx=\int \frac {{\ln \left (a\,x^n\right )}^{3/2}}{x^3} \,d x \]
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