Integrand size = 10, antiderivative size = 72 \[ \int \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\frac {3}{4} n^{3/2} \sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )-\frac {3}{2} n x \sqrt {\log \left (a x^n\right )}+x \log ^{\frac {3}{2}}\left (a x^n\right ) \]
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Time = 0.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2333, 2337, 2211, 2235} \[ \int \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\frac {3}{4} \sqrt {\pi } n^{3/2} x \left (a x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )+x \log ^{\frac {3}{2}}\left (a x^n\right )-\frac {3}{2} n x \sqrt {\log \left (a x^n\right )} \]
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Rule 2211
Rule 2235
Rule 2333
Rule 2337
Rubi steps \begin{align*} \text {integral}& = x \log ^{\frac {3}{2}}\left (a x^n\right )-\frac {1}{2} (3 n) \int \sqrt {\log \left (a x^n\right )} \, dx \\ & = -\frac {3}{2} n x \sqrt {\log \left (a x^n\right )}+x \log ^{\frac {3}{2}}\left (a x^n\right )+\frac {1}{4} \left (3 n^2\right ) \int \frac {1}{\sqrt {\log \left (a x^n\right )}} \, dx \\ & = -\frac {3}{2} n x \sqrt {\log \left (a x^n\right )}+x \log ^{\frac {3}{2}}\left (a x^n\right )+\frac {1}{4} \left (3 n x \left (a x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right ) \\ & = -\frac {3}{2} n x \sqrt {\log \left (a x^n\right )}+x \log ^{\frac {3}{2}}\left (a x^n\right )+\frac {1}{2} \left (3 n x \left (a x^n\right )^{-1/n}\right ) \text {Subst}\left (\int e^{\frac {x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right ) \\ & = \frac {3}{4} n^{3/2} \sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )-\frac {3}{2} n x \sqrt {\log \left (a x^n\right )}+x \log ^{\frac {3}{2}}\left (a x^n\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00 \[ \int \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\frac {3}{4} n^{3/2} \sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )-\frac {3}{2} n x \sqrt {\log \left (a x^n\right )}+x \log ^{\frac {3}{2}}\left (a x^n\right ) \]
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\[\int \ln \left (a \,x^{n}\right )^{\frac {3}{2}}d x\]
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Exception generated. \[ \int \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\text {Exception raised: TypeError} \]
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\[ \int \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\int \log {\left (a x^{n} \right )}^{\frac {3}{2}}\, dx \]
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\[ \int \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\int { \log \left (a x^{n}\right )^{\frac {3}{2}} \,d x } \]
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\[ \int \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\int { \log \left (a x^{n}\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\int {\ln \left (a\,x^n\right )}^{3/2} \,d x \]
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