Integrand size = 14, antiderivative size = 90 \[ \int x^2 \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\frac {1}{12} n^{3/2} \sqrt {\frac {\pi }{3}} x^3 \left (a x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )-\frac {1}{6} n x^3 \sqrt {\log \left (a x^n\right )}+\frac {1}{3} x^3 \log ^{\frac {3}{2}}\left (a x^n\right ) \]
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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, 2235} \[ \int x^2 \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\frac {1}{12} \sqrt {\frac {\pi }{3}} n^{3/2} x^3 \left (a x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )+\frac {1}{3} x^3 \log ^{\frac {3}{2}}\left (a x^n\right )-\frac {1}{6} n x^3 \sqrt {\log \left (a x^n\right )} \]
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Rule 2211
Rule 2235
Rule 2342
Rule 2347
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \log ^{\frac {3}{2}}\left (a x^n\right )-\frac {1}{2} n \int x^2 \sqrt {\log \left (a x^n\right )} \, dx \\ & = -\frac {1}{6} n x^3 \sqrt {\log \left (a x^n\right )}+\frac {1}{3} x^3 \log ^{\frac {3}{2}}\left (a x^n\right )+\frac {1}{12} n^2 \int \frac {x^2}{\sqrt {\log \left (a x^n\right )}} \, dx \\ & = -\frac {1}{6} n x^3 \sqrt {\log \left (a x^n\right )}+\frac {1}{3} x^3 \log ^{\frac {3}{2}}\left (a x^n\right )+\frac {1}{12} \left (n x^3 \left (a x^n\right )^{-3/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {3 x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right ) \\ & = -\frac {1}{6} n x^3 \sqrt {\log \left (a x^n\right )}+\frac {1}{3} x^3 \log ^{\frac {3}{2}}\left (a x^n\right )+\frac {1}{6} \left (n x^3 \left (a x^n\right )^{-3/n}\right ) \text {Subst}\left (\int e^{\frac {3 x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right ) \\ & = \frac {1}{12} n^{3/2} \sqrt {\frac {\pi }{3}} x^3 \left (a x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )-\frac {1}{6} n x^3 \sqrt {\log \left (a x^n\right )}+\frac {1}{3} x^3 \log ^{\frac {3}{2}}\left (a x^n\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84 \[ \int x^2 \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\frac {1}{36} x^3 \left (n^{3/2} \sqrt {3 \pi } \left (a x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )-6 \left (n-2 \log \left (a x^n\right )\right ) \sqrt {\log \left (a x^n\right )}\right ) \]
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\[\int x^{2} \ln \left (a \,x^{n}\right )^{\frac {3}{2}}d x\]
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Exception generated. \[ \int x^2 \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\text {Exception raised: TypeError} \]
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\[ \int x^2 \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\int x^{2} \log {\left (a x^{n} \right )}^{\frac {3}{2}}\, dx \]
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\[ \int x^2 \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\int { x^{2} \log \left (a x^{n}\right )^{\frac {3}{2}} \,d x } \]
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\[ \int x^2 \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\int { x^{2} \log \left (a x^{n}\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int x^2 \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx=\int x^2\,{\ln \left (a\,x^n\right )}^{3/2} \,d x \]
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