Integrand size = 14, antiderivative size = 89 \[ \int \frac {x^2}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\frac {4 \sqrt {3 \pi } x^3 \left (a x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{5/2}}-\frac {2 x^3}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 x^3}{n^2 \sqrt {\log \left (a x^n\right )}} \]
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Time = 0.05 (sec) , antiderivative size = 89, 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, 2235} \[ \int \frac {x^2}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\frac {4 \sqrt {3 \pi } x^3 \left (a x^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{5/2}}-\frac {4 x^3}{n^2 \sqrt {\log \left (a x^n\right )}}-\frac {2 x^3}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )} \]
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Rule 2211
Rule 2235
Rule 2343
Rule 2347
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^3}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}+\frac {2 \int \frac {x^2}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx}{n} \\ & = -\frac {2 x^3}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 x^3}{n^2 \sqrt {\log \left (a x^n\right )}}+\frac {12 \int \frac {x^2}{\sqrt {\log \left (a x^n\right )}} \, dx}{n^2} \\ & = -\frac {2 x^3}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 x^3}{n^2 \sqrt {\log \left (a x^n\right )}}+\frac {\left (12 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 )}{n^3} \\ & = -\frac {2 x^3}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 x^3}{n^2 \sqrt {\log \left (a x^n\right )}}+\frac {\left (24 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 )}{n^3} \\ & = \frac {4 \sqrt {3 \pi } 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 )}{n^{5/2}}-\frac {2 x^3}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )}-\frac {4 x^3}{n^2 \sqrt {\log \left (a x^n\right )}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.03 \[ \int \frac {x^2}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=-\frac {2 x^3 \left (a x^n\right )^{-3/n} \left (6 \sqrt {3} n \Gamma \left (\frac {1}{2},-\frac {3 \log \left (a x^n\right )}{n}\right ) \left (-\frac {\log \left (a x^n\right )}{n}\right )^{3/2}+\left (a x^n\right )^{3/n} \left (n+6 \log \left (a x^n\right )\right )\right )}{3 n^2 \log ^{\frac {3}{2}}\left (a x^n\right )} \]
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\[\int \frac {x^{2}}{\ln \left (a \,x^{n}\right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {x^2}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^2}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int \frac {x^{2}}{\log {\left (a x^{n} \right )}^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {x^2}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int { \frac {x^{2}}{\log \left (a x^{n}\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x^2}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int { \frac {x^{2}}{\log \left (a x^{n}\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\int \frac {x^2}{{\ln \left (a\,x^n\right )}^{5/2}} \,d x \]
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