Integrand size = 14, antiderivative size = 64 \[ \int x^3 \sqrt {\log \left (a x^n\right )} \, dx=-\frac {1}{16} \sqrt {n} \sqrt {\pi } x^4 \left (a x^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )+\frac {1}{4} x^4 \sqrt {\log \left (a x^n\right )} \]
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Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2342, 2347, 2211, 2235} \[ \int x^3 \sqrt {\log \left (a x^n\right )} \, dx=\frac {1}{4} x^4 \sqrt {\log \left (a x^n\right )}-\frac {1}{16} \sqrt {\pi } \sqrt {n} x^4 \left (a x^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right ) \]
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Rule 2211
Rule 2235
Rule 2342
Rule 2347
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \sqrt {\log \left (a x^n\right )}-\frac {1}{8} n \int \frac {x^3}{\sqrt {\log \left (a x^n\right )}} \, dx \\ & = \frac {1}{4} x^4 \sqrt {\log \left (a x^n\right )}-\frac {1}{8} \left (x^4 \left (a x^n\right )^{-4/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {4 x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right ) \\ & = \frac {1}{4} x^4 \sqrt {\log \left (a x^n\right )}-\frac {1}{4} \left (x^4 \left (a x^n\right )^{-4/n}\right ) \text {Subst}\left (\int e^{\frac {4 x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right ) \\ & = -\frac {1}{16} \sqrt {n} \sqrt {\pi } x^4 \left (a x^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )+\frac {1}{4} x^4 \sqrt {\log \left (a x^n\right )} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95 \[ \int x^3 \sqrt {\log \left (a x^n\right )} \, dx=\frac {1}{16} x^4 \left (-\sqrt {n} \sqrt {\pi } \left (a x^n\right )^{-4/n} \text {erfi}\left (\frac {2 \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )+4 \sqrt {\log \left (a x^n\right )}\right ) \]
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\[\int x^{3} \sqrt {\ln \left (a \,x^{n}\right )}d x\]
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Exception generated. \[ \int x^3 \sqrt {\log \left (a x^n\right )} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x^3 \sqrt {\log \left (a x^n\right )} \, dx=\int x^{3} \sqrt {\log {\left (a x^{n} \right )}}\, dx \]
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\[ \int x^3 \sqrt {\log \left (a x^n\right )} \, dx=\int { x^{3} \sqrt {\log \left (a x^{n}\right )} \,d x } \]
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\[ \int x^3 \sqrt {\log \left (a x^n\right )} \, dx=\int { x^{3} \sqrt {\log \left (a x^{n}\right )} \,d x } \]
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Timed out. \[ \int x^3 \sqrt {\log \left (a x^n\right )} \, dx=\int x^3\,\sqrt {\ln \left (a\,x^n\right )} \,d x \]
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