Optimal. Leaf size=72 \[ \frac{1}{3} x^3 \sqrt{\log \left (a x^n\right )}-\frac{1}{6} \sqrt{\frac{\pi }{3}} \sqrt{n} x^3 \left (a x^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right ) \]
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Rubi [A] time = 0.0649955, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2305, 2310, 2180, 2204} \[ \frac{1}{3} x^3 \sqrt{\log \left (a x^n\right )}-\frac{1}{6} \sqrt{\frac{\pi }{3}} \sqrt{n} x^3 \left (a x^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right ) \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2310
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int x^2 \sqrt{\log \left (a x^n\right )} \, dx &=\frac{1}{3} x^3 \sqrt{\log \left (a x^n\right )}-\frac{1}{6} n \int \frac{x^2}{\sqrt{\log \left (a x^n\right )}} \, dx\\ &=\frac{1}{3} x^3 \sqrt{\log \left (a x^n\right )}-\frac{1}{6} \left (x^3 \left (a x^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{3 x}{n}}}{\sqrt{x}} \, dx,x,\log \left (a x^n\right )\right )\\ &=\frac{1}{3} x^3 \sqrt{\log \left (a x^n\right )}-\frac{1}{3} \left (x^3 \left (a x^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int e^{\frac{3 x^2}{n}} \, dx,x,\sqrt{\log \left (a x^n\right )}\right )\\ &=-\frac{1}{6} \sqrt{n} \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}{3} x^3 \sqrt{\log \left (a x^n\right )}\\ \end{align*}
Mathematica [A] time = 0.0326051, size = 67, normalized size = 0.93 \[ \frac{1}{18} x^3 \left (6 \sqrt{\log \left (a x^n\right )}-\sqrt{3 \pi } \sqrt{n} \left (a x^n\right )^{-3/n} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.174, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\sqrt{\ln \left ( a{x}^{n} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sqrt{\log \left (a x^{n}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sqrt{\log{\left (a x^{n} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sqrt{\log \left (a x^{n}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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