Optimal. Leaf size=86 \[ \frac{x^{m+1} \sqrt{\log \left (a x^n\right )}}{m+1}-\frac{\sqrt{\pi } \sqrt{n} x^{m+1} \left (a x^n\right )^{-\frac{m+1}{n}} \text{Erfi}\left (\frac{\sqrt{m+1} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{2 (m+1)^{3/2}} \]
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Rubi [A] time = 0.0665412, antiderivative size = 86, 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} \[ \frac{x^{m+1} \sqrt{\log \left (a x^n\right )}}{m+1}-\frac{\sqrt{\pi } \sqrt{n} x^{m+1} \left (a x^n\right )^{-\frac{m+1}{n}} \text{Erfi}\left (\frac{\sqrt{m+1} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{2 (m+1)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2310
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int x^m \sqrt{\log \left (a x^n\right )} \, dx &=\frac{x^{1+m} \sqrt{\log \left (a x^n\right )}}{1+m}-\frac{n \int \frac{x^m}{\sqrt{\log \left (a x^n\right )}} \, dx}{2 (1+m)}\\ &=\frac{x^{1+m} \sqrt{\log \left (a x^n\right )}}{1+m}-\frac{\left (x^{1+m} \left (a x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{(1+m) x}{n}}}{\sqrt{x}} \, dx,x,\log \left (a x^n\right )\right )}{2 (1+m)}\\ &=\frac{x^{1+m} \sqrt{\log \left (a x^n\right )}}{1+m}-\frac{\left (x^{1+m} \left (a x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int e^{\frac{(1+m) x^2}{n}} \, dx,x,\sqrt{\log \left (a x^n\right )}\right )}{1+m}\\ &=-\frac{\sqrt{n} \sqrt{\pi } x^{1+m} \left (a x^n\right )^{-\frac{1+m}{n}} \text{erfi}\left (\frac{\sqrt{1+m} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{2 (1+m)^{3/2}}+\frac{x^{1+m} \sqrt{\log \left (a x^n\right )}}{1+m}\\ \end{align*}
Mathematica [A] time = 0.0513356, size = 86, normalized size = 1. \[ \frac{x^{m+1} \sqrt{\log \left (a x^n\right )}}{m+1}-\frac{\sqrt{\pi } \sqrt{n} x^{m+1} \left (a x^n\right )^{-\frac{m+1}{n}} \text{Erfi}\left (\frac{\sqrt{m+1} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{2 (m+1)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.178, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\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^{m} \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} \sqrt{\log \left (a x^{n}\right )}, x\right ) \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^{m} \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^{m} \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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