Optimal. Leaf size=72 \[ \frac{3}{4} \sqrt{\pi } n^{3/2} x \left (a x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )+x \log ^{\frac{3}{2}}\left (a x^n\right )-\frac{3}{2} n x \sqrt{\log \left (a x^n\right )} \]
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Rubi [A] time = 0.0347186, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {2296, 2300, 2180, 2204} \[ \frac{3}{4} \sqrt{\pi } n^{3/2} x \left (a x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )+x \log ^{\frac{3}{2}}\left (a x^n\right )-\frac{3}{2} n x \sqrt{\log \left (a x^n\right )} \]
Antiderivative was successfully verified.
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Rule 2296
Rule 2300
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \log ^{\frac{3}{2}}\left (a x^n\right ) \, dx &=x \log ^{\frac{3}{2}}\left (a x^n\right )-\frac{1}{2} (3 n) \int \sqrt{\log \left (a x^n\right )} \, dx\\ &=-\frac{3}{2} n x \sqrt{\log \left (a x^n\right )}+x \log ^{\frac{3}{2}}\left (a x^n\right )+\frac{1}{4} \left (3 n^2\right ) \int \frac{1}{\sqrt{\log \left (a x^n\right )}} \, dx\\ &=-\frac{3}{2} n x \sqrt{\log \left (a x^n\right )}+x \log ^{\frac{3}{2}}\left (a x^n\right )+\frac{1}{4} \left (3 n x \left (a x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{\sqrt{x}} \, dx,x,\log \left (a x^n\right )\right )\\ &=-\frac{3}{2} n x \sqrt{\log \left (a x^n\right )}+x \log ^{\frac{3}{2}}\left (a x^n\right )+\frac{1}{2} \left (3 n x \left (a x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int e^{\frac{x^2}{n}} \, dx,x,\sqrt{\log \left (a x^n\right )}\right )\\ &=\frac{3}{4} n^{3/2} \sqrt{\pi } x \left (a x^n\right )^{-1/n} \text{erfi}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )-\frac{3}{2} n x \sqrt{\log \left (a x^n\right )}+x \log ^{\frac{3}{2}}\left (a x^n\right )\\ \end{align*}
Mathematica [A] time = 0.0380405, size = 72, normalized size = 1. \[ \frac{3}{4} \sqrt{\pi } n^{3/2} x \left (a x^n\right )^{-1/n} \text{Erfi}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )+x \log ^{\frac{3}{2}}\left (a x^n\right )-\frac{3}{2} n x \sqrt{\log \left (a x^n\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.17, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ( a{x}^{n} \right ) \right ) ^{{\frac{3}{2}}}\, 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 \log \left (a x^{n}\right )^{\frac{3}{2}}\,{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 \log{\left (a x^{n} \right )}^{\frac{3}{2}}\, 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 \log \left (a x^{n}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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