Optimal. Leaf size=77 \[ \frac{3 \sqrt{\pi } n^{3/2} \left (a x^n\right )^{\frac{1}{n}} \text{Erf}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{4 x}-\frac{\log ^{\frac{3}{2}}\left (a x^n\right )}{x}-\frac{3 n \sqrt{\log \left (a x^n\right )}}{2 x} \]
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Rubi [A] time = 0.0665744, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2305, 2310, 2180, 2205} \[ \frac{3 \sqrt{\pi } n^{3/2} \left (a x^n\right )^{\frac{1}{n}} \text{Erf}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{4 x}-\frac{\log ^{\frac{3}{2}}\left (a x^n\right )}{x}-\frac{3 n \sqrt{\log \left (a x^n\right )}}{2 x} \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2310
Rule 2180
Rule 2205
Rubi steps
\begin{align*} \int \frac{\log ^{\frac{3}{2}}\left (a x^n\right )}{x^2} \, dx &=-\frac{\log ^{\frac{3}{2}}\left (a x^n\right )}{x}+\frac{1}{2} (3 n) \int \frac{\sqrt{\log \left (a x^n\right )}}{x^2} \, dx\\ &=-\frac{3 n \sqrt{\log \left (a x^n\right )}}{2 x}-\frac{\log ^{\frac{3}{2}}\left (a x^n\right )}{x}+\frac{1}{4} \left (3 n^2\right ) \int \frac{1}{x^2 \sqrt{\log \left (a x^n\right )}} \, dx\\ &=-\frac{3 n \sqrt{\log \left (a x^n\right )}}{2 x}-\frac{\log ^{\frac{3}{2}}\left (a x^n\right )}{x}+\frac{\left (3 n \left (a x^n\right )^{\frac{1}{n}}\right ) \operatorname{Subst}\left (\int \frac{e^{-\frac{x}{n}}}{\sqrt{x}} \, dx,x,\log \left (a x^n\right )\right )}{4 x}\\ &=-\frac{3 n \sqrt{\log \left (a x^n\right )}}{2 x}-\frac{\log ^{\frac{3}{2}}\left (a x^n\right )}{x}+\frac{\left (3 n \left (a x^n\right )^{\frac{1}{n}}\right ) \operatorname{Subst}\left (\int e^{-\frac{x^2}{n}} \, dx,x,\sqrt{\log \left (a x^n\right )}\right )}{2 x}\\ &=\frac{3 n^{3/2} \sqrt{\pi } \left (a x^n\right )^{\frac{1}{n}} \text{erf}\left (\frac{\sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{4 x}-\frac{3 n \sqrt{\log \left (a x^n\right )}}{2 x}-\frac{\log ^{\frac{3}{2}}\left (a x^n\right )}{x}\\ \end{align*}
Mathematica [A] time = 0.0562425, size = 79, normalized size = 1.03 \[ -\frac{3 n^2 \left (a x^n\right )^{\frac{1}{n}} \sqrt{\frac{\log \left (a x^n\right )}{n}} \text{Gamma}\left (\frac{1}{2},\frac{\log \left (a x^n\right )}{n}\right )+4 \log ^2\left (a x^n\right )+6 n \log \left (a x^n\right )}{4 x \sqrt{\log \left (a x^n\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.169, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \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 \frac{\log \left (a x^{n}\right )^{\frac{3}{2}}}{x^{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 \frac{\log{\left (a x^{n} \right )}^{\frac{3}{2}}}{x^{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 \frac{\log \left (a x^{n}\right )^{\frac{3}{2}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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