Optimal. Leaf size=93 \[ \frac{8 \sqrt{2 \pi } x^2 \left (a x^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{3 n^{5/2}}-\frac{8 x^2}{3 n^2 \sqrt{\log \left (a x^n\right )}}-\frac{2 x^2}{3 n \log ^{\frac{3}{2}}\left (a x^n\right )} \]
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Rubi [A] time = 0.0579577, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2306, 2310, 2180, 2204} \[ \frac{8 \sqrt{2 \pi } x^2 \left (a x^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{3 n^{5/2}}-\frac{8 x^2}{3 n^2 \sqrt{\log \left (a x^n\right )}}-\frac{2 x^2}{3 n \log ^{\frac{3}{2}}\left (a x^n\right )} \]
Antiderivative was successfully verified.
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Rule 2306
Rule 2310
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \frac{x}{\log ^{\frac{5}{2}}\left (a x^n\right )} \, dx &=-\frac{2 x^2}{3 n \log ^{\frac{3}{2}}\left (a x^n\right )}+\frac{4 \int \frac{x}{\log ^{\frac{3}{2}}\left (a x^n\right )} \, dx}{3 n}\\ &=-\frac{2 x^2}{3 n \log ^{\frac{3}{2}}\left (a x^n\right )}-\frac{8 x^2}{3 n^2 \sqrt{\log \left (a x^n\right )}}+\frac{16 \int \frac{x}{\sqrt{\log \left (a x^n\right )}} \, dx}{3 n^2}\\ &=-\frac{2 x^2}{3 n \log ^{\frac{3}{2}}\left (a x^n\right )}-\frac{8 x^2}{3 n^2 \sqrt{\log \left (a x^n\right )}}+\frac{\left (16 x^2 \left (a x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{\sqrt{x}} \, dx,x,\log \left (a x^n\right )\right )}{3 n^3}\\ &=-\frac{2 x^2}{3 n \log ^{\frac{3}{2}}\left (a x^n\right )}-\frac{8 x^2}{3 n^2 \sqrt{\log \left (a x^n\right )}}+\frac{\left (32 x^2 \left (a x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int e^{\frac{2 x^2}{n}} \, dx,x,\sqrt{\log \left (a x^n\right )}\right )}{3 n^3}\\ &=\frac{8 \sqrt{2 \pi } x^2 \left (a x^n\right )^{-2/n} \text{erfi}\left (\frac{\sqrt{2} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{3 n^{5/2}}-\frac{2 x^2}{3 n \log ^{\frac{3}{2}}\left (a x^n\right )}-\frac{8 x^2}{3 n^2 \sqrt{\log \left (a x^n\right )}}\\ \end{align*}
Mathematica [A] time = 0.0636767, size = 92, normalized size = 0.99 \[ -\frac{2 x^2 \left (a x^n\right )^{-2/n} \left (4 \sqrt{2} n \left (-\frac{\log \left (a x^n\right )}{n}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{2 \log \left (a x^n\right )}{n}\right )+\left (a x^n\right )^{2/n} \left (4 \log \left (a x^n\right )+n\right )\right )}{3 n^2 \log ^{\frac{3}{2}}\left (a x^n\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.168, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \ln \left ( a{x}^{n} \right ) \right ) ^{-{\frac{5}{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{x}{\log \left (a x^{n}\right )^{\frac{5}{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{x}{\log{\left (a x^{n} \right )}^{\frac{5}{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{x}{\log \left (a x^{n}\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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