Optimal. Leaf size=90 \[ \frac{3}{16} \sqrt{\frac{\pi }{2}} n^{3/2} x^2 \left (a x^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )+\frac{1}{2} x^2 \log ^{\frac{3}{2}}\left (a x^n\right )-\frac{3}{8} n x^2 \sqrt{\log \left (a x^n\right )} \]
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Rubi [A] time = 0.0565114, antiderivative size = 90, 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 = {2305, 2310, 2180, 2204} \[ \frac{3}{16} \sqrt{\frac{\pi }{2}} n^{3/2} x^2 \left (a x^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )+\frac{1}{2} x^2 \log ^{\frac{3}{2}}\left (a x^n\right )-\frac{3}{8} n x^2 \sqrt{\log \left (a x^n\right )} \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2310
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int x \log ^{\frac{3}{2}}\left (a x^n\right ) \, dx &=\frac{1}{2} x^2 \log ^{\frac{3}{2}}\left (a x^n\right )-\frac{1}{4} (3 n) \int x \sqrt{\log \left (a x^n\right )} \, dx\\ &=-\frac{3}{8} n x^2 \sqrt{\log \left (a x^n\right )}+\frac{1}{2} x^2 \log ^{\frac{3}{2}}\left (a x^n\right )+\frac{1}{16} \left (3 n^2\right ) \int \frac{x}{\sqrt{\log \left (a x^n\right )}} \, dx\\ &=-\frac{3}{8} n x^2 \sqrt{\log \left (a x^n\right )}+\frac{1}{2} x^2 \log ^{\frac{3}{2}}\left (a x^n\right )+\frac{1}{16} \left (3 n 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 )\\ &=-\frac{3}{8} n x^2 \sqrt{\log \left (a x^n\right )}+\frac{1}{2} x^2 \log ^{\frac{3}{2}}\left (a x^n\right )+\frac{1}{8} \left (3 n 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 )\\ &=\frac{3}{16} n^{3/2} \sqrt{\frac{\pi }{2}} 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 )-\frac{3}{8} n x^2 \sqrt{\log \left (a x^n\right )}+\frac{1}{2} x^2 \log ^{\frac{3}{2}}\left (a x^n\right )\\ \end{align*}
Mathematica [A] time = 0.0574906, size = 79, normalized size = 0.88 \[ \frac{1}{32} x^2 \left (3 \sqrt{2 \pi } n^{3/2} \left (a x^n\right )^{-2/n} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )+4 \sqrt{\log \left (a x^n\right )} \left (4 \log \left (a x^n\right )-3 n\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.173, size = 0, normalized size = 0. \begin{align*} \int x \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 x \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 x \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 x \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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