Optimal. Leaf size=83 \[ \frac{2 \sqrt{\pi } \sqrt{m+1} x^{m+1} \left (a x^n\right )^{-\frac{m+1}{n}} \text{Erfi}\left (\frac{\sqrt{m+1} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{n^{3/2}}-\frac{2 x^{m+1}}{n \sqrt{\log \left (a x^n\right )}} \]
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Rubi [A] time = 0.0679234, antiderivative size = 83, 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 = {2306, 2310, 2180, 2204} \[ \frac{2 \sqrt{\pi } \sqrt{m+1} x^{m+1} \left (a x^n\right )^{-\frac{m+1}{n}} \text{Erfi}\left (\frac{\sqrt{m+1} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{n^{3/2}}-\frac{2 x^{m+1}}{n \sqrt{\log \left (a x^n\right )}} \]
Antiderivative was successfully verified.
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Rule 2306
Rule 2310
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \frac{x^m}{\log ^{\frac{3}{2}}\left (a x^n\right )} \, dx &=-\frac{2 x^{1+m}}{n \sqrt{\log \left (a x^n\right )}}+\frac{(2 (1+m)) \int \frac{x^m}{\sqrt{\log \left (a x^n\right )}} \, dx}{n}\\ &=-\frac{2 x^{1+m}}{n \sqrt{\log \left (a x^n\right )}}+\frac{\left (2 (1+m) 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 )}{n^2}\\ &=-\frac{2 x^{1+m}}{n \sqrt{\log \left (a x^n\right )}}+\frac{\left (4 (1+m) 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 )}{n^2}\\ &=\frac{2 \sqrt{1+m} \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 )}{n^{3/2}}-\frac{2 x^{1+m}}{n \sqrt{\log \left (a x^n\right )}}\\ \end{align*}
Mathematica [A] time = 0.193792, size = 86, normalized size = 1.04 \[ \frac{2 \sqrt{\pi } \sqrt{m+1} e^{-\frac{(m+1) \left (\log \left (a x^n\right )-n \log (x)\right )}{n}} \text{Erfi}\left (\frac{\sqrt{m+1} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{n^{3/2}}-\frac{2 x^{m+1}}{n \sqrt{\log \left (a x^n\right )}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.185, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \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{x^{m}}{\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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{m}}{\log \left (a x^{n}\right )^{\frac{3}{2}}}, 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 \frac{x^{m}}{\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 \frac{x^{m}}{\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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