Optimal. Leaf size=111 \[ \frac{3 \sqrt{\pi } n^{3/2} x^{m+1} \left (a x^n\right )^{-\frac{m+1}{n}} \text{Erfi}\left (\frac{\sqrt{m+1} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{4 (m+1)^{5/2}}+\frac{x^{m+1} \log ^{\frac{3}{2}}\left (a x^n\right )}{m+1}-\frac{3 n x^{m+1} \sqrt{\log \left (a x^n\right )}}{2 (m+1)^2} \]
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Rubi [A] time = 0.125584, antiderivative size = 111, 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, 2204} \[ \frac{3 \sqrt{\pi } n^{3/2} x^{m+1} \left (a x^n\right )^{-\frac{m+1}{n}} \text{Erfi}\left (\frac{\sqrt{m+1} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )}{4 (m+1)^{5/2}}+\frac{x^{m+1} \log ^{\frac{3}{2}}\left (a x^n\right )}{m+1}-\frac{3 n x^{m+1} \sqrt{\log \left (a x^n\right )}}{2 (m+1)^2} \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2310
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int x^m \log ^{\frac{3}{2}}\left (a x^n\right ) \, dx &=\frac{x^{1+m} \log ^{\frac{3}{2}}\left (a x^n\right )}{1+m}-\frac{(3 n) \int x^m \sqrt{\log \left (a x^n\right )} \, dx}{2 (1+m)}\\ &=-\frac{3 n x^{1+m} \sqrt{\log \left (a x^n\right )}}{2 (1+m)^2}+\frac{x^{1+m} \log ^{\frac{3}{2}}\left (a x^n\right )}{1+m}+\frac{\left (3 n^2\right ) \int \frac{x^m}{\sqrt{\log \left (a x^n\right )}} \, dx}{4 (1+m)^2}\\ &=-\frac{3 n x^{1+m} \sqrt{\log \left (a x^n\right )}}{2 (1+m)^2}+\frac{x^{1+m} \log ^{\frac{3}{2}}\left (a x^n\right )}{1+m}+\frac{\left (3 n 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 )}{4 (1+m)^2}\\ &=-\frac{3 n x^{1+m} \sqrt{\log \left (a x^n\right )}}{2 (1+m)^2}+\frac{x^{1+m} \log ^{\frac{3}{2}}\left (a x^n\right )}{1+m}+\frac{\left (3 n 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 )}{2 (1+m)^2}\\ &=\frac{3 n^{3/2} \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 )}{4 (1+m)^{5/2}}-\frac{3 n x^{1+m} \sqrt{\log \left (a x^n\right )}}{2 (1+m)^2}+\frac{x^{1+m} \log ^{\frac{3}{2}}\left (a x^n\right )}{1+m}\\ \end{align*}
Mathematica [A] time = 0.194285, size = 101, normalized size = 0.91 \[ \frac{x^{m+1} \left (3 \sqrt{\pi } n^{3/2} \left (a x^n\right )^{-\frac{m+1}{n}} \text{Erfi}\left (\frac{\sqrt{m+1} \sqrt{\log \left (a x^n\right )}}{\sqrt{n}}\right )+2 \sqrt{m+1} \sqrt{\log \left (a x^n\right )} \left (2 (m+1) \log \left (a x^n\right )-3 n\right )\right )}{4 (m+1)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.206, 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 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 (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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{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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