Optimal. Leaf size=72 \[ \frac {3}{4} \sqrt {\pi } n^{3/2} x \left (a x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )+x \log ^{\frac {3}{2}}\left (a x^n\right )-\frac {3}{2} n x \sqrt {\log \left (a x^n\right )} \]
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Rubi [A] time = 0.03, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2296, 2300, 2180, 2204} \[ \frac {3}{4} \sqrt {\pi } n^{3/2} x \left (a x^n\right )^{-1/n} \text {Erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )+x \log ^{\frac {3}{2}}\left (a x^n\right )-\frac {3}{2} n x \sqrt {\log \left (a x^n\right )} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2296
Rule 2300
Rubi steps
\begin {align*} \int \log ^{\frac {3}{2}}\left (a x^n\right ) \, dx &=x \log ^{\frac {3}{2}}\left (a x^n\right )-\frac {1}{2} (3 n) \int \sqrt {\log \left (a x^n\right )} \, dx\\ &=-\frac {3}{2} n x \sqrt {\log \left (a x^n\right )}+x \log ^{\frac {3}{2}}\left (a x^n\right )+\frac {1}{4} \left (3 n^2\right ) \int \frac {1}{\sqrt {\log \left (a x^n\right )}} \, dx\\ &=-\frac {3}{2} n x \sqrt {\log \left (a x^n\right )}+x \log ^{\frac {3}{2}}\left (a x^n\right )+\frac {1}{4} \left (3 n x \left (a x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {x}{n}}}{\sqrt {x}} \, dx,x,\log \left (a x^n\right )\right )\\ &=-\frac {3}{2} n x \sqrt {\log \left (a x^n\right )}+x \log ^{\frac {3}{2}}\left (a x^n\right )+\frac {1}{2} \left (3 n x \left (a x^n\right )^{-1/n}\right ) \operatorname {Subst}\left (\int e^{\frac {x^2}{n}} \, dx,x,\sqrt {\log \left (a x^n\right )}\right )\\ &=\frac {3}{4} n^{3/2} \sqrt {\pi } x \left (a x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )-\frac {3}{2} n x \sqrt {\log \left (a x^n\right )}+x \log ^{\frac {3}{2}}\left (a x^n\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 72, normalized size = 1.00 \[ \frac {3}{4} \sqrt {\pi } n^{3/2} x \left (a x^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )+x \log ^{\frac {3}{2}}\left (a x^n\right )-\frac {3}{2} n x \sqrt {\log \left (a x^n\right )} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \log \left (a x^{n}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \ln \left (a \,x^{n}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \log \left (a x^{n}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\ln \left (a\,x^n\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \log {\left (a x^{n} \right )}^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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