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 = 0.06, antiderivative size = 90, normalized size of antiderivative = 1.00, 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 2180
Rule 2204
Rule 2305
Rule 2310
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*}
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Mathematica [A] time = 0.06, 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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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 \[ \int x \log \left (a x^{n}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int x \ln \left (a \,x^{n}\right )^{\frac {3}{2}}\, dx \]
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 \[ \int x \log \left (a x^{n}\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int x\,{\ln \left (a\,x^n\right )}^{3/2} \,d x \]
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 \[ \int x \log {\left (a x^{n} \right )}^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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