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 = 0.13, antiderivative size = 111, normalized size of antiderivative = 1.00, 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 2180
Rule 2204
Rule 2305
Rule 2310
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*}
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Mathematica [A] time = 0.19, 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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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \log \left (a x^{n}\right )^{\frac {3}{2}}, x\right ) \]
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 \[ \int x^{m} \log \left (a x^{n}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int x^{m} \ln \left (a \,x^{n}\right )^{\frac {3}{2}}\, dx \]
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 \[ \int x^{m} \log \left (a x^{n}\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int x^m\,{\ln \left (a\,x^n\right )}^{3/2} \,d x \]
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 \[ \int x^{m} \log {\left (a x^{n} \right )}^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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