Optimal. Leaf size=86 \[ \frac {x^{m+1} \sqrt {\log \left (a x^n\right )}}{m+1}-\frac {\sqrt {\pi } \sqrt {n} x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \text {erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{2 (m+1)^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 86, normalized size of antiderivative = 1.00, 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 = {2305, 2310, 2180, 2204} \[ \frac {x^{m+1} \sqrt {\log \left (a x^n\right )}}{m+1}-\frac {\sqrt {\pi } \sqrt {n} x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \text {Erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{2 (m+1)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2305
Rule 2310
Rubi steps
\begin {align*} \int x^m \sqrt {\log \left (a x^n\right )} \, dx &=\frac {x^{1+m} \sqrt {\log \left (a x^n\right )}}{1+m}-\frac {n \int \frac {x^m}{\sqrt {\log \left (a x^n\right )}} \, dx}{2 (1+m)}\\ &=\frac {x^{1+m} \sqrt {\log \left (a x^n\right )}}{1+m}-\frac {\left (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 )}{2 (1+m)}\\ &=\frac {x^{1+m} \sqrt {\log \left (a x^n\right )}}{1+m}-\frac {\left (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 )}{1+m}\\ &=-\frac {\sqrt {n} \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 )}{2 (1+m)^{3/2}}+\frac {x^{1+m} \sqrt {\log \left (a x^n\right )}}{1+m}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 86, normalized size = 1.00 \[ \frac {x^{m+1} \sqrt {\log \left (a x^n\right )}}{m+1}-\frac {\sqrt {\pi } \sqrt {n} x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \text {erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{2 (m+1)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \sqrt {\log \left (a x^{n}\right )}, x\right ) \]
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 \[ \int x^{m} \sqrt {\log \left (a x^{n}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int x^{m} \sqrt {\ln \left (a \,x^{n}\right )}\, dx \]
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 \[ \int x^{m} \sqrt {\log \left (a x^{n}\right )}\,{d x} \]
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 \[ \int x^m\,\sqrt {\ln \left (a\,x^n\right )} \,d x \]
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 \[ \int x^{m} \sqrt {\log {\left (a x^{n} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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