Optimal. Leaf size=83 \[ \frac {2 \sqrt {\pi } \sqrt {m+1} x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \text {erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{3/2}}-\frac {2 x^{m+1}}{n \sqrt {\log \left (a x^n\right )}} \]
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Rubi [A] time = 0.07, antiderivative size = 83, 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 = {2306, 2310, 2180, 2204} \[ \frac {2 \sqrt {\pi } \sqrt {m+1} x^{m+1} \left (a x^n\right )^{-\frac {m+1}{n}} \text {Erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{3/2}}-\frac {2 x^{m+1}}{n \sqrt {\log \left (a x^n\right )}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2306
Rule 2310
Rubi steps
\begin {align*} \int \frac {x^m}{\log ^{\frac {3}{2}}\left (a x^n\right )} \, dx &=-\frac {2 x^{1+m}}{n \sqrt {\log \left (a x^n\right )}}+\frac {(2 (1+m)) \int \frac {x^m}{\sqrt {\log \left (a x^n\right )}} \, dx}{n}\\ &=-\frac {2 x^{1+m}}{n \sqrt {\log \left (a x^n\right )}}+\frac {\left (2 (1+m) 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 )}{n^2}\\ &=-\frac {2 x^{1+m}}{n \sqrt {\log \left (a x^n\right )}}+\frac {\left (4 (1+m) 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 )}{n^2}\\ &=\frac {2 \sqrt {1+m} \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 )}{n^{3/2}}-\frac {2 x^{1+m}}{n \sqrt {\log \left (a x^n\right )}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 86, normalized size = 1.04 \[ \frac {2 \sqrt {\pi } \sqrt {m+1} e^{-\frac {(m+1) \left (\log \left (a x^n\right )-n \log (x)\right )}{n}} \text {erfi}\left (\frac {\sqrt {m+1} \sqrt {\log \left (a x^n\right )}}{\sqrt {n}}\right )}{n^{3/2}}-\frac {2 x^{m+1}}{n \sqrt {\log \left (a x^n\right )}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{\log \left (a x^{n}\right )^{\frac {3}{2}}}, x\right ) \]
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 \[ \int \frac {x^{m}}{\log \left (a x^{n}\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^{m}}{\ln \left (a \,x^{n}\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x^{m}}{\log \left (a x^{n}\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^m}{{\ln \left (a\,x^n\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{m}}{\log {\left (a x^{n} \right )}^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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