Optimal. Leaf size=17 \[ \frac {2 \log ^{\frac {5}{2}}\left (a x^n\right )}{5 n} \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2302, 30} \[ \frac {2 \log ^{\frac {5}{2}}\left (a x^n\right )}{5 n} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2302
Rubi steps
\begin {align*} \int \frac {\log ^{\frac {3}{2}}\left (a x^n\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int x^{3/2} \, dx,x,\log \left (a x^n\right )\right )}{n}\\ &=\frac {2 \log ^{\frac {5}{2}}\left (a x^n\right )}{5 n}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 17, normalized size = 1.00 \[ \frac {2 \log ^{\frac {5}{2}}\left (a x^n\right )}{5 n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 34, normalized size = 2.00 \[ \frac {2 \, {\left (n^{2} \log \relax (x)^{2} + 2 \, n \log \relax (a) \log \relax (x) + \log \relax (a)^{2}\right )} \sqrt {n \log \relax (x) + \log \relax (a)}}{5 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 72, normalized size = 4.24 \[ \frac {2 \, {\left (3 \, {\left (n \log \relax (x) + \log \relax (a)\right )}^{\frac {5}{2}} - 10 \, {\left (n \log \relax (x) + \log \relax (a)\right )}^{\frac {3}{2}} \log \relax (a) + 30 \, \sqrt {n \log \relax (x) + \log \relax (a)} \log \relax (a)^{2} + 10 \, {\left ({\left (n \log \relax (x) + \log \relax (a)\right )}^{\frac {3}{2}} - 3 \, \sqrt {n \log \relax (x) + \log \relax (a)} \log \relax (a)\right )} \log \relax (a)\right )}}{15 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 14, normalized size = 0.82 \[ \frac {2 \ln \left (a \,x^{n}\right )^{\frac {5}{2}}}{5 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 13, normalized size = 0.76 \[ \frac {2 \, \log \left (a x^{n}\right )^{\frac {5}{2}}}{5 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.53, size = 13, normalized size = 0.76 \[ \frac {2\,{\ln \left (a\,x^n\right )}^{5/2}}{5\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 21.92, size = 75, normalized size = 4.41 \[ \begin {cases} \frac {2 n \sqrt {n \log {\relax (x )} + \log {\relax (a )}} \log {\relax (x )}^{2}}{5} + \frac {4 \sqrt {n \log {\relax (x )} + \log {\relax (a )}} \log {\relax (a )} \log {\relax (x )}}{5} + \frac {2 \sqrt {n \log {\relax (x )} + \log {\relax (a )}} \log {\relax (a )}^{2}}{5 n} & \text {for}\: n \neq 0 \\\log {\relax (a )}^{\frac {3}{2}} \log {\relax (x )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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