Integrand size = 14, antiderivative size = 17 \[ \int \frac {1}{x \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=-\frac {2}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )} \]
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Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2339, 30} \[ \int \frac {1}{x \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=-\frac {2}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )} \]
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Rule 30
Rule 2339
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x^{5/2}} \, dx,x,\log \left (a x^n\right )\right )}{n} \\ & = -\frac {2}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=-\frac {2}{3 n \log ^{\frac {3}{2}}\left (a x^n\right )} \]
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Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(-\frac {2}{3 n \ln \left (a \,x^{n}\right )^{\frac {3}{2}}}\) | \(14\) |
default | \(-\frac {2}{3 n \ln \left (a \,x^{n}\right )^{\frac {3}{2}}}\) | \(14\) |
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Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (13) = 26\).
Time = 0.31 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.18 \[ \int \frac {1}{x \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=-\frac {2 \, \sqrt {n \log \left (x\right ) + \log \left (a\right )}}{3 \, {\left (n^{3} \log \left (x\right )^{2} + 2 \, n^{2} \log \left (a\right ) \log \left (x\right ) + n \log \left (a\right )^{2}\right )}} \]
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Time = 6.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.53 \[ \int \frac {1}{x \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=\begin {cases} - \frac {2}{3 n \log {\left (a x^{n} \right )}^{\frac {3}{2}}} & \text {for}\: n \neq 0 \\\frac {\log {\left (x \right )}}{\log {\left (a \right )}^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=-\frac {2}{3 \, n \log \left (a x^{n}\right )^{\frac {3}{2}}} \]
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none
Time = 0.31 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=-\frac {2}{3 \, {\left (n \log \left (x\right ) + \log \left (a\right )\right )}^{\frac {3}{2}} n} \]
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Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x \log ^{\frac {5}{2}}\left (a x^n\right )} \, dx=-\frac {2}{3\,n\,{\ln \left (a\,x^n\right )}^{3/2}} \]
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