Integrand size = 16, antiderivative size = 89 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x^3} \, dx=-\frac {2^{-1-p} e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \Gamma \left (1+p,\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{x^2} \]
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Time = 0.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2347, 2212} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x^3} \, dx=-\frac {2^{-p-1} e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \Gamma \left (p+1,\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{x^2} \]
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Rule 2212
Rule 2347
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c x^n\right )^{2/n} \text {Subst}\left (\int e^{-\frac {2 x}{n}} (a+b x)^p \, dx,x,\log \left (c x^n\right )\right )}{n x^2} \\ & = -\frac {2^{-1-p} e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \Gamma \left (1+p,\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{x^2} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x^3} \, dx=-\frac {2^{-1-p} e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \Gamma \left (1+p,\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{x^2} \]
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\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{p}}{x^{3}}d x\]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x^3} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{p}}{x^{3}}\, dx \]
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Exception generated. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x^3} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^p}{x^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p}{x^3} \,d x \]
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