Integrand size = 14, antiderivative size = 52 \[ \int \frac {(a+b \log (c x))^p}{x^2} \, dx=-c e^{a/b} \Gamma \left (1+p,\frac {a+b \log (c x)}{b}\right ) (a+b \log (c x))^p \left (\frac {a+b \log (c x)}{b}\right )^{-p} \]
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Time = 0.03 (sec) , antiderivative size = 52, 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 = {2346, 2212} \[ \int \frac {(a+b \log (c x))^p}{x^2} \, dx=-c e^{a/b} (a+b \log (c x))^p \left (\frac {a+b \log (c x)}{b}\right )^{-p} \Gamma \left (p+1,\frac {a+b \log (c x)}{b}\right ) \]
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Rule 2212
Rule 2346
Rubi steps \begin{align*} \text {integral}& = c \text {Subst}\left (\int e^{-x} (a+b x)^p \, dx,x,\log (c x)\right ) \\ & = -c e^{a/b} \Gamma \left (1+p,\frac {a+b \log (c x)}{b}\right ) (a+b \log (c x))^p \left (\frac {a+b \log (c x)}{b}\right )^{-p} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b \log (c x))^p}{x^2} \, dx=-c e^{a/b} \Gamma \left (1+p,\frac {a}{b}+\log (c x)\right ) \left (\frac {a}{b}+\log (c x)\right )^{-p} (a+b \log (c x))^p \]
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\[\int \frac {\left (a +b \ln \left (x c \right )\right )^{p}}{x^{2}}d x\]
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\[ \int \frac {(a+b \log (c x))^p}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x\right ) + a\right )}^{p}}{x^{2}} \,d x } \]
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\[ \int \frac {(a+b \log (c x))^p}{x^2} \, dx=\int \frac {\left (a + b \log {\left (c x \right )}\right )^{p}}{x^{2}}\, dx \]
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none
Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.77 \[ \int \frac {(a+b \log (c x))^p}{x^2} \, dx=-\frac {{\left (b \log \left (c x\right ) + a\right )}^{p + 1} c e^{\frac {a}{b}} E_{-p}\left (\frac {b \log \left (c x\right ) + a}{b}\right )}{b} \]
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\[ \int \frac {(a+b \log (c x))^p}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x\right ) + a\right )}^{p}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \log (c x))^p}{x^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x\right )\right )}^p}{x^2} \,d x \]
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