Integrand size = 14, antiderivative size = 63 \[ \int \frac {(a+b \log (c x))^p}{x^3} \, dx=-2^{-1-p} c^2 e^{\frac {2 a}{b}} \Gamma \left (1+p,\frac {2 (a+b \log (c x))}{b}\right ) (a+b \log (c x))^p \left (\frac {a+b \log (c x)}{b}\right )^{-p} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 63, 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} \[ \int \frac {(a+b \log (c x))^p}{x^3} \, dx=c^2 \left (-2^{-p-1}\right ) e^{\frac {2 a}{b}} (a+b \log (c x))^p \left (\frac {a+b \log (c x)}{b}\right )^{-p} \Gamma \left (p+1,\frac {2 (a+b \log (c x))}{b}\right ) \]
[In]
[Out]
Rule 2212
Rule 2346
Rubi steps \begin{align*} \text {integral}& = c^2 \text {Subst}\left (\int e^{-2 x} (a+b x)^p \, dx,x,\log (c x)\right ) \\ & = -2^{-1-p} c^2 e^{\frac {2 a}{b}} \Gamma \left (1+p,\frac {2 (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 = 63, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \log (c x))^p}{x^3} \, dx=-2^{-1-p} c^2 e^{\frac {2 a}{b}} \Gamma \left (1+p,\frac {2 (a+b \log (c x))}{b}\right ) (a+b \log (c x))^p \left (\frac {a+b \log (c x)}{b}\right )^{-p} \]
[In]
[Out]
\[\int \frac {\left (a +b \ln \left (x c \right )\right )^{p}}{x^{3}}d x\]
[In]
[Out]
\[ \int \frac {(a+b \log (c x))^p}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x\right ) + a\right )}^{p}}{x^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {(a+b \log (c x))^p}{x^3} \, dx=\int \frac {\left (a + b \log {\left (c x \right )}\right )^{p}}{x^{3}}\, dx \]
[In]
[Out]
none
Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.70 \[ \int \frac {(a+b \log (c x))^p}{x^3} \, dx=-\frac {{\left (b \log \left (c x\right ) + a\right )}^{p + 1} c^{2} e^{\left (\frac {2 \, a}{b}\right )} E_{-p}\left (\frac {2 \, {\left (b \log \left (c x\right ) + a\right )}}{b}\right )}{b} \]
[In]
[Out]
\[ \int \frac {(a+b \log (c x))^p}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x\right ) + a\right )}^{p}}{x^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(a+b \log (c x))^p}{x^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x\right )\right )}^p}{x^3} \,d x \]
[In]
[Out]