Integrand size = 18, antiderivative size = 26 \[ \int \frac {\left (a+b \log \left (c \sqrt {x}\right )\right )^p}{x} \, dx=\frac {2 \left (a+b \log \left (c \sqrt {x}\right )\right )^{1+p}}{b (1+p)} \]
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Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2339, 30} \[ \int \frac {\left (a+b \log \left (c \sqrt {x}\right )\right )^p}{x} \, dx=\frac {2 \left (a+b \log \left (c \sqrt {x}\right )\right )^{p+1}}{b (p+1)} \]
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Rule 30
Rule 2339
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int x^p \, dx,x,a+b \log \left (c \sqrt {x}\right )\right )}{b} \\ & = \frac {2 \left (a+b \log \left (c \sqrt {x}\right )\right )^{1+p}}{b (1+p)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \sqrt {x}\right )\right )^p}{x} \, dx=\frac {2 \left (a+b \log \left (c \sqrt {x}\right )\right )^{1+p}}{b (1+p)} \]
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Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {2 \left (a +b \ln \left (c \sqrt {x}\right )\right )^{p +1}}{b \left (p +1\right )}\) | \(25\) |
default | \(\frac {2 \left (a +b \ln \left (c \sqrt {x}\right )\right )^{p +1}}{b \left (p +1\right )}\) | \(25\) |
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none
Time = 0.34 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b \log \left (c \sqrt {x}\right )\right )^p}{x} \, dx=\frac {2 \, {\left (b \log \left (c \sqrt {x}\right ) + a\right )} {\left (b \log \left (c \sqrt {x}\right ) + a\right )}^{p}}{b p + b} \]
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Time = 2.21 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a+b \log \left (c \sqrt {x}\right )\right )^p}{x} \, dx=- \begin {cases} - a^{p} \log {\left (x \right )} & \text {for}\: b = 0 \\- \frac {2 \left (\begin {cases} \frac {\left (a + b \log {\left (c \sqrt {x} \right )}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + b \log {\left (c \sqrt {x} \right )} \right )} & \text {otherwise} \end {cases}\right )}{b} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \log \left (c \sqrt {x}\right )\right )^p}{x} \, dx=\frac {2 \, {\left (b \log \left (c \sqrt {x}\right ) + a\right )}^{p + 1}}{b {\left (p + 1\right )}} \]
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none
Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b \log \left (c \sqrt {x}\right )\right )^p}{x} \, dx=\frac {2 \, {\left (b \log \left (c\right ) + \frac {1}{2} \, b \log \left (x\right ) + a\right )}^{p + 1}}{b {\left (p + 1\right )}} \]
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Time = 0.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \log \left (c \sqrt {x}\right )\right )^p}{x} \, dx=\frac {2\,{\left (a+b\,\ln \left (c\,\sqrt {x}\right )\right )}^{p+1}}{b\,\left (p+1\right )} \]
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