Integrand size = 17, antiderivative size = 30 \[ \int \frac {1}{x \sqrt {a+b (c x)^m}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b (c x)^m}}{\sqrt {a}}\right )}{\sqrt {a} m} \]
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Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {374, 12, 272, 65, 214} \[ \int \frac {1}{x \sqrt {a+b (c x)^m}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b (c x)^m}}{\sqrt {a}}\right )}{\sqrt {a} m} \]
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Rule 12
Rule 65
Rule 214
Rule 272
Rule 374
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {c}{x \sqrt {a+b x^m}} \, dx,x,c x\right )}{c} \\ & = \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^m}} \, dx,x,c x\right ) \\ & = \frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,(c x)^m\right )}{m} \\ & = \frac {2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b (c x)^m}\right )}{b m} \\ & = -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^m}}{\sqrt {a}}\right )}{\sqrt {a} m} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {a+b (c x)^m}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b (c x)^m}}{\sqrt {a}}\right )}{\sqrt {a} m} \]
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Time = 1.42 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \left (c x \right )^{m}}}{\sqrt {a}}\right )}{m \sqrt {a}}\) | \(25\) |
default | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \left (c x \right )^{m}}}{\sqrt {a}}\right )}{m \sqrt {a}}\) | \(25\) |
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none
Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.60 \[ \int \frac {1}{x \sqrt {a+b (c x)^m}} \, dx=\left [\frac {\log \left (\frac {\left (c x\right )^{m} b - 2 \, \sqrt {\left (c x\right )^{m} b + a} \sqrt {a} + 2 \, a}{\left (c x\right )^{m}}\right )}{\sqrt {a} m}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {\left (c x\right )^{m} b + a} \sqrt {-a}}{a}\right )}{a m}\right ] \]
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\[ \int \frac {1}{x \sqrt {a+b (c x)^m}} \, dx=\int \frac {1}{x \sqrt {a + b \left (c x\right )^{m}}}\, dx \]
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\[ \int \frac {1}{x \sqrt {a+b (c x)^m}} \, dx=\int { \frac {1}{\sqrt {\left (c x\right )^{m} b + a} x} \,d x } \]
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\[ \int \frac {1}{x \sqrt {a+b (c x)^m}} \, dx=\int { \frac {1}{\sqrt {\left (c x\right )^{m} b + a} x} \,d x } \]
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Timed out. \[ \int \frac {1}{x \sqrt {a+b (c x)^m}} \, dx=\int \frac {1}{x\,\sqrt {a+b\,{\left (c\,x\right )}^m}} \,d x \]
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