Integrand size = 19, antiderivative size = 53 \[ \int \frac {\sqrt {-a+b (c x)^n}}{x} \, dx=\frac {2 \sqrt {-a+b (c x)^n}}{n}-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {-a+b (c x)^n}}{\sqrt {a}}\right )}{n} \]
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Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {374, 12, 272, 52, 65, 211} \[ \int \frac {\sqrt {-a+b (c x)^n}}{x} \, dx=\frac {2 \sqrt {b (c x)^n-a}}{n}-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {b (c x)^n-a}}{\sqrt {a}}\right )}{n} \]
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Rule 12
Rule 52
Rule 65
Rule 211
Rule 272
Rule 374
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {c \sqrt {-a+b x^n}}{x} \, dx,x,c x\right )}{c} \\ & = \text {Subst}\left (\int \frac {\sqrt {-a+b x^n}}{x} \, dx,x,c x\right ) \\ & = \frac {\text {Subst}\left (\int \frac {\sqrt {-a+b x}}{x} \, dx,x,(c x)^n\right )}{n} \\ & = \frac {2 \sqrt {-a+b (c x)^n}}{n}-\frac {a \text {Subst}\left (\int \frac {1}{x \sqrt {-a+b x}} \, dx,x,(c x)^n\right )}{n} \\ & = \frac {2 \sqrt {-a+b (c x)^n}}{n}-\frac {(2 a) \text {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b (c x)^n}\right )}{b n} \\ & = \frac {2 \sqrt {-a+b (c x)^n}}{n}-\frac {2 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {-a+b (c x)^n}}{\sqrt {a}}\right )}{n} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {-a+b (c x)^n}}{x} \, dx=\frac {2 \left (\sqrt {-a+b (c x)^n}-\sqrt {a} \arctan \left (\frac {\sqrt {-a+b (c x)^n}}{\sqrt {a}}\right )\right )}{n} \]
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Time = 1.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {2 \sqrt {-a +b \left (c x \right )^{n}}-2 \sqrt {a}\, \arctan \left (\frac {\sqrt {-a +b \left (c x \right )^{n}}}{\sqrt {a}}\right )}{n}\) | \(44\) |
default | \(\frac {2 \sqrt {-a +b \left (c x \right )^{n}}-2 \sqrt {a}\, \arctan \left (\frac {\sqrt {-a +b \left (c x \right )^{n}}}{\sqrt {a}}\right )}{n}\) | \(44\) |
risch | \(-\frac {2 \left (a -b \,{\mathrm e}^{n \ln \left (c x \right )}\right )}{n \sqrt {-a +b \,{\mathrm e}^{n \ln \left (c x \right )}}}-\frac {2 \sqrt {a}\, \arctan \left (\frac {\sqrt {-a +b \,{\mathrm e}^{n \ln \left (c x \right )}}}{\sqrt {a}}\right )}{n}\) | \(62\) |
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Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.08 \[ \int \frac {\sqrt {-a+b (c x)^n}}{x} \, dx=\left [\frac {\sqrt {-a} \log \left (\frac {\left (c x\right )^{n} b - 2 \, \sqrt {\left (c x\right )^{n} b - a} \sqrt {-a} - 2 \, a}{\left (c x\right )^{n}}\right ) + 2 \, \sqrt {\left (c x\right )^{n} b - a}}{n}, -\frac {2 \, {\left (\sqrt {a} \arctan \left (\frac {\sqrt {\left (c x\right )^{n} b - a}}{\sqrt {a}}\right ) - \sqrt {\left (c x\right )^{n} b - a}\right )}}{n}\right ] \]
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\[ \int \frac {\sqrt {-a+b (c x)^n}}{x} \, dx=\int \frac {\sqrt {- a + b \left (c x\right )^{n}}}{x}\, dx \]
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\[ \int \frac {\sqrt {-a+b (c x)^n}}{x} \, dx=\int { \frac {\sqrt {\left (c x\right )^{n} b - a}}{x} \,d x } \]
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\[ \int \frac {\sqrt {-a+b (c x)^n}}{x} \, dx=\int { \frac {\sqrt {\left (c x\right )^{n} b - a}}{x} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {-a+b (c x)^n}}{x} \, dx=\int \frac {\sqrt {b\,{\left (c\,x\right )}^n-a}}{x} \,d x \]
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