Integrand size = 18, antiderivative size = 51 \[ \int \frac {\sqrt {a+b x^n}}{c x} \, dx=\frac {2 \sqrt {a+b x^n}}{c n}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{c n} \]
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Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {12, 272, 52, 65, 214} \[ \int \frac {\sqrt {a+b x^n}}{c x} \, dx=\frac {2 \sqrt {a+b x^n}}{c n}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{c n} \]
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Rule 12
Rule 52
Rule 65
Rule 214
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sqrt {a+b x^n}}{x} \, dx}{c} \\ & = \frac {\text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^n\right )}{c n} \\ & = \frac {2 \sqrt {a+b x^n}}{c n}+\frac {a \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^n\right )}{c n} \\ & = \frac {2 \sqrt {a+b x^n}}{c n}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^n}\right )}{b c n} \\ & = \frac {2 \sqrt {a+b x^n}}{c n}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{c n} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {a+b x^n}}{c x} \, dx=\frac {2 \left (\sqrt {a+b x^n}-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )\right )}{c n} \]
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Time = 2.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(\frac {2 \sqrt {a +b \,x^{n}}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {a +b \,x^{n}}}{\sqrt {a}}\right )}{c n}\) | \(39\) |
default | \(\frac {2 \sqrt {a +b \,x^{n}}-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {a +b \,x^{n}}}{\sqrt {a}}\right )}{c n}\) | \(39\) |
risch | \(\frac {2 \sqrt {a +b \,{\mathrm e}^{n \ln \left (x \right )}}}{n c}-\frac {2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {a +b \,{\mathrm e}^{n \ln \left (x \right )}}}{\sqrt {a}}\right )}{n c}\) | \(48\) |
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none
Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.90 \[ \int \frac {\sqrt {a+b x^n}}{c x} \, dx=\left [\frac {\sqrt {a} \log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) + 2 \, \sqrt {b x^{n} + a}}{c n}, \frac {2 \, {\left (\sqrt {-a} \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-a}}{a}\right ) + \sqrt {b x^{n} + a}\right )}}{c n}\right ] \]
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Time = 0.91 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.53 \[ \int \frac {\sqrt {a+b x^n}}{c x} \, dx=\frac {- \frac {2 \sqrt {a} \operatorname {asinh}{\left (\frac {\sqrt {a} x^{- \frac {n}{2}}}{\sqrt {b}} \right )}}{n} + \frac {2 a x^{- \frac {n}{2}}}{\sqrt {b} n \sqrt {\frac {a x^{- n}}{b} + 1}} + \frac {2 \sqrt {b} x^{\frac {n}{2}}}{n \sqrt {\frac {a x^{- n}}{b} + 1}}}{c} \]
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none
Time = 0.31 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {a+b x^n}}{c x} \, dx=\frac {\frac {\sqrt {a} \log \left (\frac {\sqrt {b x^{n} + a} - \sqrt {a}}{\sqrt {b x^{n} + a} + \sqrt {a}}\right )}{n} + \frac {2 \, \sqrt {b x^{n} + a}}{n}}{c} \]
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\[ \int \frac {\sqrt {a+b x^n}}{c x} \, dx=\int { \frac {\sqrt {b x^{n} + a}}{c x} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+b x^n}}{c x} \, dx=\int \frac {\sqrt {a+b\,x^n}}{c\,x} \,d x \]
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