Integrand size = 20, antiderivative size = 64 \[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {e^{-\frac {a}{2 b n}} \sqrt {d x} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{b d n} \]
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Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2347, 2209} \[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {\sqrt {d x} e^{-\frac {a}{2 b n}} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{b d n} \]
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Rule 2209
Rule 2347
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {d x} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{2 n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{d n} \\ & = \frac {e^{-\frac {a}{2 b n}} \sqrt {d x} \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \text {Ei}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{b d n} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {e^{-\frac {a}{2 b n}} x \left (c x^n\right )^{\left .-\frac {1}{2}\right /n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c x^n\right )}{2 b n}\right )}{b n \sqrt {d x}} \]
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\[\int \frac {1}{\sqrt {d x}\, \left (a +b \ln \left (c \,x^{n}\right )\right )}d x\]
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\[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{\sqrt {d x} {\left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]
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\[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{\sqrt {d x} \left (a + b \log {\left (c x^{n} \right )}\right )}\, dx \]
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\[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{\sqrt {d x} {\left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]
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\[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{\sqrt {d x} {\left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {d x} \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{\sqrt {d\,x}\,\left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]
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