Integrand size = 28, antiderivative size = 85 \[ \int \frac {1}{x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx=\frac {\left (a+b x^n\right ) \log (x)}{a \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1369, 272, 36, 29, 31} \[ \int \frac {1}{x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx=\frac {\log (x) \left (a+b x^n\right )}{a \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Rule 29
Rule 31
Rule 36
Rule 272
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a b+b^2 x^n\right ) \int \frac {1}{x \left (a b+b^2 x^n\right )} \, dx}{\sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \\ & = \frac {\left (a b+b^2 x^n\right ) \text {Subst}\left (\int \frac {1}{x \left (a b+b^2 x\right )} \, dx,x,x^n\right )}{n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \\ & = \frac {\left (a b+b^2 x^n\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^n\right )}{a b n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {\left (b \left (a b+b^2 x^n\right )\right ) \text {Subst}\left (\int \frac {1}{a b+b^2 x} \, dx,x,x^n\right )}{a n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \\ & = \frac {\left (a+b x^n\right ) \log (x)}{a \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.53 \[ \int \frac {1}{x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx=\frac {\left (a+b x^n\right ) \left (\log \left (x^n\right )-\log \left (a n \left (a+b x^n\right )\right )\right )}{a n \sqrt {\left (a+b x^n\right )^2}} \]
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Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.78
method | result | size |
risch | \(\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, \ln \left (x \right )}{\left (a +b \,x^{n}\right ) a}-\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, \ln \left (x^{n}+\frac {a}{b}\right )}{\left (a +b \,x^{n}\right ) a n}\) | \(66\) |
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Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx=\frac {n \log \left (x\right ) - \log \left (b x^{n} + a\right )}{a n} \]
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\[ \int \frac {1}{x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx=\int \frac {1}{x \sqrt {\left (a + b x^{n}\right )^{2}}}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.32 \[ \int \frac {1}{x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx=\frac {\log \left (x\right )}{a} - \frac {\log \left (\frac {b x^{n} + a}{b}\right )}{a n} \]
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\[ \int \frac {1}{x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx=\int { \frac {1}{\sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}} x} \,d x } \]
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Timed out. \[ \int \frac {1}{x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx=\int \frac {1}{x\,\sqrt {a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n}} \,d x \]
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