Integrand size = 28, antiderivative size = 159 \[ \int \frac {1}{x \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\frac {1}{a^2 n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}+\frac {1}{2 a n \left (a+b x^n\right ) \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}+\frac {\left (a+b x^n\right ) \log (x)}{a^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a^3 n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Time = 0.05 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1369, 272, 46} \[ \int \frac {1}{x \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\frac {1}{a^2 n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}+\frac {1}{2 a n \left (a+b x^n\right ) \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}+\frac {\log (x) \left (a+b x^n\right )}{a^3 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}-\frac {\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a^3 n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Rule 46
Rule 272
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^2 \left (a b+b^2 x^n\right )\right ) \int \frac {1}{x \left (a b+b^2 x^n\right )^3} \, dx}{\sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \\ & = \frac {\left (b^2 \left (a b+b^2 x^n\right )\right ) \text {Subst}\left (\int \frac {1}{x \left (a b+b^2 x\right )^3} \, dx,x,x^n\right )}{n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \\ & = \frac {\left (b^2 \left (a b+b^2 x^n\right )\right ) \text {Subst}\left (\int \left (\frac {1}{a^3 b^3 x}-\frac {1}{a b^2 (a+b x)^3}-\frac {1}{a^2 b^2 (a+b x)^2}-\frac {1}{a^3 b^2 (a+b x)}\right ) \, dx,x,x^n\right )}{n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \\ & = \frac {1}{a^2 n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}+\frac {1}{2 a n \left (a+b x^n\right ) \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}+\frac {\left (a+b x^n\right ) \log (x)}{a^3 \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^3 n \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.50 \[ \int \frac {1}{x \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\frac {a \left (3 a+2 b x^n\right )+2 \left (a+b x^n\right )^2 \log \left (x^n\right )-2 \left (a+b x^n\right )^2 \log \left (a+b x^n\right )}{2 a^3 n \left (a+b x^n\right ) \sqrt {\left (a+b x^n\right )^2}} \]
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Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.65
method | result | size |
risch | \(\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, \ln \left (x \right )}{\left (a +b \,x^{n}\right ) a^{3}}+\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, \left (2 b \,x^{n}+3 a \right )}{2 \left (a +b \,x^{n}\right )^{3} a^{2} n}-\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, \ln \left (x^{n}+\frac {a}{b}\right )}{\left (a +b \,x^{n}\right ) a^{3} n}\) | \(104\) |
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Time = 0.30 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.67 \[ \int \frac {1}{x \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\frac {2 \, b^{2} n x^{2 \, n} \log \left (x\right ) + 2 \, a^{2} n \log \left (x\right ) + 3 \, a^{2} + 2 \, {\left (2 \, a b n \log \left (x\right ) + a b\right )} x^{n} - 2 \, {\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )} \log \left (b x^{n} + a\right )}{2 \, {\left (a^{3} b^{2} n x^{2 \, n} + 2 \, a^{4} b n x^{n} + a^{5} n\right )}} \]
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\[ \int \frac {1}{x \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\int \frac {1}{x \left (\left (a + b x^{n}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.44 \[ \int \frac {1}{x \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\frac {2 \, b x^{n} + 3 \, a}{2 \, {\left (a^{2} b^{2} n x^{2 \, n} + 2 \, a^{3} b n x^{n} + a^{4} n\right )}} + \frac {\log \left (x\right )}{a^{3}} - \frac {\log \left (\frac {b x^{n} + a}{b}\right )}{a^{3} n} \]
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\[ \int \frac {1}{x \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac {3}{2}} x} \,d x } \]
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Timed out. \[ \int \frac {1}{x \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\int \frac {1}{x\,{\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^{3/2}} \,d x \]
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