Integrand size = 32, antiderivative size = 48 \[ \int \frac {x^{-1+2 n}}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\frac {x^{2 n}}{2 a n \left (a+b x^n\right ) \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1369, 270} \[ \int \frac {x^{-1+2 n}}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\frac {x^{2 n}}{2 a n \left (a+b x^n\right ) \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Rule 270
Rule 1369
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^2 \left (a b+b^2 x^n\right )\right ) \int \frac {x^{-1+2 n}}{\left (a b+b^2 x^n\right )^3} \, dx}{\sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \\ & = \frac {x^{2 n}}{2 a n \left (a+b x^n\right ) \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.83 \[ \int \frac {x^{-1+2 n}}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\frac {\left (-a-2 b x^n\right ) \left (a+b x^n\right )}{2 b^2 n \left (\left (a+b x^n\right )^2\right )^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.77
method | result | size |
risch | \(-\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, \left (2 b \,x^{n}+a \right )}{2 \left (a +b \,x^{n}\right )^{3} b^{2} n}\) | \(37\) |
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Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.85 \[ \int \frac {x^{-1+2 n}}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=-\frac {2 \, b x^{n} + a}{2 \, {\left (b^{4} n x^{2 \, n} + 2 \, a b^{3} n x^{n} + a^{2} b^{2} n\right )}} \]
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\[ \int \frac {x^{-1+2 n}}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\int \frac {x^{2 n - 1}}{\left (\left (a + b x^{n}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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none
Time = 0.18 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.85 \[ \int \frac {x^{-1+2 n}}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=-\frac {2 \, b x^{n} + a}{2 \, {\left (b^{4} n x^{2 \, n} + 2 \, a b^{3} n x^{n} + a^{2} b^{2} n\right )}} \]
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\[ \int \frac {x^{-1+2 n}}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\int { \frac {x^{2 \, n - 1}}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^{-1+2 n}}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\int \frac {x^{2\,n-1}}{{\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^{3/2}} \,d x \]
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