Integrand size = 24, antiderivative size = 88 \[ \int \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, dx=\frac {a x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n}+\frac {b^2 x^{1+n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(1+n) \left (a b+b^2 x^n\right )} \]
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Time = 0.01 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {1357} \[ \int \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, dx=\frac {b^2 x^{n+1} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(n+1) \left (a b+b^2 x^n\right )}+\frac {a x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n} \]
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Rule 1357
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \int \left (2 a b+2 b^2 x^n\right ) \, dx}{2 a b+2 b^2 x^n} \\ & = \frac {a x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n}+\frac {b^2 x^{1+n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(1+n) \left (a b+b^2 x^n\right )} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.44 \[ \int \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, dx=\frac {x \sqrt {\left (a+b x^n\right )^2} \left (a+a n+b x^n\right )}{(1+n) \left (a+b x^n\right )} \]
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Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64
method | result | size |
risch | \(\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, a x}{a +b \,x^{n}}+\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, b x \,x^{n}}{\left (a +b \,x^{n}\right ) \left (1+n \right )}\) | \(56\) |
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Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.23 \[ \int \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, dx=\frac {b x x^{n} + {\left (a n + a\right )} x}{n + 1} \]
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\[ \int \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, dx=\int \sqrt {a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.22 \[ \int \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, dx=\frac {a {\left (n + 1\right )} x + b x x^{n}}{n + 1} \]
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Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.28 \[ \int \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, dx={\left (a x + \frac {b x^{n + 1}}{n + 1}\right )} \mathrm {sgn}\left (b x^{n} + a\right ) \]
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Timed out. \[ \int \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, dx=\int \sqrt {a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n} \,d x \]
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