Integrand size = 36, antiderivative size = 52 \[ \int \left (a^2+b^2 x^{-\frac {2}{1+2 p}}+2 a b x^{-\frac {1}{1+2 p}}\right )^p \, dx=\frac {x \left (a+b x^{\frac {1}{-1-2 p}}\right ) \left (a^2+2 a b x^{\frac {1}{-1-2 p}}+b^2 x^{-\frac {2}{1+2 p}}\right )^p}{a} \]
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Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1357, 197} \[ \int \left (a^2+b^2 x^{-\frac {2}{1+2 p}}+2 a b x^{-\frac {1}{1+2 p}}\right )^p \, dx=\frac {x \left (a+b x^{\frac {1}{-2 p-1}}\right ) \left (a^2+2 a b x^{\frac {1}{-2 p-1}}+b^2 x^{-\frac {2}{2 p+1}}\right )^p}{a} \]
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Rule 197
Rule 1357
Rubi steps \begin{align*} \text {integral}& = \left (\left (a^2+b^2 x^{-\frac {2}{1+2 p}}+2 a b x^{-\frac {1}{1+2 p}}\right )^p \left (2 a b+2 b^2 x^{-\frac {1}{1+2 p}}\right )^{-2 p}\right ) \int \left (2 a b+2 b^2 x^{-\frac {1}{1+2 p}}\right )^{2 p} \, dx \\ & = \frac {x \left (a+b x^{\frac {1}{-1-2 p}}\right ) \left (a^2+2 a b x^{\frac {1}{-1-2 p}}+b^2 x^{-\frac {2}{1+2 p}}\right )^p}{a} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.12 \[ \int \left (a^2+b^2 x^{-\frac {2}{1+2 p}}+2 a b x^{-\frac {1}{1+2 p}}\right )^p \, dx=\frac {x^{\frac {2 p}{1+2 p}} \left (b+a x^{\frac {1}{1+2 p}}\right ) \left (x^{-\frac {2}{1+2 p}} \left (b+a x^{\frac {1}{1+2 p}}\right )^2\right )^p}{a} \]
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\[\int \left (a^{2}+b^{2} x^{-\frac {2}{1+2 p}}+2 a b \,x^{-\frac {1}{1+2 p}}\right )^{p}d x\]
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none
Time = 0.27 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.52 \[ \int \left (a^2+b^2 x^{-\frac {2}{1+2 p}}+2 a b x^{-\frac {1}{1+2 p}}\right )^p \, dx=\frac {{\left (a x x^{\left (\frac {1}{2 \, p + 1}\right )} + b x\right )} \left (\frac {a^{2} x^{\frac {2}{2 \, p + 1}} + 2 \, a b x^{\left (\frac {1}{2 \, p + 1}\right )} + b^{2}}{x^{\frac {2}{2 \, p + 1}}}\right )^{p}}{a x^{\left (\frac {1}{2 \, p + 1}\right )}} \]
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Timed out. \[ \int \left (a^2+b^2 x^{-\frac {2}{1+2 p}}+2 a b x^{-\frac {1}{1+2 p}}\right )^p \, dx=\text {Timed out} \]
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\[ \int \left (a^2+b^2 x^{-\frac {2}{1+2 p}}+2 a b x^{-\frac {1}{1+2 p}}\right )^p \, dx=\int { {\left (a^{2} + \frac {b^{2}}{x^{\frac {2}{2 \, p + 1}}} + \frac {2 \, a b}{x^{\left (\frac {1}{2 \, p + 1}\right )}}\right )}^{p} \,d x } \]
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\[ \int \left (a^2+b^2 x^{-\frac {2}{1+2 p}}+2 a b x^{-\frac {1}{1+2 p}}\right )^p \, dx=\int { {\left (a^{2} + \frac {b^{2}}{x^{\frac {2}{2 \, p + 1}}} + \frac {2 \, a b}{x^{\left (\frac {1}{2 \, p + 1}\right )}}\right )}^{p} \,d x } \]
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Timed out. \[ \int \left (a^2+b^2 x^{-\frac {2}{1+2 p}}+2 a b x^{-\frac {1}{1+2 p}}\right )^p \, dx=\int {\left (a^2+\frac {b^2}{x^{\frac {2}{2\,p+1}}}+\frac {2\,a\,b}{x^{\frac {1}{2\,p+1}}}\right )}^p \,d x \]
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