Optimal. Leaf size=37 \[ \frac{x \sqrt{a x^{2 n}} \, _2F_1\left (\frac{1}{2},1+\frac{1}{n};2+\frac{1}{n};-x^n\right )}{n+1} \]
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Rubi [A] time = 0.0122416, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {15, 364} \[ \frac{x \sqrt{a x^{2 n}} \, _2F_1\left (\frac{1}{2},1+\frac{1}{n};2+\frac{1}{n};-x^n\right )}{n+1} \]
Antiderivative was successfully verified.
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Rule 15
Rule 364
Rubi steps
\begin{align*} \int \frac{\sqrt{a x^{2 n}}}{\sqrt{1+x^n}} \, dx &=\left (x^{-n} \sqrt{a x^{2 n}}\right ) \int \frac{x^n}{\sqrt{1+x^n}} \, dx\\ &=\frac{x \sqrt{a x^{2 n}} \, _2F_1\left (\frac{1}{2},1+\frac{1}{n};2+\frac{1}{n};-x^n\right )}{1+n}\\ \end{align*}
Mathematica [A] time = 0.0119701, size = 37, normalized size = 1. \[ \frac{x \sqrt{a x^{2 n}} \, _2F_1\left (\frac{1}{2},1+\frac{1}{n};2+\frac{1}{n};-x^n\right )}{n+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{a{x}^{2\,n}}{\frac{1}{\sqrt{1+{x}^{n}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x^{2 \, n}}}{\sqrt{x^{n} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x^{2 n}}}{\sqrt{x^{n} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x^{2 \, n}}}{\sqrt{x^{n} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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