Optimal. Leaf size=52 \[ \frac{4 x \sqrt{a x^{n/2}} \, _2F_1\left (\frac{1}{2},\frac{1}{4} \left (1+\frac{4}{n}\right );\frac{1}{4} \left (5+\frac{4}{n}\right );-x^n\right )}{n+4} \]
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Rubi [A] time = 0.0148895, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {15, 364} \[ \frac{4 x \sqrt{a x^{n/2}} \, _2F_1\left (\frac{1}{2},\frac{1}{4} \left (1+\frac{4}{n}\right );\frac{1}{4} \left (5+\frac{4}{n}\right );-x^n\right )}{n+4} \]
Antiderivative was successfully verified.
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Rule 15
Rule 364
Rubi steps
\begin{align*} \int \frac{\sqrt{a x^{n/2}}}{\sqrt{1+x^n}} \, dx &=\left (x^{-n/4} \sqrt{a x^{n/2}}\right ) \int \frac{x^{n/4}}{\sqrt{1+x^n}} \, dx\\ &=\frac{4 x \sqrt{a x^{n/2}} \, _2F_1\left (\frac{1}{2},\frac{1}{4} \left (1+\frac{4}{n}\right );\frac{1}{4} \left (5+\frac{4}{n}\right );-x^n\right )}{4+n}\\ \end{align*}
Mathematica [A] time = 0.0115588, size = 44, normalized size = 0.85 \[ \frac{4 x \sqrt{a x^{n/2}} \, _2F_1\left (\frac{1}{2},\frac{1}{4}+\frac{1}{n};\frac{5}{4}+\frac{1}{n};-x^n\right )}{n+4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.064, size = 37, normalized size = 0.7 \begin{align*} 4\,{\frac{x{\mbox{$_2$F$_1$}(1/2,1/4+{n}^{-1};\,5/4+{n}^{-1};\,-{x}^{n})}\sqrt{a{x}^{n/2}}}{4+n}} \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^{\frac{1}{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^{\frac{n}{2}}}}{\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^{\frac{1}{2} \, n}}}{\sqrt{x^{n} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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