Optimal. Leaf size=37 \[ \frac {x \sqrt {a x^{2 n}} \, _2F_1\left (\frac {1}{2},1+\frac {1}{n};2+\frac {1}{n};-x^n\right )}{1+n} \]
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Rubi [A]
time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, 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, 371}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 371
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*}
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Mathematica [A]
time = 0.03, size = 37, normalized size = 1.00 \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 36, normalized size = 0.97
method | result | size |
meijerg | \(\frac {x \hypergeom \left (\left [\frac {1}{2}, 1+\frac {1}{n}\right ], \left [2+\frac {1}{n}\right ], -x^{n}\right ) \sqrt {a \,x^{2 n}}}{1+n}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{2 n}}}{\sqrt {x^{n} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {a\,x^{2\,n}}}{\sqrt {x^n+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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