Optimal. Leaf size=34 \[ \frac {2 x^{1-n} \sqrt {a x^{2 n}} \sqrt {1+x^n}}{2+n} \]
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Rubi [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, antiderivative size = 80, normalized size of antiderivative = 2.35, number of steps
used = 5, number of rules used = 3, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {15, 371, 251}
\begin {gather*} \frac {2 x^{1-n} \sqrt {a x^{2 n}} \, _2F_1\left (\frac {1}{2},\frac {1}{n};1+\frac {1}{n};-x^n\right )}{n+2}+\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 251
Rule 371
Rubi steps
\begin {align*} \int \left (\frac {\sqrt {a x^{2 n}}}{\sqrt {1+x^n}}+\frac {2 x^{-n} \sqrt {a x^{2 n}}}{(2+n) \sqrt {1+x^n}}\right ) \, dx &=\frac {2 \int \frac {x^{-n} \sqrt {a x^{2 n}}}{\sqrt {1+x^n}} \, dx}{2+n}+\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 {\left (2 x^{-n} \sqrt {a x^{2 n}}\right ) \int \frac {1}{\sqrt {1+x^n}} \, dx}{2+n}\\ &=\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}+\frac {2 x^{1-n} \sqrt {a x^{2 n}} \, _2F_1\left (\frac {1}{2},\frac {1}{n};1+\frac {1}{n};-x^n\right )}{2+n}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 33, normalized size = 0.97 \begin {gather*} \frac {2 a x^{1+n} \sqrt {1+x^n}}{(2+n) \sqrt {a x^{2 n}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 30, normalized size = 0.88
method | result | size |
risch | \(\frac {2 x \sqrt {1+x^{n}}\, \sqrt {a \,x^{2 n}}\, x^{-n}}{2+n}\) | \(30\) |
meijerg | \(\frac {x \hypergeom \left (\left [\frac {1}{2}, 1+\frac {1}{n}\right ], \left [2+\frac {1}{n}\right ], -x^{n}\right ) \sqrt {a \,x^{2 n}}}{1+n}+\frac {2 \sqrt {a \,x^{2 n}}\, x^{1-n} \hypergeom \left (\left [\frac {1}{2}, \frac {1}{n}\right ], \left [1+\frac {1}{n}\right ], -x^{n}\right )}{2+n}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 18, normalized size = 0.53 \begin {gather*} \frac {2 \, \sqrt {a} \sqrt {x^{n} + 1} x}{n + 2} \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*} \frac {\int \frac {2 \sqrt {a x^{2 n}}}{\sqrt {x^{n} + 1}}\, dx + \int \frac {n \sqrt {a x^{2 n}}}{\sqrt {x^{n} + 1}}\, dx + \int \frac {2 x^{- n} \sqrt {a x^{2 n}}}{\sqrt {x^{n} + 1}}\, dx}{n + 2} \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 [B]
time = 2.89, size = 43, normalized size = 1.26 \begin {gather*} \frac {\sqrt {a\,x^{2\,n}}\,\left (\frac {2\,x}{n+2}+\frac {2\,x^{n+1}}{n+2}\right )}{x^n\,\sqrt {x^n+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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