Optimal. Leaf size=46 \[ \frac {2 x \sqrt {x^n} \, _2F_1\left (\frac {1}{2},\frac {1}{2} \left (1+\frac {2}{n}\right );\frac {1}{2} \left (3+\frac {2}{n}\right );-x^n\right )}{n+2} \]
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Rubi [A] time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1958, 15, 364} \[ \frac {2 x \sqrt {x^n} \, _2F_1\left (\frac {1}{2},\frac {1}{2} \left (1+\frac {2}{n}\right );\frac {1}{2} \left (3+\frac {2}{n}\right );-x^n\right )}{n+2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 364
Rule 1958
Rubi steps
\begin {align*} \int \sqrt {\frac {x^n}{1+x^n}} \, dx &=\int \frac {\sqrt {x^n}}{\sqrt {1+x^n}} \, dx\\ &=\left (x^{-n/2} \sqrt {x^n}\right ) \int \frac {x^{n/2}}{\sqrt {1+x^n}} \, dx\\ &=\frac {2 x \sqrt {x^n} \, _2F_1\left (\frac {1}{2},\frac {1}{2} \left (1+\frac {2}{n}\right );\frac {1}{2} \left (3+\frac {2}{n}\right );-x^n\right )}{2+n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 0.83 \[ \frac {2 x \sqrt {x^n} \, _2F_1\left (\frac {1}{2},\frac {1}{2}+\frac {1}{n};\frac {3}{2}+\frac {1}{n};-x^n\right )}{n+2} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \sqrt {\frac {x^{n}}{x^{n} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \sqrt {\frac {x^{n}}{x^{n}+1}}\, dx \]
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 \[ \int \sqrt {\frac {x^{n}}{x^{n} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {\frac {x^n}{x^n+1}} \,d x \]
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 \[ \int \sqrt {\frac {x^{n}}{x^{n} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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