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 = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, 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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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 \[ \int \frac {\sqrt {a x^{\frac {1}{2} \, n}}}{\sqrt {x^{n} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 37, normalized size = 0.71 \[ \frac {4 \sqrt {a \,x^{\frac {n}{2}}}\, x \hypergeom \left (\left [\frac {1}{2}, \frac {1}{n}+\frac {1}{4}\right ], \left [\frac {1}{n}+\frac {5}{4}\right ], -x^{n}\right )}{n +4} \]
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 \[ \int \frac {\sqrt {a x^{\frac {1}{2} \, n}}}{\sqrt {x^{n} + 1}}\,{d x} \]
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 \[ \int \frac {\sqrt {a\,x^{n/2}}}{\sqrt {x^n+1}} \,d x \]
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 \[ \int \frac {\sqrt {a x^{\frac {n}{2}}}}{\sqrt {x^{n} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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