Optimal. Leaf size=52 \[ \frac {4 x \sqrt {a x^{n/2}} \, _2F_1\left (\frac {1}{2},\frac {1}{4} \left (1+\frac {4}{n}\right );\frac {1}{4} \left (5+\frac {4}{n}\right );-x^n\right )}{4+n} \]
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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, 371}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 371
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 = 0.03, size = 44, normalized size = 0.85 \begin {gather*} \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 )}{4+n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 37, normalized size = 0.71
method | result | size |
meijerg | \(\frac {4 x \hypergeom \left (\left [\frac {1}{2}, \frac {1}{4}+\frac {1}{n}\right ], \left [\frac {5}{4}+\frac {1}{n}\right ], -x^{n}\right ) \sqrt {a \,x^{\frac {n}{2}}}}{4+n}\) | \(37\) |
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^{\frac {n}{2}}}}{\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 = -0.02 \begin {gather*} \int \frac {\sqrt {a\,x^{n/2}}}{\sqrt {x^n+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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