Integrand size = 23, antiderivative size = 52 \[ \int \frac {\sqrt {a x^{n/2}}}{\sqrt {1+x^n}} \, dx=\frac {4 x \sqrt {a x^{n/2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} \left (1+\frac {4}{n}\right ),\frac {1}{4} \left (5+\frac {4}{n}\right ),-x^n\right )}{4+n} \]
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Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {15, 371} \[ \int \frac {\sqrt {a x^{n/2}}}{\sqrt {1+x^n}} \, dx=\frac {4 x \sqrt {a x^{n/2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} \left (1+\frac {4}{n}\right ),\frac {1}{4} \left (5+\frac {4}{n}\right ),-x^n\right )}{n+4} \]
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Rule 15
Rule 371
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {a x^{n/2}}}{\sqrt {1+x^n}} \, dx=\frac {4 x \sqrt {a x^{n/2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4}+\frac {1}{n},\frac {5}{4}+\frac {1}{n},-x^n\right )}{4+n} \]
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Time = 1.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.71
method | result | size |
meijerg | \(\frac {4 x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {1}{4}+\frac {1}{n};\frac {5}{4}+\frac {1}{n};-x^{n}\right ) \sqrt {a \,x^{\frac {n}{2}}}}{4+n}\) | \(37\) |
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Exception generated. \[ \int \frac {\sqrt {a x^{n/2}}}{\sqrt {1+x^n}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\sqrt {a x^{n/2}}}{\sqrt {1+x^n}} \, dx=\int \frac {\sqrt {a x^{\frac {n}{2}}}}{\sqrt {x^{n} + 1}}\, dx \]
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\[ \int \frac {\sqrt {a x^{n/2}}}{\sqrt {1+x^n}} \, dx=\int { \frac {\sqrt {a x^{\frac {1}{2} \, n}}}{\sqrt {x^{n} + 1}} \,d x } \]
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\[ \int \frac {\sqrt {a x^{n/2}}}{\sqrt {1+x^n}} \, dx=\int { \frac {\sqrt {a x^{\frac {1}{2} \, n}}}{\sqrt {x^{n} + 1}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a x^{n/2}}}{\sqrt {1+x^n}} \, dx=\int \frac {\sqrt {a\,x^{n/2}}}{\sqrt {x^n+1}} \,d x \]
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