Integrand size = 13, antiderivative size = 48 \[ \int \frac {x}{\sqrt {a+b x^n}} \, dx=\frac {x^2 \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+\frac {2}{n},\frac {2+n}{n},-\frac {b x^n}{a}\right )}{2 a} \]
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Time = 0.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.19, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {372, 371} \[ \int \frac {x}{\sqrt {a+b x^n}} \, dx=\frac {x^2 \sqrt {\frac {b x^n}{a}+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{n},\frac {n+2}{n},-\frac {b x^n}{a}\right )}{2 \sqrt {a+b x^n}} \]
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Rule 371
Rule 372
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+\frac {b x^n}{a}} \int \frac {x}{\sqrt {1+\frac {b x^n}{a}}} \, dx}{\sqrt {a+b x^n}} \\ & = \frac {x^2 \sqrt {1+\frac {b x^n}{a}} \, _2F_1\left (\frac {1}{2},\frac {2}{n};\frac {2+n}{n};-\frac {b x^n}{a}\right )}{2 \sqrt {a+b x^n}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.19 \[ \int \frac {x}{\sqrt {a+b x^n}} \, dx=\frac {x^2 \sqrt {1+\frac {b x^n}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{n},1+\frac {2}{n},-\frac {b x^n}{a}\right )}{2 \sqrt {a+b x^n}} \]
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\[\int \frac {x}{\sqrt {a +b \,x^{n}}}d x\]
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Exception generated. \[ \int \frac {x}{\sqrt {a+b x^n}} \, dx=\text {Exception raised: TypeError} \]
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Result contains complex when optimal does not.
Time = 0.59 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.06 \[ \int \frac {x}{\sqrt {a+b x^n}} \, dx=\frac {a^{\frac {2}{n}} a^{- \frac {1}{2} - \frac {2}{n}} x^{2} \Gamma \left (\frac {2}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {2}{n} \\ 1 + \frac {2}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {2}{n}\right )} \]
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\[ \int \frac {x}{\sqrt {a+b x^n}} \, dx=\int { \frac {x}{\sqrt {b x^{n} + a}} \,d x } \]
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\[ \int \frac {x}{\sqrt {a+b x^n}} \, dx=\int { \frac {x}{\sqrt {b x^{n} + a}} \,d x } \]
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Timed out. \[ \int \frac {x}{\sqrt {a+b x^n}} \, dx=\int \frac {x}{\sqrt {a+b\,x^n}} \,d x \]
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