Integrand size = 11, antiderivative size = 39 \[ \int \frac {1}{\left (a+b x^n\right )^{5/2}} \, dx=\frac {x \operatorname {Hypergeometric2F1}\left (1,-\frac {3}{2}+\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a \left (a+b x^n\right )^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.31, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {252, 251} \[ \int \frac {1}{\left (a+b x^n\right )^{5/2}} \, dx=\frac {x \sqrt {\frac {b x^n}{a}+1} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^2 \sqrt {a+b x^n}} \]
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Rule 251
Rule 252
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+\frac {b x^n}{a}} \int \frac {1}{\left (1+\frac {b x^n}{a}\right )^{5/2}} \, dx}{a^2 \sqrt {a+b x^n}} \\ & = \frac {x \sqrt {1+\frac {b x^n}{a}} \, _2F_1\left (\frac {5}{2},\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2 \sqrt {a+b x^n}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.31 \[ \int \frac {1}{\left (a+b x^n\right )^{5/2}} \, dx=\frac {x \sqrt {1+\frac {b x^n}{a}} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^2 \sqrt {a+b x^n}} \]
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\[\int \frac {1}{\left (a +b \,x^{n}\right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{\left (a+b x^n\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Result contains complex when optimal does not.
Time = 1.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\left (a+b x^n\right )^{5/2}} \, dx=\frac {a^{\frac {1}{n}} a^{- \frac {5}{2} - \frac {1}{n}} x \Gamma \left (\frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, \frac {1}{n} \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {1}{n}\right )} \]
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\[ \int \frac {1}{\left (a+b x^n\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (a+b x^n\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{n} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 5.85 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (a+b x^n\right )^{5/2}} \, dx=\frac {x\,{\left (\frac {b\,x^n}{a}+1\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {5}{2},\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,x^n}{a}\right )}{{\left (a+b\,x^n\right )}^{5/2}} \]
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