Integrand size = 24, antiderivative size = 57 \[ \int \frac {1}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\frac {x \left (a+b x^n\right )^3 \operatorname {Hypergeometric2F1}\left (3,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^3 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1357, 251} \[ \int \frac {1}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\frac {x \left (a+b x^n\right )^3 \operatorname {Hypergeometric2F1}\left (3,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^3 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \]
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Rule 251
Rule 1357
Rubi steps \begin{align*} \text {integral}& = \frac {\left (2 a b+2 b^2 x^n\right )^3 \int \frac {1}{\left (2 a b+2 b^2 x^n\right )^3} \, dx}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \\ & = \frac {x \left (a+b x^n\right )^3 \, _2F_1\left (3,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^3 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\frac {x \left (a+b x^n\right )^3 \operatorname {Hypergeometric2F1}\left (3,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^3 \left (\left (a+b x^n\right )^2\right )^{3/2}} \]
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\[\int \frac {1}{\left (a^{2}+2 a b \,x^{n}+b^{2} x^{2 n}\right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {1}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\int \frac {1}{\left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx=\int \frac {1}{{\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^{3/2}} \,d x \]
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