Optimal. Leaf size=24 \[ \frac{x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2} \]
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Rubi [A] time = 0.0053188, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {245} \[ \frac{x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2} \]
Antiderivative was successfully verified.
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Rule 245
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^n\right )^2} \, dx &=\frac{x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0019092, size = 24, normalized size = 1. \[ \frac{x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.054, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{n} \right ) ^{-2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\left (n - 1\right )} \int \frac{1}{a b n x^{n} + a^{2} n}\,{d x} + \frac{x}{a b n x^{n} + a^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.24888, size = 257, normalized size = 10.71 \begin{align*} \frac{n x \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{1}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac{1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac{1}{n}\right )\right )} + \frac{n x \Gamma \left (\frac{1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac{1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac{1}{n}\right )\right )} - \frac{x \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{1}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac{1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac{1}{n}\right )\right )} + \frac{b n x x^{n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{1}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac{1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac{1}{n}\right )\right )} - \frac{b x x^{n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{1}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac{1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac{1}{n}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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