Optimal. Leaf size=50 \[ x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]
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Rubi [A] time = 0.0107844, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {246, 245} \[ x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 246
Rule 245
Rubi steps
\begin{align*} \int \left (a+b x^n\right )^{-1/n} \, dx &=\left (\left (a+b x^n\right )^{-1/n} \left (1+\frac{b x^n}{a}\right )^{\frac{1}{n}}\right ) \int \left (1+\frac{b x^n}{a}\right )^{-1/n} \, dx\\ &=x \left (a+b x^n\right )^{-1/n} \left (1+\frac{b x^n}{a}\right )^{\frac{1}{n}} \, _2F_1\left (\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )\\ \end{align*}
Mathematica [A] time = 0.0024129, size = 50, normalized size = 1. \[ x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left ( \sqrt [n]{a+b{x}^{n}} \right ) ^{-1}\, 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*} \int \frac{1}{{\left (b x^{n} + a\right )}^{\left (\frac{1}{n}\right )}}\,{d x} \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}{{\left (b x^{n} + a\right )}^{\left (\frac{1}{n}\right )}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 18.6021, size = 39, normalized size = 0.78 \begin{align*} \frac{a^{- \frac{1}{n}} x \Gamma \left (\frac{1}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{n}, \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 )} \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 )}^{\left (\frac{1}{n}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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