Optimal. Leaf size=59 \[ x \left (a+b x^n\right )^{-\frac{1-n}{n}} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}-1} \, _2F_1\left (\frac{1}{n}-1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]
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Rubi [A] time = 0.0125478, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {246, 245} \[ x \left (a+b x^n\right )^{-\frac{1-n}{n}} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}-1} \, _2F_1\left (\frac{1}{n}-1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 246
Rule 245
Rubi steps
\begin{align*} \int \left (a+b x^n\right )^{-\frac{1-n}{n}} \, dx &=\left (\left (a+b x^n\right )^{-\frac{1-n}{n}} \left (1+\frac{b x^n}{a}\right )^{\frac{1-n}{n}}\right ) \int \left (1+\frac{b x^n}{a}\right )^{-\frac{1-n}{n}} \, dx\\ &=x \left (a+b x^n\right )^{-\frac{1-n}{n}} \left (1+\frac{b x^n}{a}\right )^{-1+\frac{1}{n}} \, _2F_1\left (-1+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )\\ \end{align*}
Mathematica [A] time = 0.0100094, size = 53, normalized size = 0.9 \[ a x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (\frac{1}{n}-1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.074, size = 0, normalized size = 0. \begin{align*} \int \left ( \left ( a+b{x}^{n} \right ) ^{{\frac{1-n}{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{\left (b x^{n} + a\right )}^{\frac{n - 1}{n}}\,{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 ({\left (b x^{n} + a\right )}^{\frac{n - 1}{n}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 10.7146, size = 42, normalized size = 0.71 \begin{align*} \frac{a a^{- \frac{1}{n}} x \Gamma \left (\frac{1}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{n}, -1 + \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{\left (b x^{n} + a\right )}^{\frac{n - 1}{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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