Optimal. Leaf size=40 \[ \frac{(c x)^{n+1} \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{b x^n}{a}\right )}{a c (n+1)} \]
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Rubi [A] time = 0.012494, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {364} \[ \frac{(c x)^{n+1} \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{b x^n}{a}\right )}{a c (n+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rubi steps
\begin{align*} \int \frac{(c x)^n}{a+b x^n} \, dx &=\frac{(c x)^{1+n} \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{b x^n}{a}\right )}{a c (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0083094, size = 36, normalized size = 0.9 \[ \frac{x (c x)^n \, _2F_1\left (1,1+\frac{1}{n};2+\frac{1}{n};-\frac{b x^n}{a}\right )}{a (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.067, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( cx \right ) ^{n}}{a+b{x}^{n}}}\, 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*} -a c^{n} \int \frac{1}{b^{2} x^{n} + a b}\,{d x} + \frac{c^{n} x}{b} \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{\left (c x\right )^{n}}{b x^{n} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.81903, size = 41, normalized size = 1.02 \begin{align*} - \frac{c^{n} x \Phi \left (\frac{a x^{- n} e^{i \pi }}{b}, 1, \frac{e^{i \pi }}{n}\right ) \Gamma \left (\frac{1}{n}\right )}{b n^{2} \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{\left (c x\right )^{n}}{b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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