Optimal. Leaf size=44 \[ \frac{(c x)^{n+3} \, _2F_1\left (1,\frac{n+3}{n};2+\frac{3}{n};-\frac{b x^n}{a}\right )}{a c (n+3)} \]
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Rubi [A] time = 0.0132325, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {364} \[ \frac{(c x)^{n+3} \, _2F_1\left (1,\frac{n+3}{n};2+\frac{3}{n};-\frac{b x^n}{a}\right )}{a c (n+3)} \]
Antiderivative was successfully verified.
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Rule 364
Rubi steps
\begin{align*} \int \frac{(c x)^{2+n}}{a+b x^n} \, dx &=\frac{(c x)^{3+n} \, _2F_1\left (1,\frac{3+n}{n};2+\frac{3}{n};-\frac{b x^n}{a}\right )}{a c (3+n)}\\ \end{align*}
Mathematica [A] time = 0.0113546, size = 44, normalized size = 1. \[ \frac{x (c x)^{n+2} \, _2F_1\left (1,\frac{n+3}{n};\frac{n+3}{n}+1;-\frac{b x^n}{a}\right )}{a (n+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( cx \right ) ^{2+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*} \frac{c^{n + 2} x^{3}}{3 \, b} - a c^{n + 2} \int \frac{x^{2}}{b^{2} x^{n} + a b}\,{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{\left (c x\right )^{n + 2}}{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 = 3.6304, size = 97, normalized size = 2.2 \begin{align*} \frac{c^{2} c^{n} x^{3} x^{n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, 1 + \frac{3}{n}\right ) \Gamma \left (1 + \frac{3}{n}\right )}{a n \Gamma \left (2 + \frac{3}{n}\right )} + \frac{3 c^{2} c^{n} x^{3} x^{n} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, 1 + \frac{3}{n}\right ) \Gamma \left (1 + \frac{3}{n}\right )}{a n^{2} \Gamma \left (2 + \frac{3}{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 + 2}}{b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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