Optimal. Leaf size=136 \[ -\frac{c x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2} \]
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Rubi [A] time = 0.115744, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{c x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{c x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2} \]
Antiderivative was successfully verified.
[In] Int[x/(a + b*x^n + c*x^(2*n)),x]
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Rubi in Sympy [A] time = 22.1616, size = 112, normalized size = 0.82 \[ - \frac{c x^{2}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{2 c x^{n}}{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{- 4 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}} - \frac{c x^{2}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{2 c x^{n}}{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{- 4 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(a+b*x**n+c*x**(2*n)),x)
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Mathematica [A] time = 1.43682, size = 263, normalized size = 1.93 \[ -c x^2 \left (\frac{1-\left (\frac{x^n}{x^n-\frac{\sqrt{b^2-4 a c}-b}{2 c}}\right )^{-2/n} \, _2F_1\left (-\frac{2}{n},-\frac{2}{n};\frac{n-2}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}+\frac{1-4^{-1/n} \left (\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right )^{-2/n} \, _2F_1\left (-\frac{2}{n},-\frac{2}{n};\frac{n-2}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x/(a + b*x^n + c*x^(2*n)),x]
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Maple [F] time = 0.04, size = 0, normalized size = 0. \[ \int{\frac{x}{a+b{x}^{n}+c{x}^{2\,n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(a+b*x^n+c*x^(2*n)),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{c x^{2 \, n} + b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(c*x^(2*n) + b*x^n + a),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x}{c x^{2 \, n} + b x^{n} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(c*x^(2*n) + b*x^n + a),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a+b*x**n+c*x**(2*n)),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{c x^{2 \, n} + b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(c*x^(2*n) + b*x^n + a),x, algorithm="giac")
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