Optimal. Leaf size=164 \[ \frac{b \left (b^2-2 a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 a^4 n}-\frac{b \log (x) \left (b^2-2 a c\right )}{a^4}-\frac{x^{-n} \left (b^2-a c\right )}{a^3 n}+\frac{b x^{-2 n}}{2 a^2 n}-\frac{\left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right )}{a^4 n \sqrt{b^2-4 a c}}-\frac{x^{-3 n}}{3 a n} \]
[Out]
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Rubi [A] time = 0.474117, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{b \left (b^2-2 a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 a^4 n}-\frac{b \log (x) \left (b^2-2 a c\right )}{a^4}-\frac{x^{-n} \left (b^2-a c\right )}{a^3 n}+\frac{b x^{-2 n}}{2 a^2 n}-\frac{\left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right )}{a^4 n \sqrt{b^2-4 a c}}-\frac{x^{-3 n}}{3 a n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 - 3*n)/(a + b*x^n + c*x^(2*n)),x]
[Out]
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Rubi in Sympy [A] time = 70.4593, size = 151, normalized size = 0.92 \[ - \frac{x^{- 3 n}}{3 a n} + \frac{b x^{- 2 n}}{2 a^{2} n} - \frac{x^{- n} \left (- a c + b^{2}\right )}{a^{3} n} - \frac{b \left (- 2 a c + b^{2}\right ) \log{\left (x^{n} \right )}}{a^{4} n} + \frac{b \left (- 2 a c + b^{2}\right ) \log{\left (a + b x^{n} + c x^{2 n} \right )}}{2 a^{4} n} - \frac{\left (2 a^{2} c^{2} - 4 a b^{2} c + b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{n}}{\sqrt{- 4 a c + b^{2}}} \right )}}{a^{4} n \sqrt{- 4 a c + b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1-3*n)/(a+b*x**n+c*x**(2*n)),x)
[Out]
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Mathematica [A] time = 0.527123, size = 136, normalized size = 0.83 \[ \frac{a x^{-3 n} \left (-2 a^2+3 a x^n \left (b+2 c x^n\right )-6 b^2 x^{2 n}\right )-\frac{6 \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{2 a x^{-n}+b}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+3 \left (b^3-2 a b c\right ) \log \left (x^{-2 n} \left (a+b x^n\right )+c\right )}{6 a^4 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 - 3*n)/(a + b*x^n + c*x^(2*n)),x]
[Out]
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Maple [B] time = 0.256, size = 1300, normalized size = 7.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1-3*n)/(a+b*x^n+c*x^(2*n)),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \frac{{\left (3 \, a b x^{n} - 2 \, a^{2} - 6 \,{\left (b^{2} - a c\right )} x^{2 \, n}\right )} x^{-3 \, n}}{6 \, a^{3} n} + \int -\frac{b^{3} - 2 \, a b c +{\left (b^{2} c - a c^{2}\right )} x^{n}}{a^{3} c x x^{2 \, n} + a^{3} b x x^{n} + a^{4} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(-3*n - 1)/(c*x^(2*n) + b*x^n + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.295659, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{b^{2} - 4 \, a c} a^{2} b x^{n} - 6 \,{\left (b^{3} - 2 \, a b c\right )} \sqrt{b^{2} - 4 \, a c} n x^{3 \, n} \log \left (x\right ) - 2 \, \sqrt{b^{2} - 4 \, a c} a^{3} + 3 \,{\left (b^{3} - 2 \, a b c\right )} \sqrt{b^{2} - 4 \, a c} x^{3 \, n} \log \left (c x^{2 \, n} + b x^{n} + a\right ) + 3 \,{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} x^{3 \, n} \log \left (\frac{2 \, \sqrt{b^{2} - 4 \, a c} c^{2} x^{2 \, n} - b^{3} + 4 \, a b c - 2 \,{\left (b^{2} c - 4 \, a c^{2} - \sqrt{b^{2} - 4 \, a c} b c\right )} x^{n} +{\left (b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2 \, n} + b x^{n} + a}\right ) - 6 \,{\left (a b^{2} - a^{2} c\right )} \sqrt{b^{2} - 4 \, a c} x^{2 \, n}}{6 \, \sqrt{b^{2} - 4 \, a c} a^{4} n x^{3 \, n}}, \frac{3 \, \sqrt{-b^{2} + 4 \, a c} a^{2} b x^{n} - 6 \,{\left (b^{3} - 2 \, a b c\right )} \sqrt{-b^{2} + 4 \, a c} n x^{3 \, n} \log \left (x\right ) - 2 \, \sqrt{-b^{2} + 4 \, a c} a^{3} + 3 \,{\left (b^{3} - 2 \, a b c\right )} \sqrt{-b^{2} + 4 \, a c} x^{3 \, n} \log \left (c x^{2 \, n} + b x^{n} + a\right ) + 6 \,{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} x^{3 \, n} \arctan \left (-\frac{2 \, \sqrt{-b^{2} + 4 \, a c} c x^{n} + \sqrt{-b^{2} + 4 \, a c} b}{b^{2} - 4 \, a c}\right ) - 6 \,{\left (a b^{2} - a^{2} c\right )} \sqrt{-b^{2} + 4 \, a c} x^{2 \, n}}{6 \, \sqrt{-b^{2} + 4 \, a c} a^{4} n x^{3 \, n}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(-3*n - 1)/(c*x^(2*n) + b*x^n + a),x, algorithm="fricas")
[Out]
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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**(-1-3*n)/(a+b*x**n+c*x**(2*n)),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-3 \, n - 1}}{c x^{2 \, n} + b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(-3*n - 1)/(c*x^(2*n) + b*x^n + a),x, algorithm="giac")
[Out]