3.552 \(\int \frac{x^{-1+n}}{a+b x^n+c x^{2 n}} \, dx\)

Optimal. Leaf size=39 \[ -\frac{2 \tanh ^{-1}\left (\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right )}{n \sqrt{b^2-4 a c}} \]

[Out]

(-2*ArcTanh[(b + 2*c*x^n)/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*n)

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Rubi [A]  time = 0.0661677, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{2 \tanh ^{-1}\left (\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right )}{n \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + n)/(a + b*x^n + c*x^(2*n)),x]

[Out]

(-2*ArcTanh[(b + 2*c*x^n)/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]*n)

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Rubi in Sympy [A]  time = 11.0421, size = 37, normalized size = 0.95 \[ - \frac{2 \operatorname{atanh}{\left (\frac{b + 2 c x^{n}}{\sqrt{- 4 a c + b^{2}}} \right )}}{n \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+n)/(a+b*x**n+c*x**(2*n)),x)

[Out]

-2*atanh((b + 2*c*x**n)/sqrt(-4*a*c + b**2))/(n*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.0374889, size = 43, normalized size = 1.1 \[ \frac{2 \tan ^{-1}\left (\frac{b+2 c x^n}{\sqrt{4 a c-b^2}}\right )}{n \sqrt{4 a c-b^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + n)/(a + b*x^n + c*x^(2*n)),x]

[Out]

(2*ArcTan[(b + 2*c*x^n)/Sqrt[-b^2 + 4*a*c]])/(Sqrt[-b^2 + 4*a*c]*n)

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Maple [B]  time = 0.074, size = 113, normalized size = 2.9 \[ -{\frac{1}{n}\ln \left ({x}^{n}+{\frac{1}{2\,c} \left ( b\sqrt{-4\,ac+{b}^{2}}-4\,ac+{b}^{2} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}+{\frac{1}{n}\ln \left ({x}^{n}+{\frac{1}{2\,c} \left ( b\sqrt{-4\,ac+{b}^{2}}+4\,ac-{b}^{2} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+n)/(a+b*x^n+c*x^(2*n)),x)

[Out]

-1/(-4*a*c+b^2)^(1/2)/n*ln(x^n+1/2*(b*(-4*a*c+b^2)^(1/2)-4*a*c+b^2)/c/(-4*a*c+b^
2)^(1/2))+1/(-4*a*c+b^2)^(1/2)/n*ln(x^n+1/2*(b*(-4*a*c+b^2)^(1/2)+4*a*c-b^2)/c/(
-4*a*c+b^2)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{n - 1}}{c x^{2 \, n} + b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(n - 1)/(c*x^(2*n) + b*x^n + a),x, algorithm="maxima")

[Out]

integrate(x^(n - 1)/(c*x^(2*n) + b*x^n + a), x)

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Fricas [A]  time = 0.284568, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{2 \, \sqrt{b^{2} - 4 \, a c} c^{2} x^{2 \, n} - b^{3} + 4 \, a b c - 2 \,{\left (b^{2} c - 4 \, a c^{2} - \sqrt{b^{2} - 4 \, a c} b c\right )} x^{n} +{\left (b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2 \, n} + b x^{n} + a}\right )}{\sqrt{b^{2} - 4 \, a c} n}, \frac{2 \, \arctan \left (-\frac{2 \, \sqrt{-b^{2} + 4 \, a c} c x^{n} + \sqrt{-b^{2} + 4 \, a c} b}{b^{2} - 4 \, a c}\right )}{\sqrt{-b^{2} + 4 \, a c} n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(n - 1)/(c*x^(2*n) + b*x^n + a),x, algorithm="fricas")

[Out]

[log((2*sqrt(b^2 - 4*a*c)*c^2*x^(2*n) - b^3 + 4*a*b*c - 2*(b^2*c - 4*a*c^2 - sqr
t(b^2 - 4*a*c)*b*c)*x^n + (b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^(2*n) + b*x^n +
a))/(sqrt(b^2 - 4*a*c)*n), 2*arctan(-(2*sqrt(-b^2 + 4*a*c)*c*x^n + sqrt(-b^2 + 4
*a*c)*b)/(b^2 - 4*a*c))/(sqrt(-b^2 + 4*a*c)*n)]

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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+n)/(a+b*x**n+c*x**(2*n)),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.270978, size = 53, normalized size = 1.36 \[ \frac{2 \, \arctan \left (\frac{2 \, c x^{n} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(n - 1)/(c*x^(2*n) + b*x^n + a),x, algorithm="giac")

[Out]

2*arctan((2*c*x^n + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*n)