3.565 \(\int \frac{1}{x \left (a+b x^n+c x^{2 n}\right )} \, dx\)

Optimal. Leaf size=74 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right )}{a n \sqrt{b^2-4 a c}}-\frac{\log \left (a+b x^n+c x^{2 n}\right )}{2 a n}+\frac{\log (x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.151955, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \frac{b \tanh ^{-1}\left (\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right )}{a n \sqrt{b^2-4 a c}}-\frac{\log \left (a+b x^n+c x^{2 n}\right )}{2 a n}+\frac{\log (x)}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.3556, size = 66, normalized size = 0.89 \[ \frac{b \operatorname{atanh}{\left (\frac{b + 2 c x^{n}}{\sqrt{- 4 a c + b^{2}}} \right )}}{a n \sqrt{- 4 a c + b^{2}}} + \frac{\log{\left (x^{n} \right )}}{a n} - \frac{\log{\left (a + b x^{n} + c x^{2 n} \right )}}{2 a n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.176913, size = 74, normalized size = 1. \[ -\frac{\frac{2 b \tan ^{-1}\left (\frac{b+2 c x^n}{\sqrt{4 a c-b^2}}\right )}{n \sqrt{4 a c-b^2}}+\frac{\log \left (a+x^n \left (b+c x^n\right )\right )}{n}-2 \log (x)}{2 a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.112, size = 397, normalized size = 5.4 \[ 4\,{\frac{{n}^{2}\ln \left ( x \right ) ac}{4\,{a}^{2}c{n}^{2}-a{b}^{2}{n}^{2}}}-{\frac{{n}^{2}\ln \left ( x \right ){b}^{2}}{4\,{a}^{2}c{n}^{2}-a{b}^{2}{n}^{2}}}-2\,{\frac{c}{ \left ( 4\,ac-{b}^{2} \right ) n}\ln \left ({x}^{n}-1/2\,{\frac{-{b}^{2}+\sqrt{-4\,a{b}^{2}c+{b}^{4}}}{bc}} \right ) }+{\frac{{b}^{2}}{2\,a \left ( 4\,ac-{b}^{2} \right ) n}\ln \left ({x}^{n}-{\frac{1}{2\,bc} \left ( -{b}^{2}+\sqrt{-4\,a{b}^{2}c+{b}^{4}} \right ) } \right ) }+{\frac{1}{2\,a \left ( 4\,ac-{b}^{2} \right ) n}\ln \left ({x}^{n}-{\frac{1}{2\,bc} \left ( -{b}^{2}+\sqrt{-4\,a{b}^{2}c+{b}^{4}} \right ) } \right ) \sqrt{-4\,a{b}^{2}c+{b}^{4}}}-2\,{\frac{c}{ \left ( 4\,ac-{b}^{2} \right ) n}\ln \left ({x}^{n}+1/2\,{\frac{{b}^{2}+\sqrt{-4\,a{b}^{2}c+{b}^{4}}}{bc}} \right ) }+{\frac{{b}^{2}}{2\,a \left ( 4\,ac-{b}^{2} \right ) n}\ln \left ({x}^{n}+{\frac{1}{2\,bc} \left ({b}^{2}+\sqrt{-4\,a{b}^{2}c+{b}^{4}} \right ) } \right ) }-{\frac{1}{2\,a \left ( 4\,ac-{b}^{2} \right ) n}\ln \left ({x}^{n}+{\frac{1}{2\,bc} \left ({b}^{2}+\sqrt{-4\,a{b}^{2}c+{b}^{4}} \right ) } \right ) \sqrt{-4\,a{b}^{2}c+{b}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*x^n+c*x^(2*n)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.297817, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{b^{2} - 4 \, a c} n \log \left (x\right ) + b \log \left (\frac{2 \, \sqrt{b^{2} - 4 \, a c} c^{2} x^{2 \, n} + b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2} + \sqrt{b^{2} - 4 \, a c} b c\right )} x^{n} +{\left (b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2 \, n} + b x^{n} + a}\right ) - \sqrt{b^{2} - 4 \, a c} \log \left (c x^{2 \, n} + b x^{n} + a\right )}{2 \, \sqrt{b^{2} - 4 \, a c} a n}, \frac{2 \, \sqrt{-b^{2} + 4 \, a c} n \log \left (x\right ) - 2 \, b \arctan \left (-\frac{2 \, \sqrt{-b^{2} + 4 \, a c} c x^{n} + \sqrt{-b^{2} + 4 \, a c} b}{b^{2} - 4 \, a c}\right ) - \sqrt{-b^{2} + 4 \, a c} \log \left (c x^{2 \, n} + b x^{n} + a\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} a n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(2*sqrt(b^2 - 4*a*c)*n*log(x) + b*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 + sqrt(b^2 - 4*a*c)*b*c)*x^n + (b^2 - 2*a*c)*sq
rt(b^2 - 4*a*c))/(c*x^(2*n) + b*x^n + a)) - sqrt(b^2 - 4*a*c)*log(c*x^(2*n) + b*
x^n + a))/(sqrt(b^2 - 4*a*c)*a*n), 1/2*(2*sqrt(-b^2 + 4*a*c)*n*log(x) - 2*b*arct
an(-(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)*log(c*x^(2*n) + b*x^n + a))/(sqrt(-b^2 + 4*a*c)*a*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(1/x/(a+b*x**n+c*x**(2*n)),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2 \, n} + b x^{n} + a\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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