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

Optimal. Leaf size=98 \[ -\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right )}{a^2 n \sqrt{b^2-4 a c}}+\frac{b \log \left (a+b x^n+c x^{2 n}\right )}{2 a^2 n}-\frac{b \log (x)}{a^2}-\frac{x^{-n}}{a n} \]

[Out]

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

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Rubi [A]  time = 0.269766, antiderivative size = 98, 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 \[ -\frac{\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^n}{\sqrt{b^2-4 a c}}\right )}{a^2 n \sqrt{b^2-4 a c}}+\frac{b \log \left (a+b x^n+c x^{2 n}\right )}{2 a^2 n}-\frac{b \log (x)}{a^2}-\frac{x^{-n}}{a n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 44.3469, size = 90, normalized size = 0.92 \[ - \frac{x^{- n}}{a n} - \frac{b \log{\left (x^{n} \right )}}{a^{2} n} + \frac{b \log{\left (a + b x^{n} + c x^{2 n} \right )}}{2 a^{2} n} - \frac{\left (- 2 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{n}}{\sqrt{- 4 a c + b^{2}}} \right )}}{a^{2} n \sqrt{- 4 a c + b^{2}}} \]

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]

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

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Mathematica [A]  time = 0.232763, size = 87, normalized size = 0.89 \[ \frac{-\frac{2 \left (b^2-2 a c\right ) \tan ^{-1}\left (\frac{2 a x^{-n}+b}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+b \log \left (x^{-2 n} \left (a+b x^n\right )+c\right )-2 a x^{-n}}{2 a^2 n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.173, size = 658, normalized size = 6.7 \[ -{\frac{1}{an{x}^{n}}}-4\,{\frac{{n}^{2}\ln \left ( x \right ) abc}{4\,{a}^{3}c{n}^{2}-{a}^{2}{b}^{2}{n}^{2}}}+{\frac{{n}^{2}\ln \left ( x \right ){b}^{3}}{4\,{a}^{3}c{n}^{2}-{a}^{2}{b}^{2}{n}^{2}}}+2\,{\frac{bc}{a \left ( 4\,ac-{b}^{2} \right ) n}\ln \left ({x}^{n}-1/2\,{\frac{-2\,abc+{b}^{3}+\sqrt{-16\,{a}^{3}{c}^{3}+20\,{a}^{2}{b}^{2}{c}^{2}-8\,a{b}^{4}c+{b}^{6}}}{c \left ( 2\,ac-{b}^{2} \right ) }} \right ) }-{\frac{{b}^{3}}{ \left ( 8\,ac-2\,{b}^{2} \right ){a}^{2}n}\ln \left ({x}^{n}-{\frac{1}{2\,c \left ( 2\,ac-{b}^{2} \right ) } \left ( -2\,abc+{b}^{3}+\sqrt{-16\,{a}^{3}{c}^{3}+20\,{a}^{2}{b}^{2}{c}^{2}-8\,a{b}^{4}c+{b}^{6}} \right ) } \right ) }+{\frac{1}{ \left ( 8\,ac-2\,{b}^{2} \right ){a}^{2}n}\ln \left ({x}^{n}-{\frac{1}{2\,c \left ( 2\,ac-{b}^{2} \right ) } \left ( -2\,abc+{b}^{3}+\sqrt{-16\,{a}^{3}{c}^{3}+20\,{a}^{2}{b}^{2}{c}^{2}-8\,a{b}^{4}c+{b}^{6}} \right ) } \right ) \sqrt{-16\,{a}^{3}{c}^{3}+20\,{a}^{2}{b}^{2}{c}^{2}-8\,a{b}^{4}c+{b}^{6}}}+2\,{\frac{bc}{a \left ( 4\,ac-{b}^{2} \right ) n}\ln \left ({x}^{n}+1/2\,{\frac{2\,abc-{b}^{3}+\sqrt{-16\,{a}^{3}{c}^{3}+20\,{a}^{2}{b}^{2}{c}^{2}-8\,a{b}^{4}c+{b}^{6}}}{c \left ( 2\,ac-{b}^{2} \right ) }} \right ) }-{\frac{{b}^{3}}{ \left ( 8\,ac-2\,{b}^{2} \right ){a}^{2}n}\ln \left ({x}^{n}+{\frac{1}{2\,c \left ( 2\,ac-{b}^{2} \right ) } \left ( 2\,abc-{b}^{3}+\sqrt{-16\,{a}^{3}{c}^{3}+20\,{a}^{2}{b}^{2}{c}^{2}-8\,a{b}^{4}c+{b}^{6}} \right ) } \right ) }-{\frac{1}{ \left ( 8\,ac-2\,{b}^{2} \right ){a}^{2}n}\ln \left ({x}^{n}+{\frac{1}{2\,c \left ( 2\,ac-{b}^{2} \right ) } \left ( 2\,abc-{b}^{3}+\sqrt{-16\,{a}^{3}{c}^{3}+20\,{a}^{2}{b}^{2}{c}^{2}-8\,a{b}^{4}c+{b}^{6}} \right ) } \right ) \sqrt{-16\,{a}^{3}{c}^{3}+20\,{a}^{2}{b}^{2}{c}^{2}-8\,a{b}^{4}c+{b}^{6}}} \]

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/a/n/(x^n)-4/(4*a^3*c*n^2-a^2*b^2*n^2)*n^2*ln(x)*a*b*c+1/(4*a^3*c*n^2-a^2*b^2*
n^2)*n^2*ln(x)*b^3+2/(4*a*c-b^2)/a/n*ln(x^n-1/2*(-2*a*b*c+b^3+(-16*a^3*c^3+20*a^
2*b^2*c^2-8*a*b^4*c+b^6)^(1/2))/c/(2*a*c-b^2))*b*c-1/2/(4*a*c-b^2)/a^2/n*ln(x^n-
1/2*(-2*a*b*c+b^3+(-16*a^3*c^3+20*a^2*b^2*c^2-8*a*b^4*c+b^6)^(1/2))/c/(2*a*c-b^2
))*b^3+1/2/(4*a*c-b^2)/a^2/n*ln(x^n-1/2*(-2*a*b*c+b^3+(-16*a^3*c^3+20*a^2*b^2*c^
2-8*a*b^4*c+b^6)^(1/2))/c/(2*a*c-b^2))*(-16*a^3*c^3+20*a^2*b^2*c^2-8*a*b^4*c+b^6
)^(1/2)+2/(4*a*c-b^2)/a/n*ln(x^n+1/2*(2*a*b*c-b^3+(-16*a^3*c^3+20*a^2*b^2*c^2-8*
a*b^4*c+b^6)^(1/2))/c/(2*a*c-b^2))*b*c-1/2/(4*a*c-b^2)/a^2/n*ln(x^n+1/2*(2*a*b*c
-b^3+(-16*a^3*c^3+20*a^2*b^2*c^2-8*a*b^4*c+b^6)^(1/2))/c/(2*a*c-b^2))*b^3-1/2/(4
*a*c-b^2)/a^2/n*ln(x^n+1/2*(2*a*b*c-b^3+(-16*a^3*c^3+20*a^2*b^2*c^2-8*a*b^4*c+b^
6)^(1/2))/c/(2*a*c-b^2))*(-16*a^3*c^3+20*a^2*b^2*c^2-8*a*b^4*c+b^6)^(1/2)

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

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]

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

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Fricas [A]  time = 0.284065, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{b^{2} - 4 \, a c} b n x^{n} \log \left (x\right ) - \sqrt{b^{2} - 4 \, a c} b x^{n} \log \left (c x^{2 \, n} + b x^{n} + a\right ) +{\left (b^{2} - 2 \, a c\right )} x^{n} \log \left (\frac{2 \, \sqrt{b^{2} - 4 \, a c} c^{2} x^{2 \, n} + b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2} + \sqrt{b^{2} - 4 \, a c} b c\right )} x^{n} +{\left (b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2 \, n} + b x^{n} + a}\right ) + 2 \, \sqrt{b^{2} - 4 \, a c} a}{2 \, \sqrt{b^{2} - 4 \, a c} a^{2} n x^{n}}, -\frac{2 \, \sqrt{-b^{2} + 4 \, a c} b n x^{n} \log \left (x\right ) - \sqrt{-b^{2} + 4 \, a c} b x^{n} \log \left (c x^{2 \, n} + b x^{n} + a\right ) - 2 \,{\left (b^{2} - 2 \, a c\right )} x^{n} \arctan \left (-\frac{2 \, \sqrt{-b^{2} + 4 \, a c} c x^{n} + \sqrt{-b^{2} + 4 \, a c} b}{b^{2} - 4 \, a c}\right ) + 2 \, \sqrt{-b^{2} + 4 \, a c} a}{2 \, \sqrt{-b^{2} + 4 \, a c} a^{2} n x^{n}}\right ] \]

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]

[-1/2*(2*sqrt(b^2 - 4*a*c)*b*n*x^n*log(x) - sqrt(b^2 - 4*a*c)*b*x^n*log(c*x^(2*n
) + b*x^n + a) + (b^2 - 2*a*c)*x^n*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)*sqrt(b
^2 - 4*a*c))/(c*x^(2*n) + b*x^n + a)) + 2*sqrt(b^2 - 4*a*c)*a)/(sqrt(b^2 - 4*a*c
)*a^2*n*x^n), -1/2*(2*sqrt(-b^2 + 4*a*c)*b*n*x^n*log(x) - sqrt(-b^2 + 4*a*c)*b*x
^n*log(c*x^(2*n) + b*x^n + a) - 2*(b^2 - 2*a*c)*x^n*arctan(-(2*sqrt(-b^2 + 4*a*c
)*c*x^n + sqrt(-b^2 + 4*a*c)*b)/(b^2 - 4*a*c)) + 2*sqrt(-b^2 + 4*a*c)*a)/(sqrt(-
b^2 + 4*a*c)*a^2*n*x^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 [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="giac")

[Out]

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