3.555 \(\int \frac{x^{-1-3 n}}{a+b x^n+c x^{2 n}} \, dx\)

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]

-1/(3*a*n*x^(3*n)) + b/(2*a^2*n*x^(2*n)) - (b^2 - a*c)/(a^3*n*x^n) - ((b^4 - 4*a
*b^2*c + 2*a^2*c^2)*ArcTanh[(b + 2*c*x^n)/Sqrt[b^2 - 4*a*c]])/(a^4*Sqrt[b^2 - 4*
a*c]*n) - (b*(b^2 - 2*a*c)*Log[x])/a^4 + (b*(b^2 - 2*a*c)*Log[a + b*x^n + c*x^(2
*n)])/(2*a^4*n)

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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]

-1/(3*a*n*x^(3*n)) + b/(2*a^2*n*x^(2*n)) - (b^2 - a*c)/(a^3*n*x^n) - ((b^4 - 4*a
*b^2*c + 2*a^2*c^2)*ArcTanh[(b + 2*c*x^n)/Sqrt[b^2 - 4*a*c]])/(a^4*Sqrt[b^2 - 4*
a*c]*n) - (b*(b^2 - 2*a*c)*Log[x])/a^4 + (b*(b^2 - 2*a*c)*Log[a + b*x^n + c*x^(2
*n)])/(2*a^4*n)

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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]

-x**(-3*n)/(3*a*n) + b*x**(-2*n)/(2*a**2*n) - x**(-n)*(-a*c + b**2)/(a**3*n) - b
*(-2*a*c + b**2)*log(x**n)/(a**4*n) + b*(-2*a*c + b**2)*log(a + b*x**n + c*x**(2
*n))/(2*a**4*n) - (2*a**2*c**2 - 4*a*b**2*c + b**4)*atanh((b + 2*c*x**n)/sqrt(-4
*a*c + b**2))/(a**4*n*sqrt(-4*a*c + b**2))

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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]

((a*(-2*a^2 - 6*b^2*x^(2*n) + 3*a*x^n*(b + 2*c*x^n)))/x^(3*n) - (6*(b^4 - 4*a*b^
2*c + 2*a^2*c^2)*ArcTan[(b + (2*a)/x^n)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c]
+ 3*(b^3 - 2*a*b*c)*Log[c + (a + b*x^n)/x^(2*n)])/(6*a^4*n)

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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]

1/a^2/n/(x^n)*c-1/a^3/n/(x^n)*b^2+1/2*b/a^2/n/(x^n)^2-1/3/a/n/(x^n)^3+8/(4*a^5*c
*n^2-a^4*b^2*n^2)*n^2*ln(x)*a^2*b*c^2-6/(4*a^5*c*n^2-a^4*b^2*n^2)*n^2*ln(x)*a*b^
3*c+1/(4*a^5*c*n^2-a^4*b^2*n^2)*n^2*ln(x)*b^5-4/a^2/(4*a*c-b^2)/n*ln(x^n+1/2*(2*
a^2*b*c^2-4*a*b^3*c+b^5+(-16*a^5*c^5+68*a^4*b^2*c^4-96*a^3*b^4*c^3+52*a^2*b^6*c^
2-12*a*b^8*c+b^10)^(1/2))/c/(2*a^2*c^2-4*a*b^2*c+b^4))*b*c^2+3/a^3/(4*a*c-b^2)/n
*ln(x^n+1/2*(2*a^2*b*c^2-4*a*b^3*c+b^5+(-16*a^5*c^5+68*a^4*b^2*c^4-96*a^3*b^4*c^
3+52*a^2*b^6*c^2-12*a*b^8*c+b^10)^(1/2))/c/(2*a^2*c^2-4*a*b^2*c+b^4))*b^3*c-1/2/
a^4/(4*a*c-b^2)/n*ln(x^n+1/2*(2*a^2*b*c^2-4*a*b^3*c+b^5+(-16*a^5*c^5+68*a^4*b^2*
c^4-96*a^3*b^4*c^3+52*a^2*b^6*c^2-12*a*b^8*c+b^10)^(1/2))/c/(2*a^2*c^2-4*a*b^2*c
+b^4))*b^5+1/2/a^4/(4*a*c-b^2)/n*ln(x^n+1/2*(2*a^2*b*c^2-4*a*b^3*c+b^5+(-16*a^5*
c^5+68*a^4*b^2*c^4-96*a^3*b^4*c^3+52*a^2*b^6*c^2-12*a*b^8*c+b^10)^(1/2))/c/(2*a^
2*c^2-4*a*b^2*c+b^4))*(-16*a^5*c^5+68*a^4*b^2*c^4-96*a^3*b^4*c^3+52*a^2*b^6*c^2-
12*a*b^8*c+b^10)^(1/2)-4/a^2/(4*a*c-b^2)/n*ln(x^n-1/2*(-2*a^2*b*c^2+4*a*b^3*c-b^
5+(-16*a^5*c^5+68*a^4*b^2*c^4-96*a^3*b^4*c^3+52*a^2*b^6*c^2-12*a*b^8*c+b^10)^(1/
2))/c/(2*a^2*c^2-4*a*b^2*c+b^4))*b*c^2+3/a^3/(4*a*c-b^2)/n*ln(x^n-1/2*(-2*a^2*b*
c^2+4*a*b^3*c-b^5+(-16*a^5*c^5+68*a^4*b^2*c^4-96*a^3*b^4*c^3+52*a^2*b^6*c^2-12*a
*b^8*c+b^10)^(1/2))/c/(2*a^2*c^2-4*a*b^2*c+b^4))*b^3*c-1/2/a^4/(4*a*c-b^2)/n*ln(
x^n-1/2*(-2*a^2*b*c^2+4*a*b^3*c-b^5+(-16*a^5*c^5+68*a^4*b^2*c^4-96*a^3*b^4*c^3+5
2*a^2*b^6*c^2-12*a*b^8*c+b^10)^(1/2))/c/(2*a^2*c^2-4*a*b^2*c+b^4))*b^5-1/2/a^4/(
4*a*c-b^2)/n*ln(x^n-1/2*(-2*a^2*b*c^2+4*a*b^3*c-b^5+(-16*a^5*c^5+68*a^4*b^2*c^4-
96*a^3*b^4*c^3+52*a^2*b^6*c^2-12*a*b^8*c+b^10)^(1/2))/c/(2*a^2*c^2-4*a*b^2*c+b^4
))*(-16*a^5*c^5+68*a^4*b^2*c^4-96*a^3*b^4*c^3+52*a^2*b^6*c^2-12*a*b^8*c+b^10)^(1
/2)

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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]

1/6*(3*a*b*x^n - 2*a^2 - 6*(b^2 - a*c)*x^(2*n))*x^(-3*n)/(a^3*n) + integrate(-(b
^3 - 2*a*b*c + (b^2*c - a*c^2)*x^n)/(a^3*c*x*x^(2*n) + a^3*b*x*x^n + a^4*x), x)

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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]

[1/6*(3*sqrt(b^2 - 4*a*c)*a^2*b*x^n - 6*(b^3 - 2*a*b*c)*sqrt(b^2 - 4*a*c)*n*x^(3
*n)*log(x) - 2*sqrt(b^2 - 4*a*c)*a^3 + 3*(b^3 - 2*a*b*c)*sqrt(b^2 - 4*a*c)*x^(3*
n)*log(c*x^(2*n) + b*x^n + a) + 3*(b^4 - 4*a*b^2*c + 2*a^2*c^2)*x^(3*n)*log((2*s
qrt(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)) - 6*(
a*b^2 - a^2*c)*sqrt(b^2 - 4*a*c)*x^(2*n))/(sqrt(b^2 - 4*a*c)*a^4*n*x^(3*n)), 1/6
*(3*sqrt(-b^2 + 4*a*c)*a^2*b*x^n - 6*(b^3 - 2*a*b*c)*sqrt(-b^2 + 4*a*c)*n*x^(3*n
)*log(x) - 2*sqrt(-b^2 + 4*a*c)*a^3 + 3*(b^3 - 2*a*b*c)*sqrt(-b^2 + 4*a*c)*x^(3*
n)*log(c*x^(2*n) + b*x^n + a) + 6*(b^4 - 4*a*b^2*c + 2*a^2*c^2)*x^(3*n)*arctan(-
(2*sqrt(-b^2 + 4*a*c)*c*x^n + sqrt(-b^2 + 4*a*c)*b)/(b^2 - 4*a*c)) - 6*(a*b^2 -
a^2*c)*sqrt(-b^2 + 4*a*c)*x^(2*n))/(sqrt(-b^2 + 4*a*c)*a^4*n*x^(3*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**(-1-3*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. \[ \int \frac{x^{-3 \, n - 1}}{c x^{2 \, n} + b x^{n} + a}\,{d x} \]

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]

integrate(x^(-3*n - 1)/(c*x^(2*n) + b*x^n + a), x)