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

Optimal. Leaf size=610 \[ \frac{2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{n \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{2^{2/3} c^{2/3} \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{n \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x^{n/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} n \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x^{n/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} n \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{2^{2/3} \sqrt{3} c^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{n \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{2^{2/3} \sqrt{3} c^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{n \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}} \]

[Out]

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

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Rubi [A]  time = 2.16576, antiderivative size = 610, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{n \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{2^{2/3} c^{2/3} \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{n \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x^{n/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} n \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x^{n/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} n \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{2^{2/3} \sqrt{3} c^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{n \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{2^{2/3} \sqrt{3} c^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{n \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + n/3)/(a + b*x^n + c*x^(2*n)),x]

[Out]

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

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Rubi in Sympy [A]  time = 170.553, size = 559, normalized size = 0.92 \[ - \frac{2^{\frac{2}{3}} c^{\frac{2}{3}} \log{\left (\sqrt [3]{2} \sqrt [3]{c} x^{\frac{n}{3}} + \sqrt [3]{b + \sqrt{- 4 a c + b^{2}}} \right )}}{n \left (b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}} \sqrt{- 4 a c + b^{2}}} + \frac{2^{\frac{2}{3}} c^{\frac{2}{3}} \log{\left (c^{\frac{2}{3}} x^{\frac{2 n}{3}} - \frac{2^{\frac{2}{3}} \sqrt [3]{c} x^{\frac{n}{3}} \sqrt [3]{b + \sqrt{- 4 a c + b^{2}}}}{2} + \frac{\sqrt [3]{2} \left (b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}}}{2} \right )}}{2 n \left (b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}} \sqrt{- 4 a c + b^{2}}} + \frac{2^{\frac{2}{3}} \sqrt{3} c^{\frac{2}{3}} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} \sqrt [3]{c} x^{\frac{n}{3}}}{3 \sqrt [3]{b + \sqrt{- 4 a c + b^{2}}}} + \frac{1}{3}\right ) \right )}}{n \left (b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}} \sqrt{- 4 a c + b^{2}}} + \frac{2^{\frac{2}{3}} c^{\frac{2}{3}} \log{\left (\sqrt [3]{2} \sqrt [3]{c} x^{\frac{n}{3}} + \sqrt [3]{b - \sqrt{- 4 a c + b^{2}}} \right )}}{n \left (b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}} \sqrt{- 4 a c + b^{2}}} - \frac{2^{\frac{2}{3}} c^{\frac{2}{3}} \log{\left (c^{\frac{2}{3}} x^{\frac{2 n}{3}} - \frac{2^{\frac{2}{3}} \sqrt [3]{c} x^{\frac{n}{3}} \sqrt [3]{b - \sqrt{- 4 a c + b^{2}}}}{2} + \frac{\sqrt [3]{2} \left (b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}}}{2} \right )}}{2 n \left (b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}} \sqrt{- 4 a c + b^{2}}} - \frac{2^{\frac{2}{3}} \sqrt{3} c^{\frac{2}{3}} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} \sqrt [3]{c} x^{\frac{n}{3}}}{3 \sqrt [3]{b - \sqrt{- 4 a c + b^{2}}}} + \frac{1}{3}\right ) \right )}}{n \left (b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+1/3*n)/(a+b*x**n+c*x**(2*n)),x)

[Out]

-2**(2/3)*c**(2/3)*log(2**(1/3)*c**(1/3)*x**(n/3) + (b + sqrt(-4*a*c + b**2))**(
1/3))/(n*(b + sqrt(-4*a*c + b**2))**(2/3)*sqrt(-4*a*c + b**2)) + 2**(2/3)*c**(2/
3)*log(c**(2/3)*x**(2*n/3) - 2**(2/3)*c**(1/3)*x**(n/3)*(b + sqrt(-4*a*c + b**2)
)**(1/3)/2 + 2**(1/3)*(b + sqrt(-4*a*c + b**2))**(2/3)/2)/(2*n*(b + sqrt(-4*a*c
+ b**2))**(2/3)*sqrt(-4*a*c + b**2)) + 2**(2/3)*sqrt(3)*c**(2/3)*atan(sqrt(3)*(-
2*2**(1/3)*c**(1/3)*x**(n/3)/(3*(b + sqrt(-4*a*c + b**2))**(1/3)) + 1/3))/(n*(b
+ sqrt(-4*a*c + b**2))**(2/3)*sqrt(-4*a*c + b**2)) + 2**(2/3)*c**(2/3)*log(2**(1
/3)*c**(1/3)*x**(n/3) + (b - sqrt(-4*a*c + b**2))**(1/3))/(n*(b - sqrt(-4*a*c +
b**2))**(2/3)*sqrt(-4*a*c + b**2)) - 2**(2/3)*c**(2/3)*log(c**(2/3)*x**(2*n/3) -
2**(2/3)*c**(1/3)*x**(n/3)*(b - sqrt(-4*a*c + b**2))**(1/3)/2 + 2**(1/3)*(b - s
qrt(-4*a*c + b**2))**(2/3)/2)/(2*n*(b - sqrt(-4*a*c + b**2))**(2/3)*sqrt(-4*a*c
+ b**2)) - 2**(2/3)*sqrt(3)*c**(2/3)*atan(sqrt(3)*(-2*2**(1/3)*c**(1/3)*x**(n/3)
/(3*(b - sqrt(-4*a*c + b**2))**(1/3)) + 1/3))/(n*(b - sqrt(-4*a*c + b**2))**(2/3
)*sqrt(-4*a*c + b**2))

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Mathematica [C]  time = 0.0679407, size = 62, normalized size = 0.1 \[ \frac{\text{RootSum}\left [\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{3 \log \left (x^{n/3}-\text{$\#$1}\right )-n \log (x)}{2 \text{$\#$1}^5 c+\text{$\#$1}^2 b}\&\right ]}{3 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + n/3)/(a + b*x^n + c*x^(2*n)),x]

[Out]

RootSum[a + b*#1^3 + c*#1^6 & , (-(n*Log[x]) + 3*Log[x^(n/3) - #1])/(b*#1^2 + 2*
c*#1^5) & ]/(3*n)

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Maple [C]  time = 0.348, size = 260, normalized size = 0.4 \[ \sum _{{\it \_R}={\it RootOf} \left ( \left ( 64\,{a}^{5}{c}^{3}{n}^{6}-48\,{a}^{4}{b}^{2}{c}^{2}{n}^{6}+12\,{a}^{3}{b}^{4}c{n}^{6}-{a}^{2}{b}^{6}{n}^{6} \right ){{\it \_Z}}^{6}+ \left ( 16\,{a}^{2}b{c}^{2}{n}^{3}-8\,a{b}^{3}c{n}^{3}+{b}^{5}{n}^{3} \right ){{\it \_Z}}^{3}+{c}^{2} \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{3}}}+ \left ( -16\,{\frac{{n}^{4}b{a}^{4}{c}^{2}}{2\,a{c}^{2}-{b}^{2}c}}+8\,{\frac{{n}^{4}{b}^{3}{a}^{3}c}{2\,a{c}^{2}-{b}^{2}c}}-{\frac{{n}^{4}{b}^{5}{a}^{2}}{2\,a{c}^{2}-{b}^{2}c}} \right ){{\it \_R}}^{4}+ \left ( 4\,{\frac{{a}^{2}{c}^{2}n}{2\,a{c}^{2}-{b}^{2}c}}-5\,{\frac{a{b}^{2}cn}{2\,a{c}^{2}-{b}^{2}c}}+{\frac{{b}^{4}n}{2\,a{c}^{2}-{b}^{2}c}} \right ){\it \_R} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+1/3*n)/(a+b*x^n+c*x^(2*n)),x)

[Out]

sum(_R*ln(x^(1/3*n)+(-16/(2*a*c^2-b^2*c)*n^4*b*a^4*c^2+8/(2*a*c^2-b^2*c)*n^4*b^3
*a^3*c-1/(2*a*c^2-b^2*c)*n^4*b^5*a^2)*_R^4+(4/(2*a*c^2-b^2*c)*n*a^2*c^2-5/(2*a*c
^2-b^2*c)*n*b^2*a*c+1/(2*a*c^2-b^2*c)*n*b^4)*_R),_R=RootOf((64*a^5*c^3*n^6-48*a^
4*b^2*c^2*n^6+12*a^3*b^4*c*n^6-a^2*b^6*n^6)*_Z^6+(16*a^2*b*c^2*n^3-8*a*b^3*c*n^3
+b^5*n^3)*_Z^3+c^2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{1}{3} \, n - 1}}{c x^{2 \, n} + b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(1/3*n - 1)/(c*x^(2*n) + b*x^n + a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.429887, size = 6064, normalized size = 9.94 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(1/3*n - 1)/(c*x^(2*n) + b*x^n + a),x, algorithm="fricas")

[Out]

2*sqrt(3)*(1/2)^(1/3)*(((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^
2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + b)/((a^2*b^2
- 4*a^3*c)*n^3))^(1/3)*arctan(-(1/2)^(1/3)*(sqrt(3)*(a^2*b^5 - 8*a^3*b^3*c + 16*
a^4*b*c^2)*n^4*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*
a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - sqrt(3)*(b^4 - 6*a*b^2*c + 8*a^2*c^2)*n)*(((a^
2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c
+ 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3)/(4*(
b^2*c - 2*a*c^2)*x*x^(1/3*n - 1) - 2*sqrt(2)*x*sqrt((2*(b^4*c^2 - 4*a*b^2*c^3 +
4*a^2*c^4)*x^2*x^(2/3*n - 2) - (1/2)^(1/3)*((a^2*b^7*c - 10*a^3*b^5*c^2 + 32*a^4
*b^3*c^3 - 32*a^5*b*c^4)*n^4*x*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12
*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - (b^6*c - 8*a*b^4*c^2 + 20*a^2*
b^2*c^3 - 16*a^3*c^4)*n*x)*x^(1/3*n - 1)*(((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4
*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^
6)) + b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3) - (1/2)^(2/3)*((a^2*b^9 - 14*a^3*b^7*c
+ 72*a^4*b^5*c^2 - 160*a^5*b^3*c^3 + 128*a^6*b*c^4)*n^5*sqrt((b^4 - 4*a*b^2*c +
4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - (b^8
- 10*a*b^6*c + 36*a^2*b^4*c^2 - 56*a^3*b^2*c^3 + 32*a^4*c^4)*n^2)*(((a^2*b^2 -
4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^
6*b^2*c^2 - 64*a^7*c^3)*n^6)) + b)/((a^2*b^2 - 4*a^3*c)*n^3))^(2/3))/x^2) - (1/2
)^(1/3)*((a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*n^4*sqrt((b^4 - 4*a*b^2*c + 4*a^
2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - (b^4 - 6*
a*b^2*c + 8*a^2*c^2)*n)*(((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*
c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + b)/((a^2*b^
2 - 4*a^3*c)*n^3))^(1/3))) - 2*sqrt(3)*(1/2)^(1/3)*(-((a^2*b^2 - 4*a^3*c)*n^3*sq
rt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*
a^7*c^3)*n^6)) - b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3)*arctan(-(1/2)^(1/3)*(sqrt(3
)*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*n^4*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/
((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + sqrt(3)*(b^4 - 6
*a*b^2*c + 8*a^2*c^2)*n)*(-((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^
2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - b)/((a^2*
b^2 - 4*a^3*c)*n^3))^(1/3)/(4*(b^2*c - 2*a*c^2)*x*x^(1/3*n - 1) - 2*sqrt(2)*x*sq
rt((2*(b^4*c^2 - 4*a*b^2*c^3 + 4*a^2*c^4)*x^2*x^(2/3*n - 2) + (1/2)^(1/3)*((a^2*
b^7*c - 10*a^3*b^5*c^2 + 32*a^4*b^3*c^3 - 32*a^5*b*c^4)*n^4*x*sqrt((b^4 - 4*a*b^
2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) +
(b^6*c - 8*a*b^4*c^2 + 20*a^2*b^2*c^3 - 16*a^3*c^4)*n*x)*x^(1/3*n - 1)*(-((a^2*
b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c +
48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3) + (1/2
)^(2/3)*((a^2*b^9 - 14*a^3*b^7*c + 72*a^4*b^5*c^2 - 160*a^5*b^3*c^3 + 128*a^6*b*
c^4)*n^5*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^
2*c^2 - 64*a^7*c^3)*n^6)) + (b^8 - 10*a*b^6*c + 36*a^2*b^4*c^2 - 56*a^3*b^2*c^3
+ 32*a^4*c^4)*n^2)*(-((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)
/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - b)/((a^2*b^2 -
4*a^3*c)*n^3))^(2/3))/x^2) + (1/2)^(1/3)*((a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)
*n^4*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^
2 - 64*a^7*c^3)*n^6)) + (b^4 - 6*a*b^2*c + 8*a^2*c^2)*n)*(-((a^2*b^2 - 4*a^3*c)*
n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2
- 64*a^7*c^3)*n^6)) - b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3))) + (1/2)^(1/3)*(((a^
2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c
+ 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3)*log(
-(2*(b^2*c - 2*a*c^2)*x*x^(1/3*n - 1) + (1/2)^(1/3)*((a^2*b^5 - 8*a^3*b^3*c + 16
*a^4*b*c^2)*n^4*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48
*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - (b^4 - 6*a*b^2*c + 8*a^2*c^2)*n)*(((a^2*b^2 -
4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a
^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3))/x) + (1/2)
^(1/3)*(-((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 -
12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - b)/((a^2*b^2 - 4*a^3*c)*n^3
))^(1/3)*log(-(2*(b^2*c - 2*a*c^2)*x*x^(1/3*n - 1) - (1/2)^(1/3)*((a^2*b^5 - 8*a
^3*b^3*c + 16*a^4*b*c^2)*n^4*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a
^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + (b^4 - 6*a*b^2*c + 8*a^2*c^2)*n)
*(-((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^
5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/
3))/x) - 1/2*(1/2)^(1/3)*(((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2
*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + b)/((a^2*b
^2 - 4*a^3*c)*n^3))^(1/3)*log(8*(2*(b^4*c^2 - 4*a*b^2*c^3 + 4*a^2*c^4)*x^2*x^(2/
3*n - 2) - (1/2)^(1/3)*((a^2*b^7*c - 10*a^3*b^5*c^2 + 32*a^4*b^3*c^3 - 32*a^5*b*
c^4)*n^4*x*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*
b^2*c^2 - 64*a^7*c^3)*n^6)) - (b^6*c - 8*a*b^4*c^2 + 20*a^2*b^2*c^3 - 16*a^3*c^4
)*n*x)*x^(1/3*n - 1)*(((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2
)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + b)/((a^2*b^2 -
4*a^3*c)*n^3))^(1/3) - (1/2)^(2/3)*((a^2*b^9 - 14*a^3*b^7*c + 72*a^4*b^5*c^2 -
160*a^5*b^3*c^3 + 128*a^6*b*c^4)*n^5*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^
6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - (b^8 - 10*a*b^6*c + 36*a
^2*b^4*c^2 - 56*a^3*b^2*c^3 + 32*a^4*c^4)*n^2)*(((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b
^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c
^3)*n^6)) + b)/((a^2*b^2 - 4*a^3*c)*n^3))^(2/3))/x^2) - 1/2*(1/2)^(1/3)*(-((a^2*
b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c +
48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3)*log(8*
(2*(b^4*c^2 - 4*a*b^2*c^3 + 4*a^2*c^4)*x^2*x^(2/3*n - 2) + (1/2)^(1/3)*((a^2*b^7
*c - 10*a^3*b^5*c^2 + 32*a^4*b^3*c^3 - 32*a^5*b*c^4)*n^4*x*sqrt((b^4 - 4*a*b^2*c
+ 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) + (b
^6*c - 8*a*b^4*c^2 + 20*a^2*b^2*c^3 - 16*a^3*c^4)*n*x)*x^(1/3*n - 1)*(-((a^2*b^2
- 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48
*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - b)/((a^2*b^2 - 4*a^3*c)*n^3))^(1/3) + (1/2)^(
2/3)*((a^2*b^9 - 14*a^3*b^7*c + 72*a^4*b^5*c^2 - 160*a^5*b^3*c^3 + 128*a^6*b*c^4
)*n^5*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c
^2 - 64*a^7*c^3)*n^6)) + (b^8 - 10*a*b^6*c + 36*a^2*b^4*c^2 - 56*a^3*b^2*c^3 + 3
2*a^4*c^4)*n^2)*(-((a^2*b^2 - 4*a^3*c)*n^3*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/((
a^4*b^6 - 12*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)*n^6)) - b)/((a^2*b^2 - 4*a
^3*c)*n^3))^(2/3))/x^2)

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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+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. \[ \int \frac{x^{\frac{1}{3} \, n - 1}}{c x^{2 \, n} + b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(1/3*n - 1)/(c*x^(2*n) + b*x^n + a),x, algorithm="giac")

[Out]

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