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

Optimal. Leaf size=699 \[ \frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{\sqrt [3]{2} a^{4/3} n \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{\sqrt [3]{2} a^{4/3} n \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \log \left (2^{2/3} a^{2/3} x^{-2 n/3}-\sqrt [3]{2} \sqrt [3]{a} x^{-n/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}\right )}{2 \sqrt [3]{2} a^{4/3} n \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \log \left (2^{2/3} a^{2/3} x^{-2 n/3}-\sqrt [3]{2} \sqrt [3]{a} x^{-n/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}\right )}{2 \sqrt [3]{2} a^{4/3} n \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{\sqrt{3} \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt [3]{2} a^{4/3} n \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\sqrt{3} \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{\sqrt [3]{2} a^{4/3} n \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{3 x^{-n/3}}{a n} \]

[Out]

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

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Rubi [A]  time = 3.15106, antiderivative size = 699, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{\sqrt [3]{2} a^{4/3} n \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{a} x^{-n/3}\right )}{\sqrt [3]{2} a^{4/3} n \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \log \left (2^{2/3} a^{2/3} x^{-2 n/3}-\sqrt [3]{2} \sqrt [3]{a} x^{-n/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}\right )}{2 \sqrt [3]{2} a^{4/3} n \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \log \left (2^{2/3} a^{2/3} x^{-2 n/3}-\sqrt [3]{2} \sqrt [3]{a} x^{-n/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}\right )}{2 \sqrt [3]{2} a^{4/3} n \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{\sqrt{3} \left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt [3]{2} a^{4/3} n \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{\sqrt{3} \left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{\sqrt [3]{2} a^{4/3} n \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{3 x^{-n/3}}{a n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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Mathematica [C]  time = 0.115379, size = 107, normalized size = 0.15 \[ -\frac{9 x^{-n/3}-\text{RootSum}\left [\text{$\#$1}^6 a+\text{$\#$1}^3 b+c\&,\frac{3 \text{$\#$1}^3 b \log \left (x^{-n/3}-\text{$\#$1}\right )+\text{$\#$1}^3 b n \log (x)+3 c \log \left (x^{-n/3}-\text{$\#$1}\right )+c n \log (x)}{2 \text{$\#$1}^5 a+\text{$\#$1}^2 b}\&\right ]}{3 a n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.585, size = 534, normalized size = 0.8 \[ -3\,{\frac{1}{an{x}^{n/3}}}+\sum _{{\it \_R}={\it RootOf} \left ( \left ( 64\,{a}^{7}{c}^{3}{n}^{6}-48\,{a}^{6}{b}^{2}{c}^{2}{n}^{6}+12\,{a}^{5}{b}^{4}c{n}^{6}-{a}^{4}{b}^{6}{n}^{6} \right ){{\it \_Z}}^{6}+ \left ( -32\,{a}^{3}b{c}^{3}{n}^{3}+32\,{a}^{2}{b}^{3}{c}^{2}{n}^{3}-10\,a{b}^{5}c{n}^{3}+{b}^{7}{n}^{3} \right ){{\it \_Z}}^{3}+{c}^{4} \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{3}}}+ \left ( -64\,{\frac{{a}^{8}{n}^{5}{c}^{4}}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}}+112\,{\frac{{n}^{5}{b}^{2}{a}^{7}{c}^{3}}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}}-60\,{\frac{{n}^{5}{b}^{4}{a}^{6}{c}^{2}}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}}+13\,{\frac{{n}^{5}{b}^{6}{a}^{5}c}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}}-{\frac{{n}^{5}{b}^{8}{a}^{4}}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}} \right ){{\it \_R}}^{5}+ \left ( 28\,{\frac{b{n}^{2}{a}^{4}{c}^{4}}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}}-63\,{\frac{{b}^{3}{n}^{2}{a}^{3}{c}^{3}}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}}+42\,{\frac{{b}^{5}{n}^{2}{a}^{2}{c}^{2}}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}}-11\,{\frac{{n}^{2}{b}^{7}ac}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}}+{\frac{{n}^{2}{b}^{9}}{2\,{a}^{2}{c}^{5}-4\,a{b}^{2}{c}^{4}+{b}^{4}{c}^{3}}} \right ){{\it \_R}}^{2} \right ) \]

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]

-3/a/n/(x^(1/3*n))+sum(_R*ln(x^(1/3*n)+(-64/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^5*
a^8*c^4+112/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^5*b^2*a^7*c^3-60/(2*a^2*c^5-4*a*b^
2*c^4+b^4*c^3)*n^5*b^4*a^6*c^2+13/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^5*b^6*a^5*c-
1/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^5*b^8*a^4)*_R^5+(28/(2*a^2*c^5-4*a*b^2*c^4+b
^4*c^3)*n^2*b*a^4*c^4-63/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^2*b^3*a^3*c^3+42/(2*a
^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^2*b^5*a^2*c^2-11/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n
^2*b^7*a*c+1/(2*a^2*c^5-4*a*b^2*c^4+b^4*c^3)*n^2*b^9)*_R^2),_R=RootOf((64*a^7*c^
3*n^6-48*a^6*b^2*c^2*n^6+12*a^5*b^4*c*n^6-a^4*b^6*n^6)*_Z^6+(-32*a^3*b*c^3*n^3+3
2*a^2*b^3*c^2*n^3-10*a*b^5*c*n^3+b^7*n^3)*_Z^3+c^4))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{3 \, x^{-\frac{1}{3} \, n}}{a n} - \int \frac{c x^{\frac{5}{3} \, n} + b x^{\frac{2}{3} \, n}}{a c x x^{2 \, n} + a b x x^{n} + a^{2} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*x^(-1/3*n)/(a*n) - integrate((c*x^(5/3*n) + b*x^(2/3*n))/(a*c*x*x^(2*n) + a*b
*x*x^n + a^2*x), x)

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Fricas [A]  time = 0.560027, size = 7922, normalized size = 11.33 \[ \text{result too large to display} \]

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]

-1/2*(4*sqrt(3)*(1/2)^(1/3)*a*n*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c
+ 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^1
0*b^2*c^2 - 64*a^11*c^3)*n^6)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3)
*arctan((1/2)^(1/3)*(sqrt(3)*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*n^4*sqrt((b^
8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*
b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) - sqrt(3)*(b^6 - 8*a*b^4*c + 18*a^2
*b^2*c^2 - 8*a^3*c^3)*n)*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^
2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c
^2 - 64*a^11*c^3)*n^6)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3)/(4*(b^
4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*x*x^(-1/3*n - 1) + 2*sqrt(2)*x*sqrt((2*(b^8*c^2 -
8*a*b^6*c^3 + 20*a^2*b^4*c^4 - 16*a^3*b^2*c^5 + 4*a^4*c^6)*x^2*x^(-2/3*n - 2) -
(1/2)^(1/3)*((a^4*b^9*c - 12*a^5*b^7*c^2 + 50*a^6*b^5*c^3 - 80*a^7*b^3*c^4 + 32
*a^8*b*c^5)*n^4*x*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^
4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) - (b^10*c
- 12*a*b^8*c^2 + 52*a^2*b^6*c^3 - 96*a^3*b^4*c^4 + 68*a^4*b^2*c^5 - 16*a^5*c^6)
*n*x)*x^(-1/3*n - 1)*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^
4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 -
64*a^11*c^3)*n^6)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3) - (1/2)^(2
/3)*((a^4*b^11 - 16*a^5*b^9*c + 98*a^6*b^7*c^2 - 280*a^7*b^5*c^3 + 352*a^8*b^3*c
^4 - 128*a^9*b*c^5)*n^5*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3
+ 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) - (
b^12 - 14*a*b^10*c + 76*a^2*b^8*c^2 - 200*a^3*b^6*c^3 + 260*a^4*b^4*c^4 - 152*a^
5*b^2*c^5 + 32*a^6*c^6)*n^2)*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 2
0*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b
^2*c^2 - 64*a^11*c^3)*n^6)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(2/3))/x
^2) - (1/2)^(1/3)*((a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*n^4*sqrt((b^8 - 8*a*b^
6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48
*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) - (b^6 - 8*a*b^4*c + 18*a^2*b^2*c^2 - 8*a^3*c
^3)*n)*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3
*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*
n^6)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3))) - 4*sqrt(3)*(1/2)^(1/3
)*a*n*(-((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3
*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*
n^6)) - b^3 + 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3)*arctan((1/2)^(1/3)*(sqrt
(3)*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*n^4*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^
4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 -
64*a^11*c^3)*n^6)) + sqrt(3)*(b^6 - 8*a*b^4*c + 18*a^2*b^2*c^2 - 8*a^3*c^3)*n)*
(-((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c
^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6))
- b^3 + 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3)/(4*(b^4*c - 4*a*b^2*c^2 + 2*a^
2*c^3)*x*x^(-1/3*n - 1) + 2*sqrt(2)*x*sqrt((2*(b^8*c^2 - 8*a*b^6*c^3 + 20*a^2*b^
4*c^4 - 16*a^3*b^2*c^5 + 4*a^4*c^6)*x^2*x^(-2/3*n - 2) + (1/2)^(1/3)*((a^4*b^9*c
- 12*a^5*b^7*c^2 + 50*a^6*b^5*c^3 - 80*a^7*b^3*c^4 + 32*a^8*b*c^5)*n^4*x*sqrt((
b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^
9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) + (b^10*c - 12*a*b^8*c^2 + 52*a^2
*b^6*c^3 - 96*a^3*b^4*c^4 + 68*a^4*b^2*c^5 - 16*a^5*c^6)*n*x)*x^(-1/3*n - 1)*(-(
(a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3
+ 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) - b
^3 + 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3) + (1/2)^(2/3)*((a^4*b^11 - 16*a^5
*b^9*c + 98*a^6*b^7*c^2 - 280*a^7*b^5*c^3 + 352*a^8*b^3*c^4 - 128*a^9*b*c^5)*n^5
*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6
- 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) + (b^12 - 14*a*b^10*c + 76
*a^2*b^8*c^2 - 200*a^3*b^6*c^3 + 260*a^4*b^4*c^4 - 152*a^5*b^2*c^5 + 32*a^6*c^6)
*n^2)*(-((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3
*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*
n^6)) - b^3 + 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(2/3))/x^2) + (1/2)^(1/3)*((a^
4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*n^4*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 -
16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^1
1*c^3)*n^6)) + (b^6 - 8*a*b^4*c + 18*a^2*b^2*c^2 - 8*a^3*c^3)*n)*(-((a^4*b^2 - 4
*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)
/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) - b^3 + 2*a*b*c
)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3))) - 2*(1/2)^(1/3)*a*n*(((a^4*b^2 - 4*a^5*c)*n
^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^
6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) + b^3 - 2*a*b*c)/((a^4*b
^2 - 4*a^5*c)*n^3))^(1/3)*log((2*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*x*x^(-1/3*n -
1) + (1/2)^(1/3)*((a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*n^4*sqrt((b^8 - 8*a*b^
6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48
*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) - (b^6 - 8*a*b^4*c + 18*a^2*b^2*c^2 - 8*a^3*c
^3)*n)*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3
*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*
n^6)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3))/x) - 2*(1/2)^(1/3)*a*n*
(-((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c
^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6))
- b^3 + 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3)*log((2*(b^4*c - 4*a*b^2*c^2 +
2*a^2*c^3)*x*x^(-1/3*n - 1) - (1/2)^(1/3)*((a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2
)*n^4*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8
*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) + (b^6 - 8*a*b^4*c +
18*a^2*b^2*c^2 - 8*a^3*c^3)*n)*(-((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c
+ 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^1
0*b^2*c^2 - 64*a^11*c^3)*n^6)) - b^3 + 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3)
)/x) + (1/2)^(1/3)*a*n*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c + 20*a^2*
b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2
- 64*a^11*c^3)*n^6)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3)*log(8*(2
*(b^8*c^2 - 8*a*b^6*c^3 + 20*a^2*b^4*c^4 - 16*a^3*b^2*c^5 + 4*a^4*c^6)*x^2*x^(-2
/3*n - 2) - (1/2)^(1/3)*((a^4*b^9*c - 12*a^5*b^7*c^2 + 50*a^6*b^5*c^3 - 80*a^7*b
^3*c^4 + 32*a^8*b*c^5)*n^4*x*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2
*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)
) - (b^10*c - 12*a*b^8*c^2 + 52*a^2*b^6*c^3 - 96*a^3*b^4*c^4 + 68*a^4*b^2*c^5 -
16*a^5*c^6)*n*x)*x^(-1/3*n - 1)*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*c
+ 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^1
0*b^2*c^2 - 64*a^11*c^3)*n^6)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/3)
- (1/2)^(2/3)*((a^4*b^11 - 16*a^5*b^9*c + 98*a^6*b^7*c^2 - 280*a^7*b^5*c^3 + 35
2*a^8*b^3*c^4 - 128*a^9*b*c^5)*n^5*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a
^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3
)*n^6)) - (b^12 - 14*a*b^10*c + 76*a^2*b^8*c^2 - 200*a^3*b^6*c^3 + 260*a^4*b^4*c
^4 - 152*a^5*b^2*c^5 + 32*a^6*c^6)*n^2)*(((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*
a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c
+ 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) + b^3 - 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3
))^(2/3))/x^2) + (1/2)^(1/3)*a*n*(-((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8 - 8*a*b^6*
c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a
^10*b^2*c^2 - 64*a^11*c^3)*n^6)) - b^3 + 2*a*b*c)/((a^4*b^2 - 4*a^5*c)*n^3))^(1/
3)*log(8*(2*(b^8*c^2 - 8*a*b^6*c^3 + 20*a^2*b^4*c^4 - 16*a^3*b^2*c^5 + 4*a^4*c^6
)*x^2*x^(-2/3*n - 2) + (1/2)^(1/3)*((a^4*b^9*c - 12*a^5*b^7*c^2 + 50*a^6*b^5*c^3
- 80*a^7*b^3*c^4 + 32*a^8*b*c^5)*n^4*x*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 -
16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^1
1*c^3)*n^6)) + (b^10*c - 12*a*b^8*c^2 + 52*a^2*b^6*c^3 - 96*a^3*b^4*c^4 + 68*a^4
*b^2*c^5 - 16*a^5*c^6)*n*x)*x^(-1/3*n - 1)*(-((a^4*b^2 - 4*a^5*c)*n^3*sqrt((b^8
- 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^
4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) - b^3 + 2*a*b*c)/((a^4*b^2 - 4*a^5*c)
*n^3))^(1/3) + (1/2)^(2/3)*((a^4*b^11 - 16*a^5*b^9*c + 98*a^6*b^7*c^2 - 280*a^7*
b^5*c^3 + 352*a^8*b^3*c^4 - 128*a^9*b*c^5)*n^5*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^
4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 -
64*a^11*c^3)*n^6)) + (b^12 - 14*a*b^10*c + 76*a^2*b^8*c^2 - 200*a^3*b^6*c^3 + 2
60*a^4*b^4*c^4 - 152*a^5*b^2*c^5 + 32*a^6*c^6)*n^2)*(-((a^4*b^2 - 4*a^5*c)*n^3*s
qrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/((a^8*b^6 -
12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)*n^6)) - b^3 + 2*a*b*c)/((a^4*b^2 -
4*a^5*c)*n^3))^(2/3))/x^2) + 6*x*x^(-1/3*n - 1))/(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(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)