3.150 \(\int \frac{1}{x^3 \left (a+b x^3+c x^6\right )} \, dx\)

Optimal. Leaf size=612 \[ \frac{c^{2/3} \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} a \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{c^{2/3} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} a \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{c^{2/3} \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} a \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{c^{2/3} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} a \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}+\frac{c^{2/3} \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} a \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{c^{2/3} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} a \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{1}{2 a x^2} \]

[Out]

-1/(2*a*x^2) + (c^(2/3)*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTan[(1 - (2*2^(1/3)*c^(1/3)
*x)/(b - Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(1/3)*Sqrt[3]*a*(b - Sqrt[b^2 -
4*a*c])^(2/3)) + (c^(2/3)*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTan[(1 - (2*2^(1/3)*c^(1/
3)*x)/(b + Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(1/3)*Sqrt[3]*a*(b + Sqrt[b^2
- 4*a*c])^(2/3)) - (c^(2/3)*(1 + b/Sqrt[b^2 - 4*a*c])*Log[(b - Sqrt[b^2 - 4*a*c]
)^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(1/3)*a*(b - Sqrt[b^2 - 4*a*c])^(2/3)) - (c^(
2/3)*(1 - b/Sqrt[b^2 - 4*a*c])*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/
3)*x])/(3*2^(1/3)*a*(b + Sqrt[b^2 - 4*a*c])^(2/3)) + (c^(2/3)*(1 + b/Sqrt[b^2 -
4*a*c])*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 + 2^(2/3)*c^(2/3)*x^2])/(6*2^(1/3)*a*(b - Sqrt[b^2 - 4*a*c])^(2/3))
+ (c^(2/3)*(1 - b/Sqrt[b^2 - 4*a*c])*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 + 2^(2/3)*c^(2/3)*x^2])/(6*2^(1/3)*a*(b
+ Sqrt[b^2 - 4*a*c])^(2/3))

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Rubi [A]  time = 1.94986, antiderivative size = 612, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444 \[ \frac{c^{2/3} \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} a \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{c^{2/3} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} a \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{c^{2/3} \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} a \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{c^{2/3} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} a \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}+\frac{c^{2/3} \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} a \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{c^{2/3} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} a \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{1}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x^3 + c*x^6)),x]

[Out]

-1/(2*a*x^2) + (c^(2/3)*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTan[(1 - (2*2^(1/3)*c^(1/3)
*x)/(b - Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(1/3)*Sqrt[3]*a*(b - Sqrt[b^2 -
4*a*c])^(2/3)) + (c^(2/3)*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTan[(1 - (2*2^(1/3)*c^(1/
3)*x)/(b + Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(1/3)*Sqrt[3]*a*(b + Sqrt[b^2
- 4*a*c])^(2/3)) - (c^(2/3)*(1 + b/Sqrt[b^2 - 4*a*c])*Log[(b - Sqrt[b^2 - 4*a*c]
)^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(1/3)*a*(b - Sqrt[b^2 - 4*a*c])^(2/3)) - (c^(
2/3)*(1 - b/Sqrt[b^2 - 4*a*c])*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/
3)*x])/(3*2^(1/3)*a*(b + Sqrt[b^2 - 4*a*c])^(2/3)) + (c^(2/3)*(1 + b/Sqrt[b^2 -
4*a*c])*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 + 2^(2/3)*c^(2/3)*x^2])/(6*2^(1/3)*a*(b - Sqrt[b^2 - 4*a*c])^(2/3))
+ (c^(2/3)*(1 - b/Sqrt[b^2 - 4*a*c])*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 + 2^(2/3)*c^(2/3)*x^2])/(6*2^(1/3)*a*(b
+ Sqrt[b^2 - 4*a*c])^(2/3))

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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(1/x**3/(c*x**6+b*x**3+a),x)

[Out]

Timed out

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Mathematica [C]  time = 0.0563509, size = 75, normalized size = 0.12 \[ -\frac{\text{RootSum}\left [\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^3 c \log (x-\text{$\#$1})+b \log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1}^2 b}\&\right ]}{3 a}-\frac{1}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x^3 + c*x^6)),x]

[Out]

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

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Maple [C]  time = 0.009, size = 62, normalized size = 0.1 \[{\frac{1}{3\,a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}c+{{\it \_Z}}^{3}b+a \right ) }{\frac{ \left ( -{{\it \_R}}^{3}c-b \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}c+{{\it \_R}}^{2}b}}}-{\frac{1}{2\,a{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(c*x^6+b*x^3+a),x)

[Out]

1/3/a*sum((-_R^3*c-b)/(2*_R^5*c+_R^2*b)*ln(x-_R),_R=RootOf(_Z^6*c+_Z^3*b+a))-1/2
/a/x^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{\int \frac{c x^{3} + b}{c x^{6} + b x^{3} + a}\,{d x}}{a} - \frac{1}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^6 + b*x^3 + a)*x^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.494311, size = 7645, normalized size = 12.49 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^6 + b*x^3 + a)*x^3),x, algorithm="fricas")

[Out]

-1/6*(4*sqrt(3)*(1/2)^(1/3)*a*x^2*(-(b^4 - 3*a*b^2*c + a^2*c^2 + (a^5*b^2 - 4*a^
6*c)*sqrt((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)
/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^2 - 4*a^6*c
))^(1/3)*arctan(-(1/2)^(1/3)*(sqrt(3)*(a^5*b^6 - 10*a^6*b^4*c + 32*a^7*b^2*c^2 -
32*a^8*c^3)*sqrt((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*
b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)) - sqrt(3)*(
b^8 - 9*a*b^6*c + 25*a^2*b^4*c^2 - 20*a^3*b^2*c^3))*(-(b^4 - 3*a*b^2*c + a^2*c^2
+ (a^5*b^2 - 4*a^6*c)*sqrt((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3
+ 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/
(a^5*b^2 - 4*a^6*c))^(1/3)/(4*(b^5*c^2 - 5*a*b^3*c^3 + 5*a^2*b*c^4)*x + 4*sqrt(1
/2)*(b^5*c^2 - 5*a*b^3*c^3 + 5*a^2*b*c^4)*sqrt((2*(b^5*c^4 - 5*a*b^3*c^5 + 5*a^2
*b*c^6)*x^2 + (1/2)^(2/3)*(b^11 - 13*a*b^9*c + 63*a^2*b^7*c^2 - 138*a^3*b^5*c^3
+ 130*a^4*b^3*c^4 - 40*a^5*b*c^5 - (a^5*b^9 - 14*a^6*b^7*c + 72*a^7*b^5*c^2 - 16
0*a^8*b^3*c^3 + 128*a^9*b*c^4)*sqrt((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3
*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13
*c^3)))*(-(b^4 - 3*a*b^2*c + a^2*c^2 + (a^5*b^2 - 4*a^6*c)*sqrt((b^10 - 10*a*b^8
*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c
+ 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^2 - 4*a^6*c))^(2/3) + (1/2)^(1/3)*((a
^5*b^6*c^2 - 10*a^6*b^4*c^3 + 32*a^7*b^2*c^4 - 32*a^8*c^5)*x*sqrt((b^10 - 10*a*b
^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4
*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)) - (b^8*c^2 - 9*a*b^6*c^3 + 25*a^2*b^4*c^4 -
20*a^3*b^2*c^5)*x)*(-(b^4 - 3*a*b^2*c + a^2*c^2 + (a^5*b^2 - 4*a^6*c)*sqrt((b^1
0 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 1
2*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^2 - 4*a^6*c))^(1/3))/(b^5
*c^4 - 5*a*b^3*c^5 + 5*a^2*b*c^6)) - (1/2)^(1/3)*(b^8 - 9*a*b^6*c + 25*a^2*b^4*c
^2 - 20*a^3*b^2*c^3 - (a^5*b^6 - 10*a^6*b^4*c + 32*a^7*b^2*c^2 - 32*a^8*c^3)*sqr
t((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b
^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))*(-(b^4 - 3*a*b^2*c + a^2*c
^2 + (a^5*b^2 - 4*a^6*c)*sqrt((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c
^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3))
)/(a^5*b^2 - 4*a^6*c))^(1/3))) - 4*sqrt(3)*(1/2)^(1/3)*a*x^2*(-(b^4 - 3*a*b^2*c
+ a^2*c^2 - (a^5*b^2 - 4*a^6*c)*sqrt((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^
3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^1
3*c^3)))/(a^5*b^2 - 4*a^6*c))^(1/3)*arctan(-(1/2)^(1/3)*(sqrt(3)*(a^5*b^6 - 10*a
^6*b^4*c + 32*a^7*b^2*c^2 - 32*a^8*c^3)*sqrt((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2
- 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2
- 64*a^13*c^3)) + sqrt(3)*(b^8 - 9*a*b^6*c + 25*a^2*b^4*c^2 - 20*a^3*b^2*c^3))*(
-(b^4 - 3*a*b^2*c + a^2*c^2 - (a^5*b^2 - 4*a^6*c)*sqrt((b^10 - 10*a*b^8*c + 35*a
^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^1
2*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^2 - 4*a^6*c))^(1/3)/(4*(b^5*c^2 - 5*a*b^3*c^3
+ 5*a^2*b*c^4)*x + 4*sqrt(1/2)*(b^5*c^2 - 5*a*b^3*c^3 + 5*a^2*b*c^4)*sqrt((2*(b^
5*c^4 - 5*a*b^3*c^5 + 5*a^2*b*c^6)*x^2 + (1/2)^(2/3)*(b^11 - 13*a*b^9*c + 63*a^2
*b^7*c^2 - 138*a^3*b^5*c^3 + 130*a^4*b^3*c^4 - 40*a^5*b*c^5 + (a^5*b^9 - 14*a^6*
b^7*c + 72*a^7*b^5*c^2 - 160*a^8*b^3*c^3 + 128*a^9*b*c^4)*sqrt((b^10 - 10*a*b^8*
c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c
+ 48*a^12*b^2*c^2 - 64*a^13*c^3)))*(-(b^4 - 3*a*b^2*c + a^2*c^2 - (a^5*b^2 - 4*a
^6*c)*sqrt((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4
)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^2 - 4*a^6*
c))^(2/3) - (1/2)^(1/3)*((a^5*b^6*c^2 - 10*a^6*b^4*c^3 + 32*a^7*b^2*c^4 - 32*a^8
*c^5)*x*sqrt((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*b^2*c
^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)) + (b^8*c^2 - 9*a
*b^6*c^3 + 25*a^2*b^4*c^4 - 20*a^3*b^2*c^5)*x)*(-(b^4 - 3*a*b^2*c + a^2*c^2 - (a
^5*b^2 - 4*a^6*c)*sqrt((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25
*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*
b^2 - 4*a^6*c))^(1/3))/(b^5*c^4 - 5*a*b^3*c^5 + 5*a^2*b*c^6)) - (1/2)^(1/3)*(b^8
- 9*a*b^6*c + 25*a^2*b^4*c^2 - 20*a^3*b^2*c^3 + (a^5*b^6 - 10*a^6*b^4*c + 32*a^
7*b^2*c^2 - 32*a^8*c^3)*sqrt((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^
3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))
*(-(b^4 - 3*a*b^2*c + a^2*c^2 - (a^5*b^2 - 4*a^6*c)*sqrt((b^10 - 10*a*b^8*c + 35
*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a
^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^2 - 4*a^6*c))^(1/3))) + (1/2)^(1/3)*a*x^2*(-
(b^4 - 3*a*b^2*c + a^2*c^2 + (a^5*b^2 - 4*a^6*c)*sqrt((b^10 - 10*a*b^8*c + 35*a^
2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12
*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^2 - 4*a^6*c))^(1/3)*log(2*(b^5*c^4 - 5*a*b^3*c^
5 + 5*a^2*b*c^6)*x^2 + (1/2)^(2/3)*(b^11 - 13*a*b^9*c + 63*a^2*b^7*c^2 - 138*a^3
*b^5*c^3 + 130*a^4*b^3*c^4 - 40*a^5*b*c^5 - (a^5*b^9 - 14*a^6*b^7*c + 72*a^7*b^5
*c^2 - 160*a^8*b^3*c^3 + 128*a^9*b*c^4)*sqrt((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2
- 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2
- 64*a^13*c^3)))*(-(b^4 - 3*a*b^2*c + a^2*c^2 + (a^5*b^2 - 4*a^6*c)*sqrt((b^10 -
10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a
^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^2 - 4*a^6*c))^(2/3) + (1/2)^
(1/3)*((a^5*b^6*c^2 - 10*a^6*b^4*c^3 + 32*a^7*b^2*c^4 - 32*a^8*c^5)*x*sqrt((b^10
- 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12
*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)) - (b^8*c^2 - 9*a*b^6*c^3 + 25*a^2*
b^4*c^4 - 20*a^3*b^2*c^5)*x)*(-(b^4 - 3*a*b^2*c + a^2*c^2 + (a^5*b^2 - 4*a^6*c)*
sqrt((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^1
0*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^2 - 4*a^6*c))^(1
/3)) + (1/2)^(1/3)*a*x^2*(-(b^4 - 3*a*b^2*c + a^2*c^2 - (a^5*b^2 - 4*a^6*c)*sqrt
((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b^
6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^2 - 4*a^6*c))^(1/3)*
log(2*(b^5*c^4 - 5*a*b^3*c^5 + 5*a^2*b*c^6)*x^2 + (1/2)^(2/3)*(b^11 - 13*a*b^9*c
+ 63*a^2*b^7*c^2 - 138*a^3*b^5*c^3 + 130*a^4*b^3*c^4 - 40*a^5*b*c^5 + (a^5*b^9
- 14*a^6*b^7*c + 72*a^7*b^5*c^2 - 160*a^8*b^3*c^3 + 128*a^9*b*c^4)*sqrt((b^10 -
10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^
11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))*(-(b^4 - 3*a*b^2*c + a^2*c^2 - (a^5*
b^2 - 4*a^6*c)*sqrt((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^
4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^2
- 4*a^6*c))^(2/3) - (1/2)^(1/3)*((a^5*b^6*c^2 - 10*a^6*b^4*c^3 + 32*a^7*b^2*c^4
- 32*a^8*c^5)*x*sqrt((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*
a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)) + (b^8*
c^2 - 9*a*b^6*c^3 + 25*a^2*b^4*c^4 - 20*a^3*b^2*c^5)*x)*(-(b^4 - 3*a*b^2*c + a^2
*c^2 - (a^5*b^2 - 4*a^6*c)*sqrt((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4
*c^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3
)))/(a^5*b^2 - 4*a^6*c))^(1/3)) - 2*(1/2)^(1/3)*a*x^2*(-(b^4 - 3*a*b^2*c + a^2*c
^2 + (a^5*b^2 - 4*a^6*c)*sqrt((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c
^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3))
)/(a^5*b^2 - 4*a^6*c))^(1/3)*log(2*(b^5*c^2 - 5*a*b^3*c^3 + 5*a^2*b*c^4)*x + (1/
2)^(1/3)*(b^8 - 9*a*b^6*c + 25*a^2*b^4*c^2 - 20*a^3*b^2*c^3 - (a^5*b^6 - 10*a^6*
b^4*c + 32*a^7*b^2*c^2 - 32*a^8*c^3)*sqrt((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 -
50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 6
4*a^13*c^3)))*(-(b^4 - 3*a*b^2*c + a^2*c^2 + (a^5*b^2 - 4*a^6*c)*sqrt((b^10 - 10
*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11
*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^2 - 4*a^6*c))^(1/3)) - 2*(1/2)^
(1/3)*a*x^2*(-(b^4 - 3*a*b^2*c + a^2*c^2 - (a^5*b^2 - 4*a^6*c)*sqrt((b^10 - 10*a
*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b
^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^2 - 4*a^6*c))^(1/3)*log(2*(b^5*c^
2 - 5*a*b^3*c^3 + 5*a^2*b*c^4)*x + (1/2)^(1/3)*(b^8 - 9*a*b^6*c + 25*a^2*b^4*c^2
- 20*a^3*b^2*c^3 + (a^5*b^6 - 10*a^6*b^4*c + 32*a^7*b^2*c^2 - 32*a^8*c^3)*sqrt(
(b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3 + 25*a^4*b^2*c^4)/(a^10*b^6
- 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))*(-(b^4 - 3*a*b^2*c + a^2*c^2
- (a^5*b^2 - 4*a^6*c)*sqrt((b^10 - 10*a*b^8*c + 35*a^2*b^6*c^2 - 50*a^3*b^4*c^3
+ 25*a^4*b^2*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/
(a^5*b^2 - 4*a^6*c))^(1/3)) + 3)/(a*x^2)

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Sympy [A]  time = 18.5479, size = 241, normalized size = 0.39 \[ \operatorname{RootSum}{\left (t^{6} \left (46656 a^{8} c^{3} - 34992 a^{7} b^{2} c^{2} + 8748 a^{6} b^{4} c - 729 a^{5} b^{6}\right ) + t^{3} \left (- 432 a^{4} c^{4} + 1512 a^{3} b^{2} c^{3} - 1107 a^{2} b^{4} c^{2} + 297 a b^{6} c - 27 b^{8}\right ) + c^{5}, \left ( t \mapsto t \log{\left (x + \frac{- 2592 t^{4} a^{8} c^{3} + 2592 t^{4} a^{7} b^{2} c^{2} - 810 t^{4} a^{6} b^{4} c + 81 t^{4} a^{5} b^{6} + 12 t a^{4} c^{4} - 75 t a^{3} b^{2} c^{3} + 78 t a^{2} b^{4} c^{2} - 27 t a b^{6} c + 3 t b^{8}}{5 a^{2} b c^{4} - 5 a b^{3} c^{3} + b^{5} c^{2}} \right )} \right )\right )} - \frac{1}{2 a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(c*x**6+b*x**3+a),x)

[Out]

RootSum(_t**6*(46656*a**8*c**3 - 34992*a**7*b**2*c**2 + 8748*a**6*b**4*c - 729*a
**5*b**6) + _t**3*(-432*a**4*c**4 + 1512*a**3*b**2*c**3 - 1107*a**2*b**4*c**2 +
297*a*b**6*c - 27*b**8) + c**5, Lambda(_t, _t*log(x + (-2592*_t**4*a**8*c**3 + 2
592*_t**4*a**7*b**2*c**2 - 810*_t**4*a**6*b**4*c + 81*_t**4*a**5*b**6 + 12*_t*a*
*4*c**4 - 75*_t*a**3*b**2*c**3 + 78*_t*a**2*b**4*c**2 - 27*_t*a*b**6*c + 3*_t*b*
*8)/(5*a**2*b*c**4 - 5*a*b**3*c**3 + b**5*c**2)))) - 1/(2*a*x**2)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{6} + b x^{3} + a\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^6 + b*x^3 + a)*x^3),x, algorithm="giac")

[Out]

integrate(1/((c*x^6 + b*x^3 + a)*x^3), x)