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

Optimal. Leaf size=558 \[ -\frac{c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}+\frac{2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{2^{2/3} c^{2/3} \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{2^{2/3} c^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{2^{2/3} c^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}} \]

[Out]

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

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Rubi [A]  time = 1.26151, antiderivative size = 558, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{3 \sqrt [3]{2} \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}+\frac{2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}-\frac{2^{2/3} c^{2/3} \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}}-\frac{2^{2/3} c^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )^{2/3}}+\frac{2^{2/3} c^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3 + c*x^6)^(-1),x]

[Out]

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

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Rubi in Sympy [A]  time = 133.387, size = 529, normalized size = 0.95 \[ - \frac{2^{\frac{2}{3}} c^{\frac{2}{3}} \log{\left (\sqrt [3]{2} \sqrt [3]{c} x + \sqrt [3]{b + \sqrt{- 4 a c + b^{2}}} \right )}}{3 \left (b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}} \sqrt{- 4 a c + b^{2}}} + \frac{2^{\frac{2}{3}} c^{\frac{2}{3}} \log{\left (c^{\frac{2}{3}} x^{2} - \frac{2^{\frac{2}{3}} \sqrt [3]{c} x \sqrt [3]{b + \sqrt{- 4 a c + b^{2}}}}{2} + \frac{\sqrt [3]{2} \left (b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}}}{2} \right )}}{6 \left (b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}} \sqrt{- 4 a c + b^{2}}} + \frac{2^{\frac{2}{3}} \sqrt{3} c^{\frac{2}{3}} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{3 \sqrt [3]{b + \sqrt{- 4 a c + b^{2}}}} + \frac{1}{3}\right ) \right )}}{3 \left (b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}} \sqrt{- 4 a c + b^{2}}} + \frac{2^{\frac{2}{3}} c^{\frac{2}{3}} \log{\left (\sqrt [3]{2} \sqrt [3]{c} x + \sqrt [3]{b - \sqrt{- 4 a c + b^{2}}} \right )}}{3 \left (b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}} \sqrt{- 4 a c + b^{2}}} - \frac{2^{\frac{2}{3}} c^{\frac{2}{3}} \log{\left (c^{\frac{2}{3}} x^{2} - \frac{2^{\frac{2}{3}} \sqrt [3]{c} x \sqrt [3]{b - \sqrt{- 4 a c + b^{2}}}}{2} + \frac{\sqrt [3]{2} \left (b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}}}{2} \right )}}{6 \left (b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}} \sqrt{- 4 a c + b^{2}}} - \frac{2^{\frac{2}{3}} \sqrt{3} c^{\frac{2}{3}} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{3 \sqrt [3]{b - \sqrt{- 4 a c + b^{2}}}} + \frac{1}{3}\right ) \right )}}{3 \left (b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2**(2/3)*c**(2/3)*log(2**(1/3)*c**(1/3)*x + (b + sqrt(-4*a*c + b**2))**(1/3))/(
3*(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 - 2**(2/3)*c**(1/3)*x*(b + sqrt(-4*a*c + b**2))**(1/3)/2 + 2**(1/3
)*(b + sqrt(-4*a*c + b**2))**(2/3)/2)/(6*(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/
(3*(b + sqrt(-4*a*c + b**2))**(1/3)) + 1/3))/(3*(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 + (b - sqrt(-4
*a*c + b**2))**(1/3))/(3*(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 - 2**(2/3)*c**(1/3)*x*(b - sqrt(-4*a*c + b*
*2))**(1/3)/2 + 2**(1/3)*(b - sqrt(-4*a*c + b**2))**(2/3)/2)/(6*(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/(3*(b - sqrt(-4*a*c + b**2))**(1/3)) + 1/3))/(3*(b - sqrt
(-4*a*c + b**2))**(2/3)*sqrt(-4*a*c + b**2))

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Mathematica [C]  time = 0.0297783, size = 45, normalized size = 0.08 \[ \frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1}^2 b}\&\right ] \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.004, size = 40, normalized size = 0.1 \[{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}c+{{\it \_Z}}^{3}b+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}c+{{\it \_R}}^{2}b}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^6+b*x^3+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(c*x^6 + b*x^3 + a), x)

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Fricas [A]  time = 0.358855, size = 5284, normalized size = 9.47 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(3)*(1/2)^(1/3)*(((a^2*b^2 - 4*a^3*c)*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)) + b)/(a^2*b^2 - 4*a^3*c
))^(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)*sqr
t((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)) - sqrt(3)*(b^4 - 6*a*b^2*c + 8*a^2*c^2))*(((a^2*b^2 - 4*a^3*c)*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)
) + b)/(a^2*b^2 - 4*a^3*c))^(1/3)/(4*(b^2*c - 2*a*c^2)*x + 4*sqrt(1/2)*(b^2*c -
2*a*c^2)*sqrt((2*(b^2*c^2 - 2*a*c^3)*x^2 + (1/2)^(2/3)*(b^6 - 8*a*b^4*c + 20*a^2
*b^2*c^2 - 16*a^3*c^3 - (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^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 - 6
4*a^7*c^3)))*(((a^2*b^2 - 4*a^3*c)*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)) + b)/(a^2*b^2 - 4*a^3*c))^(2/3) -
(1/2)^(1/3)*((a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*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)) - (b^4*c -
6*a*b^2*c^2 + 8*a^2*c^3)*x)*(((a^2*b^2 - 4*a^3*c)*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)) + b)/(a^2*b^2 - 4*a
^3*c))^(1/3))/(b^2*c^2 - 2*a*c^3)) + (1/2)^(1/3)*(b^4 - 6*a*b^2*c + 8*a^2*c^2 -
(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*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)))*(((a^2*b^2 - 4*a^3*c)*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)) + b)/(a^2*b^2 - 4*a^3*c))^(1/3))) - 2/3*sqrt(3)*(1/2)^(1/3)*(-((a^2*b^2 - 4*
a^3*c)*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)) - b)/(a^2*b^2 - 4*a^3*c))^(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)*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)) + sqrt(3)*(b^4 - 6*a*b^2*c + 8*
a^2*c^2))*(-((a^2*b^2 - 4*a^3*c)*sqrt((b^4 - 4*a*b^2*c + 4*a^2*c^2)/(a^4*b^6 - 1
2*a^5*b^4*c + 48*a^6*b^2*c^2 - 64*a^7*c^3)) - b)/(a^2*b^2 - 4*a^3*c))^(1/3)/(4*(
b^2*c - 2*a*c^2)*x + 4*sqrt(1/2)*(b^2*c - 2*a*c^2)*sqrt((2*(b^2*c^2 - 2*a*c^3)*x
^2 + (1/2)^(2/3)*(b^6 - 8*a*b^4*c + 20*a^2*b^2*c^2 - 16*a^3*c^3 + (a^2*b^7 - 12*
a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^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)))*(-((a^2*b^2 - 4*a^3*c)*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)) - b)/(a^2*b^2 - 4*a^3*c))^(2/3) + (1/2)^(1/3)*((a^2*b^5*c - 8*a^3*b^3*c
^2 + 16*a^4*b*c^3)*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)) + (b^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*x)*(-((a^2*b
^2 - 4*a^3*c)*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)) - b)/(a^2*b^2 - 4*a^3*c))^(1/3))/(b^2*c^2 - 2*a*c^3)) +
(1/2)^(1/3)*(b^4 - 6*a*b^2*c + 8*a^2*c^2 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^
2)*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)))*(-((a^2*b^2 - 4*a^3*c)*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)) - b)/(a^2*b^2 - 4*a^3*c))^(1/3)
)) - 1/6*(1/2)^(1/3)*(((a^2*b^2 - 4*a^3*c)*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)) + b)/(a^2*b^2 - 4*a^3*c))^
(1/3)*log(-2*(b^2*c^2 - 2*a*c^3)*x^2 - (1/2)^(2/3)*(b^6 - 8*a*b^4*c + 20*a^2*b^2
*c^2 - 16*a^3*c^3 - (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*sqr
t((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)))*(((a^2*b^2 - 4*a^3*c)*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)) + b)/(a^2*b^2 - 4*a^3*c))^(2/3) + (1/2
)^(1/3)*((a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*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)) - (b^4*c - 6*a*
b^2*c^2 + 8*a^2*c^3)*x)*(((a^2*b^2 - 4*a^3*c)*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)) + b)/(a^2*b^2 - 4*a^3*c
))^(1/3)) - 1/6*(1/2)^(1/3)*(-((a^2*b^2 - 4*a^3*c)*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)) - b)/(a^2*b^2 - 4*
a^3*c))^(1/3)*log(-2*(b^2*c^2 - 2*a*c^3)*x^2 - (1/2)^(2/3)*(b^6 - 8*a*b^4*c + 20
*a^2*b^2*c^2 - 16*a^3*c^3 + (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*
c^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)))*(-((a^2*b^2 - 4*a^3*c)*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)) - b)/(a^2*b^2 - 4*a^3*c))^(2/
3) - (1/2)^(1/3)*((a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*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)) + (b^4
*c - 6*a*b^2*c^2 + 8*a^2*c^3)*x)*(-((a^2*b^2 - 4*a^3*c)*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)) - b)/(a^2*b^2
- 4*a^3*c))^(1/3)) + 1/3*(1/2)^(1/3)*(((a^2*b^2 - 4*a^3*c)*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)) + b)/(a^2
*b^2 - 4*a^3*c))^(1/3)*log(-2*(b^2*c - 2*a*c^2)*x + (1/2)^(1/3)*(b^4 - 6*a*b^2*c
+ 8*a^2*c^2 - (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*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)))*(((a^2*b^2 - 4
*a^3*c)*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)) + b)/(a^2*b^2 - 4*a^3*c))^(1/3)) + 1/3*(1/2)^(1/3)*(-((a^2*b^
2 - 4*a^3*c)*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)) - b)/(a^2*b^2 - 4*a^3*c))^(1/3)*log(-2*(b^2*c - 2*a*c^2)
*x + (1/2)^(1/3)*(b^4 - 6*a*b^2*c + 8*a^2*c^2 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*
b*c^2)*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)))*(-((a^2*b^2 - 4*a^3*c)*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)) - b)/(a^2*b^2 - 4*a^3*c))^(
1/3))

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Sympy [A]  time = 9.29303, size = 155, normalized size = 0.28 \[ \operatorname{RootSum}{\left (t^{6} \left (46656 a^{5} c^{3} - 34992 a^{4} b^{2} c^{2} + 8748 a^{3} b^{4} c - 729 a^{2} b^{6}\right ) + t^{3} \left (432 a^{2} b c^{2} - 216 a b^{3} c + 27 b^{5}\right ) + c^{2}, \left ( t \mapsto t \log{\left (x + \frac{- 1296 t^{4} a^{4} b c^{2} + 648 t^{4} a^{3} b^{3} c - 81 t^{4} a^{2} b^{5} + 12 t a^{2} c^{2} - 15 t a b^{2} c + 3 t b^{4}}{2 a c^{2} - b^{2} c} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(_t**6*(46656*a**5*c**3 - 34992*a**4*b**2*c**2 + 8748*a**3*b**4*c - 729*a
**2*b**6) + _t**3*(432*a**2*b*c**2 - 216*a*b**3*c + 27*b**5) + c**2, Lambda(_t,
_t*log(x + (-1296*_t**4*a**4*b*c**2 + 648*_t**4*a**3*b**3*c - 81*_t**4*a**2*b**5
+ 12*_t*a**2*c**2 - 15*_t*a*b**2*c + 3*_t*b**4)/(2*a*c**2 - b**2*c))))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(c*x^6 + b*x^3 + a), x)