3.145 \(\int \frac{x^4}{a+b x^3+c x^6} \, dx\)

Optimal. Leaf size=558 \[ -\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left (\sqrt{b^2-4 a c}+b\right )^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}-\frac{\left (\sqrt{b^2-4 a c}+b\right )^{2/3} \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt{b^2-4 a c}}-\frac{\left (\sqrt{b^2-4 a c}+b\right )^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt{b^2-4 a c}} \]

[Out]

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

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Rubi [A]  time = 1.27545, antiderivative size = 558, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389 \[ -\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left (\sqrt{b^2-4 a c}+b\right )^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}-\frac{\left (\sqrt{b^2-4 a c}+b\right )^{2/3} \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt{b^2-4 a c}}+\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt{b^2-4 a c}}-\frac{\left (\sqrt{b^2-4 a c}+b\right )^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^3 + c*x^6),x]

[Out]

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

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Rubi in Sympy [A]  time = 149.312, size = 529, normalized size = 0.95 \[ \frac{\sqrt [3]{2} \left (b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}} \log{\left (\sqrt [3]{2} \sqrt [3]{c} x + \sqrt [3]{b - \sqrt{- 4 a c + b^{2}}} \right )}}{6 c^{\frac{2}{3}} \sqrt{- 4 a c + b^{2}}} - \frac{\sqrt [3]{2} \left (b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}} \log{\left (c^{\frac{2}{3}} x^{2} - \frac{2^{\frac{2}{3}} \sqrt [3]{c} x \sqrt [3]{b - \sqrt{- 4 a c + b^{2}}}}{2} + \frac{\sqrt [3]{2} \left (b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}}}{2} \right )}}{12 c^{\frac{2}{3}} \sqrt{- 4 a c + b^{2}}} + \frac{\sqrt [3]{2} \sqrt{3} \left (b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{3 \sqrt [3]{b - \sqrt{- 4 a c + b^{2}}}} + \frac{1}{3}\right ) \right )}}{6 c^{\frac{2}{3}} \sqrt{- 4 a c + b^{2}}} - \frac{\sqrt [3]{2} \left (b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}} \log{\left (\sqrt [3]{2} \sqrt [3]{c} x + \sqrt [3]{b + \sqrt{- 4 a c + b^{2}}} \right )}}{6 c^{\frac{2}{3}} \sqrt{- 4 a c + b^{2}}} + \frac{\sqrt [3]{2} \left (b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}} \log{\left (c^{\frac{2}{3}} x^{2} - \frac{2^{\frac{2}{3}} \sqrt [3]{c} x \sqrt [3]{b + \sqrt{- 4 a c + b^{2}}}}{2} + \frac{\sqrt [3]{2} \left (b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}}}{2} \right )}}{12 c^{\frac{2}{3}} \sqrt{- 4 a c + b^{2}}} - \frac{\sqrt [3]{2} \sqrt{3} \left (b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{3 \sqrt [3]{b + \sqrt{- 4 a c + b^{2}}}} + \frac{1}{3}\right ) \right )}}{6 c^{\frac{2}{3}} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(c*x**6+b*x**3+a),x)

[Out]

2**(1/3)*(b - sqrt(-4*a*c + b**2))**(2/3)*log(2**(1/3)*c**(1/3)*x + (b - sqrt(-4
*a*c + b**2))**(1/3))/(6*c**(2/3)*sqrt(-4*a*c + b**2)) - 2**(1/3)*(b - sqrt(-4*a
*c + b**2))**(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)/(12*c**(2/3)*sqrt(-
4*a*c + b**2)) + 2**(1/3)*sqrt(3)*(b - sqrt(-4*a*c + b**2))**(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))/(6*c**(2/3)
*sqrt(-4*a*c + b**2)) - 2**(1/3)*(b + sqrt(-4*a*c + b**2))**(2/3)*log(2**(1/3)*c
**(1/3)*x + (b + sqrt(-4*a*c + b**2))**(1/3))/(6*c**(2/3)*sqrt(-4*a*c + b**2)) +
2**(1/3)*(b + sqrt(-4*a*c + b**2))**(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)/(12*c**(2/3)*sqrt(-4*a*c + b**2)) - 2**(1/3)*sqrt(3)*(b + sqrt(-4*a*c + b**
2))**(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))/(6*c**(2/3)*sqrt(-4*a*c + b**2))

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Mathematica [C]  time = 0.0301178, size = 44, normalized size = 0.08 \[ \frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})}{2 \text{$\#$1}^3 c+b}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*x^3 + c*x^6),x]

[Out]

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

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Maple [C]  time = 0.005, size = 43, normalized size = 0.1 \[{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}c+{{\it \_Z}}^{3}b+a \right ) }{\frac{{{\it \_R}}^{4}\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}c+{{\it \_R}}^{2}b}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(c*x^6+b*x^3+a),x)

[Out]

1/3*sum(_R^4/(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. \[ \int \frac{x^{4}}{c x^{6} + b x^{3} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(c*x^6 + b*x^3 + a),x, algorithm="maxima")

[Out]

integrate(x^4/(c*x^6 + b*x^3 + a), x)

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Fricas [A]  time = 0.329221, size = 5165, normalized size = 9.26 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(c*x^6 + b*x^3 + a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.78376, size = 175, normalized size = 0.31 \[ \operatorname{RootSum}{\left (t^{6} \left (46656 a^{3} c^{5} - 34992 a^{2} b^{2} c^{4} + 8748 a b^{4} c^{3} - 729 b^{6} c^{2}\right ) + t^{3} \left (- 432 a^{2} b c^{2} + 216 a b^{3} c - 27 b^{5}\right ) + a^{2}, \left ( t \mapsto t \log{\left (x + \frac{15552 t^{5} a^{3} c^{5} - 11664 t^{5} a^{2} b^{2} c^{4} + 2916 t^{5} a b^{4} c^{3} - 243 t^{5} b^{6} c^{2} - 108 t^{2} a^{2} b c^{2} + 63 t^{2} a b^{3} c - 9 t^{2} b^{5}}{2 a^{2} c - a b^{2}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(c*x**6+b*x**3+a),x)

[Out]

RootSum(_t**6*(46656*a**3*c**5 - 34992*a**2*b**2*c**4 + 8748*a*b**4*c**3 - 729*b
**6*c**2) + _t**3*(-432*a**2*b*c**2 + 216*a*b**3*c - 27*b**5) + a**2, Lambda(_t,
_t*log(x + (15552*_t**5*a**3*c**5 - 11664*_t**5*a**2*b**2*c**4 + 2916*_t**5*a*b
**4*c**3 - 243*_t**5*b**6*c**2 - 108*_t**2*a**2*b*c**2 + 63*_t**2*a*b**3*c - 9*_
t**2*b**5)/(2*a**2*c - a*b**2))))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{c x^{6} + b x^{3} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(c*x^6 + b*x^3 + a),x, algorithm="giac")

[Out]

integrate(x^4/(c*x^6 + b*x^3 + a), x)