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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 1.29445, 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 \[ \frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{\sqrt{b^2-4 a c}+b} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{\sqrt{b^2-4 a c}+b} \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3 \sqrt [3]{2} \sqrt [3]{c} \sqrt{b^2-4 a c}}+\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} \sqrt [3]{c} \sqrt{b^2-4 a c}}-\frac{\sqrt [3]{\sqrt{b^2-4 a c}+b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3} \sqrt [3]{c} \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x^3 + c*x^6),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 139.29, size = 529, normalized size = 0.95 \[ - \frac{2^{\frac{2}{3}} \sqrt [3]{b - \sqrt{- 4 a c + b^{2}}} \log{\left (\sqrt [3]{2} \sqrt [3]{c} x + \sqrt [3]{b - \sqrt{- 4 a c + b^{2}}} \right )}}{6 \sqrt [3]{c} \sqrt{- 4 a c + b^{2}}} + \frac{2^{\frac{2}{3}} \sqrt [3]{b - \sqrt{- 4 a c + b^{2}}} \log{\left (c^{\frac{2}{3}} x^{2} - \frac{2^{\frac{2}{3}} \sqrt [3]{c} x \sqrt [3]{b - \sqrt{- 4 a c + b^{2}}}}{2} + \frac{\sqrt [3]{2} \left (b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}}}{2} \right )}}{12 \sqrt [3]{c} \sqrt{- 4 a c + b^{2}}} + \frac{2^{\frac{2}{3}} \sqrt{3} \sqrt [3]{b - \sqrt{- 4 a c + b^{2}}} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{3 \sqrt [3]{b - \sqrt{- 4 a c + b^{2}}}} + \frac{1}{3}\right ) \right )}}{6 \sqrt [3]{c} \sqrt{- 4 a c + b^{2}}} + \frac{2^{\frac{2}{3}} \sqrt [3]{b + \sqrt{- 4 a c + b^{2}}} \log{\left (\sqrt [3]{2} \sqrt [3]{c} x + \sqrt [3]{b + \sqrt{- 4 a c + b^{2}}} \right )}}{6 \sqrt [3]{c} \sqrt{- 4 a c + b^{2}}} - \frac{2^{\frac{2}{3}} \sqrt [3]{b + \sqrt{- 4 a c + b^{2}}} \log{\left (c^{\frac{2}{3}} x^{2} - \frac{2^{\frac{2}{3}} \sqrt [3]{c} x \sqrt [3]{b + \sqrt{- 4 a c + b^{2}}}}{2} + \frac{\sqrt [3]{2} \left (b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{2}{3}}}{2} \right )}}{12 \sqrt [3]{c} \sqrt{- 4 a c + b^{2}}} - \frac{2^{\frac{2}{3}} \sqrt{3} \sqrt [3]{b + \sqrt{- 4 a c + b^{2}}} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{3 \sqrt [3]{b + \sqrt{- 4 a c + b^{2}}}} + \frac{1}{3}\right ) \right )}}{6 \sqrt [3]{c} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.0273745, size = 42, normalized size = 0.08 \[ \frac{1}{3} \text{RootSum}\left [\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{2 \text{$\#$1}^3 c+b}\&\right ] \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.004, size = 43, normalized size = 0.1 \[{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}c+{{\it \_Z}}^{3}b+a \right ) }{\frac{{{\it \_R}}^{3}\ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}c+{{\it \_R}}^{2}b}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(c*x^6+b*x^3+a),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^3/(c*x^6 + b*x^3 + a), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.293198, size = 3260, normalized size = 5.84 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 6.32845, size = 122, normalized size = 0.22 \[ \operatorname{RootSum}{\left (t^{6} \left (46656 a^{3} c^{4} - 34992 a^{2} b^{2} c^{3} + 8748 a b^{4} c^{2} - 729 b^{6} c\right ) + t^{3} \left (432 a^{2} c^{2} - 216 a b^{2} c + 27 b^{4}\right ) + a, \left ( t \mapsto t \log{\left (x + \frac{2592 t^{4} a^{2} c^{3} - 1296 t^{4} a b^{2} c^{2} + 162 t^{4} b^{4} c + 12 t a c - 3 t b^{2}}{b} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(_t**6*(46656*a**3*c**4 - 34992*a**2*b**2*c**3 + 8748*a*b**4*c**2 - 729*b
**6*c) + _t**3*(432*a**2*c**2 - 216*a*b**2*c + 27*b**4) + a, Lambda(_t, _t*log(x
+ (2592*_t**4*a**2*c**3 - 1296*_t**4*a*b**2*c**2 + 162*_t**4*b**4*c + 12*_t*a*c
- 3*_t*b**2)/b)))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^3/(c*x^6 + b*x^3 + a), x)