3.2.47 \(\int \frac {x}{a+b x^3+c x^6} \, dx\) [147]

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

[Out]

-1/3*2^(1/3)*c^(1/3)*ln(2^(1/3)*c^(1/3)*x+(b-(-4*a*c+b^2)^(1/2))^(1/3))/(b-(-4*a*c+b^2)^(1/2))^(1/3)/(-4*a*c+b
^2)^(1/2)+1/6*c^(1/3)*ln(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b-(-4*a*c+b^2)^(1/2))^(1/3)+(b-(-4*a*c+b^2)^(1
/2))^(2/3))*2^(1/3)/(b-(-4*a*c+b^2)^(1/2))^(1/3)/(-4*a*c+b^2)^(1/2)-1/3*2^(1/3)*c^(1/3)*arctan(1/3*(1-2*2^(1/3
)*c^(1/3)*x/(b-(-4*a*c+b^2)^(1/2))^(1/3))*3^(1/2))*3^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/3)/(-4*a*c+b^2)^(1/2)+1/3
*2^(1/3)*c^(1/3)*ln(2^(1/3)*c^(1/3)*x+(b+(-4*a*c+b^2)^(1/2))^(1/3))/(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^
(1/3)-1/6*c^(1/3)*ln(2^(2/3)*c^(2/3)*x^2-2^(1/3)*c^(1/3)*x*(b+(-4*a*c+b^2)^(1/2))^(1/3)+(b+(-4*a*c+b^2)^(1/2))
^(2/3))*2^(1/3)/(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/3)+1/3*2^(1/3)*c^(1/3)*arctan(1/3*(1-2*2^(1/3)*c^
(1/3)*x/(b+(-4*a*c+b^2)^(1/2))^(1/3))*3^(1/2))*3^(1/2)/(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/3)

________________________________________________________________________________________

Rubi [A]
time = 0.27, antiderivative size = 558, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1389, 298, 31, 648, 631, 210, 642} \begin {gather*} -\frac {\sqrt [3]{2} \sqrt [3]{c} \text {ArcTan}\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} \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{2} \sqrt [3]{c} \text {ArcTan}\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} \sqrt [3]{\sqrt {b^2-4 a c}+b}}+\frac {\sqrt [3]{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 )}{3\ 2^{2/3} \sqrt {b^2-4 a c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\sqrt [3]{c} \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\ 2^{2/3} \sqrt {b^2-4 a c} \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\sqrt [3]{2} \sqrt [3]{c} \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} \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{2} \sqrt [3]{c} \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} \sqrt [3]{\sqrt {b^2-4 a c}+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^3 + c*x^6),x]

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1389

Int[((d_.)*(x_))^(m_.)/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]
}, Dist[c/q, Int[(d*x)^m/(b/2 - q/2 + c*x^n), x], x] - Dist[c/q, Int[(d*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; F
reeQ[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {x}{a+b x^3+c x^6} \, dx &=\frac {c \int \frac {x}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^3} \, dx}{\sqrt {b^2-4 a c}}-\frac {c \int \frac {x}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^3} \, dx}{\sqrt {b^2-4 a c}}\\ &=-\frac {\left (\sqrt [3]{2} c^{2/3}\right ) \int \frac {1}{\frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3 \sqrt {b^2-4 a c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\left (\sqrt [3]{2} c^{2/3}\right ) \int \frac {\frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x}{\frac {\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3 \sqrt {b^2-4 a c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\left (\sqrt [3]{2} c^{2/3}\right ) \int \frac {1}{\frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3 \sqrt {b^2-4 a c} \sqrt [3]{b+\sqrt {b^2-4 a c}}}-\frac {\left (\sqrt [3]{2} c^{2/3}\right ) \int \frac {\frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x}{\frac {\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3 \sqrt {b^2-4 a c} \sqrt [3]{b+\sqrt {b^2-4 a c}}}\\ &=-\frac {\sqrt [3]{2} \sqrt [3]{c} \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} \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{2} \sqrt [3]{c} \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} \sqrt [3]{b+\sqrt {b^2-4 a c}}}+\frac {c^{2/3} \int \frac {1}{\frac {\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{2 \sqrt {b^2-4 a c}}-\frac {c^{2/3} \int \frac {1}{\frac {\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{2 \sqrt {b^2-4 a c}}+\frac {\sqrt [3]{c} \int \frac {-\frac {\sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac {\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3\ 2^{2/3} \sqrt {b^2-4 a c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\sqrt [3]{c} \int \frac {-\frac {\sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac {\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3\ 2^{2/3} \sqrt {b^2-4 a c} \sqrt [3]{b+\sqrt {b^2-4 a c}}}\\ &=-\frac {\sqrt [3]{2} \sqrt [3]{c} \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} \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{2} \sqrt [3]{c} \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} \sqrt [3]{b+\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{c} \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{3\ 2^{2/3} \sqrt {b^2-4 a c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\sqrt [3]{c} \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{3\ 2^{2/3} \sqrt {b^2-4 a c} \sqrt [3]{b+\sqrt {b^2-4 a c}}}+\frac {\left (\sqrt [3]{2} \sqrt [3]{c}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\left (\sqrt [3]{2} \sqrt [3]{c}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt [3]{b+\sqrt {b^2-4 a c}}}\\ &=-\frac {\sqrt [3]{2} \sqrt [3]{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} \sqrt {b^2-4 a c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{2} \sqrt [3]{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} \sqrt {b^2-4 a c} \sqrt [3]{b+\sqrt {b^2-4 a c}}}-\frac {\sqrt [3]{2} \sqrt [3]{c} \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} \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{2} \sqrt [3]{c} \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} \sqrt [3]{b+\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{c} \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{3\ 2^{2/3} \sqrt {b^2-4 a c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\sqrt [3]{c} \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{3\ 2^{2/3} \sqrt {b^2-4 a c} \sqrt [3]{b+\sqrt {b^2-4 a c}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.01, size = 43, normalized size = 0.08 \begin {gather*} \frac {1}{3} \text {RootSum}\left [a+b \text {$\#$1}^3+c \text {$\#$1}^6\&,\frac {\log (x-\text {$\#$1})}{b \text {$\#$1}+2 c \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^3 + c*x^6),x]

[Out]

RootSum[a + b*#1^3 + c*#1^6 & , Log[x - #1]/(b*#1 + 2*c*#1^4) & ]/3

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.03, size = 41, normalized size = 0.07

method	result	size
default	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{6}+b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\textit {\_R} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} c +b \,\textit {\_R}^{2}}\right )}{3}\)	\(41\)
risch	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{6}+b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\textit {\_R} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} c +b \,\textit {\_R}^{2}}\right )}{3}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(c*x^6+b*x^3+a),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(c*x^6 + b*x^3 + a), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2980 vs. \(2 (421) = 842\).
time = 0.43, size = 2980, normalized size = 5.34 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.81, size = 158, normalized size = 0.28 \begin {gather*} \operatorname {RootSum} {\left (t^{6} \cdot \left (46656 a^{4} c^{3} - 34992 a^{3} b^{2} c^{2} + 8748 a^{2} b^{4} c - 729 a b^{6}\right ) + t^{3} \left (- 432 a^{2} c^{2} + 216 a b^{2} c - 27 b^{4}\right ) + c, \left ( t \mapsto t \log {\left (x + \frac {- 15552 t^{5} a^{4} c^{3} + 11664 t^{5} a^{3} b^{2} c^{2} - 2916 t^{5} a^{2} b^{4} c + 243 t^{5} a b^{6} + 72 t^{2} a^{2} c^{2} - 54 t^{2} a b^{2} c + 9 t^{2} b^{4}}{b c} \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(_t**6*(46656*a**4*c**3 - 34992*a**3*b**2*c**2 + 8748*a**2*b**4*c - 729*a*b**6) + _t**3*(-432*a**2*c**2
+ 216*a*b**2*c - 27*b**4) + c, Lambda(_t, _t*log(x + (-15552*_t**5*a**4*c**3 + 11664*_t**5*a**3*b**2*c**2 - 2
916*_t**5*a**2*b**4*c + 243*_t**5*a*b**6 + 72*_t**2*a**2*c**2 - 54*_t**2*a*b**2*c + 9*_t**2*b**4)/(b*c))))

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/(c*x^6 + b*x^3 + a), x)

________________________________________________________________________________________

Mupad [B]
time = 5.39, size = 1543, normalized size = 2.77 \begin {gather*} \ln \left (c^4\,x-\frac {\left (27\,c^3\,x\,\left (8\,a^2\,c^2-6\,a\,b^2\,c+b^4\right )+\frac {27\,2^{1/3}\,a\,b\,c^3\,{\left (4\,a\,c-b^2\right )}^2\,{\left (\frac {b\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+b^4+16\,a^2\,c^2-8\,a\,b^2\,c}{a\,{\left (4\,a\,c-b^2\right )}^3}\right )}^{2/3}}{2}\right )\,\left (b\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+b^4+16\,a^2\,c^2-8\,a\,b^2\,c\right )}{54\,a\,{\left (4\,a\,c-b^2\right )}^3}\right )\,{\left (-\frac {b\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+b^4+16\,a^2\,c^2-8\,a\,b^2\,c}{54\,\left (-64\,a^4\,c^3+48\,a^3\,b^2\,c^2-12\,a^2\,b^4\,c+a\,b^6\right )}\right )}^{1/3}+\ln \left (c^4\,x+\frac {\left (27\,c^3\,x\,\left (8\,a^2\,c^2-6\,a\,b^2\,c+b^4\right )+\frac {27\,2^{1/3}\,a\,b\,c^3\,{\left (4\,a\,c-b^2\right )}^2\,{\left (-\frac {b\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^4-16\,a^2\,c^2+8\,a\,b^2\,c}{a\,{\left (4\,a\,c-b^2\right )}^3}\right )}^{2/3}}{2}\right )\,\left (b\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^4-16\,a^2\,c^2+8\,a\,b^2\,c\right )}{54\,a\,{\left (4\,a\,c-b^2\right )}^3}\right )\,{\left (\frac {b\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^4-16\,a^2\,c^2+8\,a\,b^2\,c}{54\,\left (-64\,a^4\,c^3+48\,a^3\,b^2\,c^2-12\,a^2\,b^4\,c+a\,b^6\right )}\right )}^{1/3}-\ln \left (c^4\,x-\frac {\left (27\,c^3\,x\,\left (8\,a^2\,c^2-6\,a\,b^2\,c+b^4\right )+\frac {27\,2^{1/3}\,a\,b\,c^3\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (4\,a\,c-b^2\right )}^2\,{\left (\frac {b\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+b^4+16\,a^2\,c^2-8\,a\,b^2\,c}{a\,{\left (4\,a\,c-b^2\right )}^3}\right )}^{2/3}}{4}\right )\,\left (b\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+b^4+16\,a^2\,c^2-8\,a\,b^2\,c\right )}{54\,a\,{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {b\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+b^4+16\,a^2\,c^2-8\,a\,b^2\,c}{54\,\left (-64\,a^4\,c^3+48\,a^3\,b^2\,c^2-12\,a^2\,b^4\,c+a\,b^6\right )}\right )}^{1/3}+\ln \left (c^4\,x-\frac {\left (27\,c^3\,x\,\left (8\,a^2\,c^2-6\,a\,b^2\,c+b^4\right )-\frac {27\,2^{1/3}\,a\,b\,c^3\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (4\,a\,c-b^2\right )}^2\,{\left (\frac {b\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+b^4+16\,a^2\,c^2-8\,a\,b^2\,c}{a\,{\left (4\,a\,c-b^2\right )}^3}\right )}^{2/3}}{4}\right )\,\left (b\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+b^4+16\,a^2\,c^2-8\,a\,b^2\,c\right )}{54\,a\,{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {b\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+b^4+16\,a^2\,c^2-8\,a\,b^2\,c}{54\,\left (-64\,a^4\,c^3+48\,a^3\,b^2\,c^2-12\,a^2\,b^4\,c+a\,b^6\right )}\right )}^{1/3}-\ln \left (c^4\,x+\frac {\left (27\,c^3\,x\,\left (8\,a^2\,c^2-6\,a\,b^2\,c+b^4\right )+\frac {27\,2^{1/3}\,a\,b\,c^3\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (4\,a\,c-b^2\right )}^2\,{\left (-\frac {b\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^4-16\,a^2\,c^2+8\,a\,b^2\,c}{a\,{\left (4\,a\,c-b^2\right )}^3}\right )}^{2/3}}{4}\right )\,\left (b\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^4-16\,a^2\,c^2+8\,a\,b^2\,c\right )}{54\,a\,{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {b\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^4-16\,a^2\,c^2+8\,a\,b^2\,c}{54\,\left (-64\,a^4\,c^3+48\,a^3\,b^2\,c^2-12\,a^2\,b^4\,c+a\,b^6\right )}\right )}^{1/3}+\ln \left (c^4\,x+\frac {\left (27\,c^3\,x\,\left (8\,a^2\,c^2-6\,a\,b^2\,c+b^4\right )-\frac {27\,2^{1/3}\,a\,b\,c^3\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (4\,a\,c-b^2\right )}^2\,{\left (-\frac {b\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^4-16\,a^2\,c^2+8\,a\,b^2\,c}{a\,{\left (4\,a\,c-b^2\right )}^3}\right )}^{2/3}}{4}\right )\,\left (b\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^4-16\,a^2\,c^2+8\,a\,b^2\,c\right )}{54\,a\,{\left (4\,a\,c-b^2\right )}^3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {b\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^4-16\,a^2\,c^2+8\,a\,b^2\,c}{54\,\left (-64\,a^4\,c^3+48\,a^3\,b^2\,c^2-12\,a^2\,b^4\,c+a\,b^6\right )}\right )}^{1/3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^3 + c*x^6),x)

[Out]

log(c^4*x - ((27*c^3*x*(b^4 + 8*a^2*c^2 - 6*a*b^2*c) + (27*2^(1/3)*a*b*c^3*(4*a*c - b^2)^2*((b*(-(4*a*c - b^2)
^3)^(1/2) + b^4 + 16*a^2*c^2 - 8*a*b^2*c)/(a*(4*a*c - b^2)^3))^(2/3))/2)*(b*(-(4*a*c - b^2)^3)^(1/2) + b^4 + 1
6*a^2*c^2 - 8*a*b^2*c))/(54*a*(4*a*c - b^2)^3))*(-(b*(-(4*a*c - b^2)^3)^(1/2) + b^4 + 16*a^2*c^2 - 8*a*b^2*c)/
(54*(a*b^6 - 64*a^4*c^3 - 12*a^2*b^4*c + 48*a^3*b^2*c^2)))^(1/3) + log(c^4*x + ((27*c^3*x*(b^4 + 8*a^2*c^2 - 6
*a*b^2*c) + (27*2^(1/3)*a*b*c^3*(4*a*c - b^2)^2*(-(b*(-(4*a*c - b^2)^3)^(1/2) - b^4 - 16*a^2*c^2 + 8*a*b^2*c)/
(a*(4*a*c - b^2)^3))^(2/3))/2)*(b*(-(4*a*c - b^2)^3)^(1/2) - b^4 - 16*a^2*c^2 + 8*a*b^2*c))/(54*a*(4*a*c - b^2
)^3))*((b*(-(4*a*c - b^2)^3)^(1/2) - b^4 - 16*a^2*c^2 + 8*a*b^2*c)/(54*(a*b^6 - 64*a^4*c^3 - 12*a^2*b^4*c + 48
*a^3*b^2*c^2)))^(1/3) - log(c^4*x - ((27*c^3*x*(b^4 + 8*a^2*c^2 - 6*a*b^2*c) + (27*2^(1/3)*a*b*c^3*(3^(1/2)*1i
- 1)*(4*a*c - b^2)^2*((b*(-(4*a*c - b^2)^3)^(1/2) + b^4 + 16*a^2*c^2 - 8*a*b^2*c)/(a*(4*a*c - b^2)^3))^(2/3))
/4)*(b*(-(4*a*c - b^2)^3)^(1/2) + b^4 + 16*a^2*c^2 - 8*a*b^2*c))/(54*a*(4*a*c - b^2)^3))*((3^(1/2)*1i)/2 + 1/2
)*(-(b*(-(4*a*c - b^2)^3)^(1/2) + b^4 + 16*a^2*c^2 - 8*a*b^2*c)/(54*(a*b^6 - 64*a^4*c^3 - 12*a^2*b^4*c + 48*a^
3*b^2*c^2)))^(1/3) + log(c^4*x - ((27*c^3*x*(b^4 + 8*a^2*c^2 - 6*a*b^2*c) - (27*2^(1/3)*a*b*c^3*(3^(1/2)*1i +
1)*(4*a*c - b^2)^2*((b*(-(4*a*c - b^2)^3)^(1/2) + b^4 + 16*a^2*c^2 - 8*a*b^2*c)/(a*(4*a*c - b^2)^3))^(2/3))/4)
*(b*(-(4*a*c - b^2)^3)^(1/2) + b^4 + 16*a^2*c^2 - 8*a*b^2*c))/(54*a*(4*a*c - b^2)^3))*((3^(1/2)*1i)/2 - 1/2)*(
-(b*(-(4*a*c - b^2)^3)^(1/2) + b^4 + 16*a^2*c^2 - 8*a*b^2*c)/(54*(a*b^6 - 64*a^4*c^3 - 12*a^2*b^4*c + 48*a^3*b
^2*c^2)))^(1/3) - log(c^4*x + ((27*c^3*x*(b^4 + 8*a^2*c^2 - 6*a*b^2*c) + (27*2^(1/3)*a*b*c^3*(3^(1/2)*1i - 1)*
(4*a*c - b^2)^2*(-(b*(-(4*a*c - b^2)^3)^(1/2) - b^4 - 16*a^2*c^2 + 8*a*b^2*c)/(a*(4*a*c - b^2)^3))^(2/3))/4)*(
b*(-(4*a*c - b^2)^3)^(1/2) - b^4 - 16*a^2*c^2 + 8*a*b^2*c))/(54*a*(4*a*c - b^2)^3))*((3^(1/2)*1i)/2 + 1/2)*((b
*(-(4*a*c - b^2)^3)^(1/2) - b^4 - 16*a^2*c^2 + 8*a*b^2*c)/(54*(a*b^6 - 64*a^4*c^3 - 12*a^2*b^4*c + 48*a^3*b^2*
c^2)))^(1/3) + log(c^4*x + ((27*c^3*x*(b^4 + 8*a^2*c^2 - 6*a*b^2*c) - (27*2^(1/3)*a*b*c^3*(3^(1/2)*1i + 1)*(4*
a*c - b^2)^2*(-(b*(-(4*a*c - b^2)^3)^(1/2) - b^4 - 16*a^2*c^2 + 8*a*b^2*c)/(a*(4*a*c - b^2)^3))^(2/3))/4)*(b*(
-(4*a*c - b^2)^3)^(1/2) - b^4 - 16*a^2*c^2 + 8*a*b^2*c))/(54*a*(4*a*c - b^2)^3))*((3^(1/2)*1i)/2 - 1/2)*((b*(-
(4*a*c - b^2)^3)^(1/2) - b^4 - 16*a^2*c^2 + 8*a*b^2*c)/(54*(a*b^6 - 64*a^4*c^3 - 12*a^2*b^4*c + 48*a^3*b^2*c^2
)))^(1/3)

________________________________________________________________________________________