\(\int \frac {1}{x^2 (a+b x^3+c x^6)} \, dx\) [149]

Optimal result

Integrand size = 18, antiderivative size = 610 \[ \int \frac {1}{x^2 \left (a+b x^3+c x^6\right )} \, dx=-\frac {1}{a x}+\frac {\sqrt [3]{c} \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} a \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} a \sqrt [3]{b+\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{c} \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} a \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} a \sqrt [3]{b+\sqrt {b^2-4 a c}}}-\frac {\sqrt [3]{c} \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} a \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\sqrt [3]{c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} a \sqrt [3]{b+\sqrt {b^2-4 a c}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1382, 1524, 298, 31, 648, 631, 210, 642} \[ \int \frac {1}{x^2 \left (a+b x^3+c x^6\right )} \, dx=\frac {\sqrt [3]{c} \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} a \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} a \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\sqrt [3]{c} \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt {b^2-4 a c}}+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} a \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\sqrt [3]{c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt {b^2-4 a c}+b}+\left (\sqrt {b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} a \sqrt [3]{\sqrt {b^2-4 a c}+b}}+\frac {\sqrt [3]{c} \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} a \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} a \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {1}{a x} \]

[In]

Int[1/(x^2*(a + b*x^3 + c*x^6)),x]

[Out]

-(1/(a*x)) + (c^(1/3)*(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^(2/3)*Sqrt[3]*a*(b - Sqrt[b^2 - 4*a*c])^(1/3)) + (c^(1/3)*(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^(2/3)*Sqrt[3]*a*(b + Sqrt[b^2 - 4*a*c])^(1
/3)) + (c^(1/3)*(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^(2/3)*a
*(b - Sqrt[b^2 - 4*a*c])^(1/3)) + (c^(1/3)*(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^(2/3)*a*(b + Sqrt[b^2 - 4*a*c])^(1/3)) - (c^(1/3)*(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^(2/3)*a*(b
- Sqrt[b^2 - 4*a*c])^(1/3)) - (c^(1/3)*(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^(2/3)*a*(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 1382

Int[((d_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(d*x)^(m + 1)*((a +
b*x^n + c*x^(2*n))^(p + 1)/(a*d*(m + 1))), x] - Dist[1/(a*d^n*(m + 1)), Int[(d*x)^(m + n)*(b*(m + n*(p + 1) +
1) + c*(m + 2*n*(p + 1) + 1)*x^n)*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[n2, 2
*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntegerQ[p]

Rule 1524

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

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.12 \[ \int \frac {1}{x^2 \left (a+b x^3+c x^6\right )} \, dx=-\frac {1}{a x}-\frac {\text {RootSum}\left [a+b \text {$\#$1}^3+c \text {$\#$1}^6\&,\frac {b \log (x-\text {$\#$1})+c \log (x-\text {$\#$1}) \text {$\#$1}^3}{b \text {$\#$1}+2 c \text {$\#$1}^4}\&\right ]}{3 a} \]

[In]

Integrate[1/(x^2*(a + b*x^3 + c*x^6)),x]

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.10

method	result	size
default	\(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{6}+\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (\textit {\_R}^{4} c +\textit {\_R} b \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} c +b \,\textit {\_R}^{2}}}{3 a}-\frac {1}{a x}\)	\(61\)
risch	\(-\frac {1}{a x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 c^{3} a^{7}-48 b^{2} c^{2} a^{6}+12 b^{4} c \,a^{5}-b^{6} a^{4}\right ) \textit {\_Z}^{6}+\left (-32 b \,c^{3} a^{3}+32 b^{3} c^{2} a^{2}-10 b^{5} c a +b^{7}\right ) \textit {\_Z}^{3}+c^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (224 c^{3} a^{7}-176 b^{2} c^{2} a^{6}+46 b^{4} c \,a^{5}-4 b^{6} a^{4}\right ) \textit {\_R}^{6}+\left (-100 b \,c^{3} a^{3}+97 b^{3} c^{2} a^{2}-30 b^{5} c a +3 b^{7}\right ) \textit {\_R}^{3}+3 c^{4}\right ) x +\left (16 a^{6} c^{3}-24 a^{5} b^{2} c^{2}+9 a^{4} b^{4} c -a^{3} b^{6}\right ) \textit {\_R}^{5}-a^{2} b \,c^{3} \textit {\_R}^{2}\right )\right )}{3}\)	\(239\)

[In]

int(1/x^2/(c*x^6+b*x^3+a),x,method=_RETURNVERBOSE)

[Out]

-1/3/a*sum((_R^4*c+_R*b)/(2*_R^5*c+_R^2*b)*ln(x-_R),_R=RootOf(_Z^6*c+_Z^3*b+a))-1/a/x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3225 vs. \(2 (471) = 942\).

Time = 0.39 (sec) , antiderivative size = 3225, normalized size of antiderivative = 5.29 \[ \int \frac {1}{x^2 \left (a+b x^3+c x^6\right )} \, dx=\text {Too large to display} \]

[In]

integrate(1/x^2/(c*x^6+b*x^3+a),x, algorithm="fricas")

[Out]

1/6*(2*(1/2)^(1/3)*a*x*((b^3 - 2*a*b*c + (a^4*b^2 - 4*a^5*c)*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b
^2*c^3 + 4*a^4*c^4)/(a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)))/(a^4*b^2 - 4*a^5*c))^(1/3)*log(
(1/2)^(2/3)*(b^9 - 11*a*b^7*c + 42*a^2*b^5*c^2 - 62*a^3*b^3*c^3 + 24*a^4*b*c^4 - (a^4*b^8 - 13*a^5*b^6*c + 60*
a^6*b^4*c^2 - 112*a^7*b^2*c^3 + 64*a^8*c^4)*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^
4)/(a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)))*((b^3 - 2*a*b*c + (a^4*b^2 - 4*a^5*c)*sqrt((b^8
- 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11
*c^3)))/(a^4*b^2 - 4*a^5*c))^(2/3) + 2*(b^4*c^3 - 4*a*b^2*c^4 + 2*a^2*c^5)*x) + 2*(1/2)^(1/3)*a*x*((b^3 - 2*a*
b*c - (a^4*b^2 - 4*a^5*c)*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(a^8*b^6 - 12*a
^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)))/(a^4*b^2 - 4*a^5*c))^(1/3)*log((1/2)^(2/3)*(b^9 - 11*a*b^7*c + 42*
a^2*b^5*c^2 - 62*a^3*b^3*c^3 + 24*a^4*b*c^4 + (a^4*b^8 - 13*a^5*b^6*c + 60*a^6*b^4*c^2 - 112*a^7*b^2*c^3 + 64*
a^8*c^4)*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(a^8*b^6 - 12*a^9*b^4*c + 48*a^1
0*b^2*c^2 - 64*a^11*c^3)))*((b^3 - 2*a*b*c - (a^4*b^2 - 4*a^5*c)*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a
^3*b^2*c^3 + 4*a^4*c^4)/(a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)))/(a^4*b^2 - 4*a^5*c))^(2/3)
+ 2*(b^4*c^3 - 4*a*b^2*c^4 + 2*a^2*c^5)*x) + (1/2)^(1/3)*(sqrt(-3)*a*x - a*x)*((b^3 - 2*a*b*c + (a^4*b^2 - 4*a
^5*c)*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b
^2*c^2 - 64*a^11*c^3)))/(a^4*b^2 - 4*a^5*c))^(1/3)*log(-(1/2)^(2/3)*(b^9 - 11*a*b^7*c + 42*a^2*b^5*c^2 - 62*a^
3*b^3*c^3 + 24*a^4*b*c^4 + sqrt(-3)*(b^9 - 11*a*b^7*c + 42*a^2*b^5*c^2 - 62*a^3*b^3*c^3 + 24*a^4*b*c^4) - (a^4
*b^8 - 13*a^5*b^6*c + 60*a^6*b^4*c^2 - 112*a^7*b^2*c^3 + 64*a^8*c^4 + sqrt(-3)*(a^4*b^8 - 13*a^5*b^6*c + 60*a^
6*b^4*c^2 - 112*a^7*b^2*c^3 + 64*a^8*c^4))*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4
)/(a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)))*((b^3 - 2*a*b*c + (a^4*b^2 - 4*a^5*c)*sqrt((b^8 -
8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*
c^3)))/(a^4*b^2 - 4*a^5*c))^(2/3) + 4*(b^4*c^3 - 4*a*b^2*c^4 + 2*a^2*c^5)*x) - (1/2)^(1/3)*(sqrt(-3)*a*x + a*x
)*((b^3 - 2*a*b*c + (a^4*b^2 - 4*a^5*c)*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(
a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)))/(a^4*b^2 - 4*a^5*c))^(1/3)*log(-(1/2)^(2/3)*(b^9 - 1
1*a*b^7*c + 42*a^2*b^5*c^2 - 62*a^3*b^3*c^3 + 24*a^4*b*c^4 - sqrt(-3)*(b^9 - 11*a*b^7*c + 42*a^2*b^5*c^2 - 62*
a^3*b^3*c^3 + 24*a^4*b*c^4) - (a^4*b^8 - 13*a^5*b^6*c + 60*a^6*b^4*c^2 - 112*a^7*b^2*c^3 + 64*a^8*c^4 - sqrt(-
3)*(a^4*b^8 - 13*a^5*b^6*c + 60*a^6*b^4*c^2 - 112*a^7*b^2*c^3 + 64*a^8*c^4))*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^
4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)))*((b^3 - 2*a*b*c
+ (a^4*b^2 - 4*a^5*c)*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(a^8*b^6 - 12*a^9*
b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)))/(a^4*b^2 - 4*a^5*c))^(2/3) + 4*(b^4*c^3 - 4*a*b^2*c^4 + 2*a^2*c^5)*x)
+ (1/2)^(1/3)*(sqrt(-3)*a*x - a*x)*((b^3 - 2*a*b*c - (a^4*b^2 - 4*a^5*c)*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c
^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)))/(a^4*b^2 - 4*a^5*c
))^(1/3)*log(-(1/2)^(2/3)*(b^9 - 11*a*b^7*c + 42*a^2*b^5*c^2 - 62*a^3*b^3*c^3 + 24*a^4*b*c^4 + sqrt(-3)*(b^9 -
11*a*b^7*c + 42*a^2*b^5*c^2 - 62*a^3*b^3*c^3 + 24*a^4*b*c^4) + (a^4*b^8 - 13*a^5*b^6*c + 60*a^6*b^4*c^2 - 112
*a^7*b^2*c^3 + 64*a^8*c^4 + sqrt(-3)*(a^4*b^8 - 13*a^5*b^6*c + 60*a^6*b^4*c^2 - 112*a^7*b^2*c^3 + 64*a^8*c^4))
*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^
2 - 64*a^11*c^3)))*((b^3 - 2*a*b*c - (a^4*b^2 - 4*a^5*c)*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c
^3 + 4*a^4*c^4)/(a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)))/(a^4*b^2 - 4*a^5*c))^(2/3) + 4*(b^4
*c^3 - 4*a*b^2*c^4 + 2*a^2*c^5)*x) - (1/2)^(1/3)*(sqrt(-3)*a*x + a*x)*((b^3 - 2*a*b*c - (a^4*b^2 - 4*a^5*c)*sq
rt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 -
64*a^11*c^3)))/(a^4*b^2 - 4*a^5*c))^(1/3)*log(-(1/2)^(2/3)*(b^9 - 11*a*b^7*c + 42*a^2*b^5*c^2 - 62*a^3*b^3*c^
3 + 24*a^4*b*c^4 - sqrt(-3)*(b^9 - 11*a*b^7*c + 42*a^2*b^5*c^2 - 62*a^3*b^3*c^3 + 24*a^4*b*c^4) + (a^4*b^8 - 1
3*a^5*b^6*c + 60*a^6*b^4*c^2 - 112*a^7*b^2*c^3 + 64*a^8*c^4 - sqrt(-3)*(a^4*b^8 - 13*a^5*b^6*c + 60*a^6*b^4*c^
2 - 112*a^7*b^2*c^3 + 64*a^8*c^4))*sqrt((b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(a^8*b
^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)))*((b^3 - 2*a*b*c - (a^4*b^2 - 4*a^5*c)*sqrt((b^8 - 8*a*b^6
*c + 20*a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 4*a^4*c^4)/(a^8*b^6 - 12*a^9*b^4*c + 48*a^10*b^2*c^2 - 64*a^11*c^3)))/(
a^4*b^2 - 4*a^5*c))^(2/3) + 4*(b^4*c^3 - 4*a*b^2*c^4 + 2*a^2*c^5)*x) - 6)/(a*x)

Sympy [A] (verification not implemented)

Time = 2.22 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.41 \[ \int \frac {1}{x^2 \left (a+b x^3+c x^6\right )} \, dx=\operatorname {RootSum} {\left (t^{6} \cdot \left (46656 a^{7} c^{3} - 34992 a^{6} b^{2} c^{2} + 8748 a^{5} b^{4} c - 729 a^{4} b^{6}\right ) + t^{3} \left (- 864 a^{3} b c^{3} + 864 a^{2} b^{3} c^{2} - 270 a b^{5} c + 27 b^{7}\right ) + c^{4}, \left ( t \mapsto t \log {\left (x + \frac {- 15552 t^{5} a^{8} c^{4} + 27216 t^{5} a^{7} b^{2} c^{3} - 14580 t^{5} a^{6} b^{4} c^{2} + 3159 t^{5} a^{5} b^{6} c - 243 t^{5} a^{4} b^{8} + 252 t^{2} a^{4} b c^{4} - 567 t^{2} a^{3} b^{3} c^{3} + 378 t^{2} a^{2} b^{5} c^{2} - 99 t^{2} a b^{7} c + 9 t^{2} b^{9}}{2 a^{2} c^{5} - 4 a b^{2} c^{4} + b^{4} c^{3}} \right )} \right )\right )} - \frac {1}{a x} \]

[In]

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

[Out]

RootSum(_t**6*(46656*a**7*c**3 - 34992*a**6*b**2*c**2 + 8748*a**5*b**4*c - 729*a**4*b**6) + _t**3*(-864*a**3*b
*c**3 + 864*a**2*b**3*c**2 - 270*a*b**5*c + 27*b**7) + c**4, Lambda(_t, _t*log(x + (-15552*_t**5*a**8*c**4 + 2
7216*_t**5*a**7*b**2*c**3 - 14580*_t**5*a**6*b**4*c**2 + 3159*_t**5*a**5*b**6*c - 243*_t**5*a**4*b**8 + 252*_t
**2*a**4*b*c**4 - 567*_t**2*a**3*b**3*c**3 + 378*_t**2*a**2*b**5*c**2 - 99*_t**2*a*b**7*c + 9*_t**2*b**9)/(2*a
**2*c**5 - 4*a*b**2*c**4 + b**4*c**3)))) - 1/(a*x)

Maxima [F]

\[ \int \frac {1}{x^2 \left (a+b x^3+c x^6\right )} \, dx=\int { \frac {1}{{\left (c x^{6} + b x^{3} + a\right )} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(c*x^6+b*x^3+a),x, algorithm="maxima")

[Out]

-integrate((c*x^4 + b*x)/(c*x^6 + b*x^3 + a), x)/a - 1/(a*x)

Giac [F]

\[ \int \frac {1}{x^2 \left (a+b x^3+c x^6\right )} \, dx=\int { \frac {1}{{\left (c x^{6} + b x^{3} + a\right )} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(c*x^6+b*x^3+a),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 11.56 (sec) , antiderivative size = 2978, normalized size of antiderivative = 4.88 \[ \int \frac {1}{x^2 \left (a+b x^3+c x^6\right )} \, dx=\text {Too large to display} \]

[In]

int(1/(x^2*(a + b*x^3 + c*x^6)),x)

[Out]

log(36*a^9*c^6 + 9*a^7*b^4*c^4 - 45*a^8*b^2*c^5 - (2^(2/3)*(27*a^7*c^3*x*(b^6 - 8*a^3*c^3 + 18*a^2*b^2*c^2 - 8
*a*b^4*c) + (27*2^(1/3)*a^10*b*c^3*(4*a*c - b^2)^2*(-(b^7 + b^4*(-(4*a*c - b^2)^3)^(1/2) - 32*a^3*b*c^3 + 32*a
^2*b^3*c^2 + 2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c
- b^2)^3))^(2/3))/2)*(-(b^7 + b^4*(-(4*a*c - b^2)^3)^(1/2) - 32*a^3*b*c^3 + 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4*a
*c - b^2)^3)^(1/2) - 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(1/3))/6)*((b^7 +
b^4*(-(4*a*c - b^2)^3)^(1/2) - 32*a^3*b*c^3 + 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*
c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(54*(a^4*b^6 - 64*a^7*c^3 - 12*a^5*b^4*c + 48*a^6*b^2*c^2)))^(1/3) + l
og(36*a^9*c^6 + 9*a^7*b^4*c^4 - 45*a^8*b^2*c^5 - (2^(2/3)*(27*a^7*c^3*x*(b^6 - 8*a^3*c^3 + 18*a^2*b^2*c^2 - 8*
a*b^4*c) + (27*2^(1/3)*a^10*b*c^3*(4*a*c - b^2)^2*((b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 32*a^3*b*c^3 - 32*a^2
*b^3*c^2 + 2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c -
b^2)^3))^(2/3))/2)*((b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 32*a^3*b*c^3 - 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4*a*c
- b^2)^3)^(1/2) + 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(1/3))/6)*(-(b^4*(-(
4*a*c - b^2)^3)^(1/2) - b^7 + 32*a^3*b*c^3 - 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + 10*a*b^5*c
- 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(54*(a^4*b^6 - 64*a^7*c^3 - 12*a^5*b^4*c + 48*a^6*b^2*c^2)))^(1/3) - 1/(
a*x) + log(36*a^9*c^6 + 9*a^7*b^4*c^4 - 45*a^8*b^2*c^5 - (2^(2/3)*(3^(1/2)*1i - 1)*(27*a^7*c^3*x*(b^6 - 8*a^3*
c^3 + 18*a^2*b^2*c^2 - 8*a*b^4*c) - (27*2^(1/3)*a^10*b*c^3*(3^(1/2)*1i + 1)*(4*a*c - b^2)^2*(-(b^7 + b^4*(-(4*
a*c - b^2)^3)^(1/2) - 32*a^3*b*c^3 + 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c - 4*a*b^
2*c*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(2/3))/4)*(-(b^7 + b^4*(-(4*a*c - b^2)^3)^(1/2) - 32*a^3*
b*c^3 + 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))
/(a^4*(4*a*c - b^2)^3))^(1/3))/12)*((3^(1/2)*1i)/2 - 1/2)*((b^7 + b^4*(-(4*a*c - b^2)^3)^(1/2) - 32*a^3*b*c^3
+ 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(54*(
a^4*b^6 - 64*a^7*c^3 - 12*a^5*b^4*c + 48*a^6*b^2*c^2)))^(1/3) - log(36*a^9*c^6 + 9*a^7*b^4*c^4 - 45*a^8*b^2*c^
5 + (2^(2/3)*(3^(1/2)*1i + 1)*(27*a^7*c^3*x*(b^6 - 8*a^3*c^3 + 18*a^2*b^2*c^2 - 8*a*b^4*c) + (27*2^(1/3)*a^10*
b*c^3*(3^(1/2)*1i - 1)*(4*a*c - b^2)^2*(-(b^7 + b^4*(-(4*a*c - b^2)^3)^(1/2) - 32*a^3*b*c^3 + 32*a^2*b^3*c^2 +
2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^
(2/3))/4)*(-(b^7 + b^4*(-(4*a*c - b^2)^3)^(1/2) - 32*a^3*b*c^3 + 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4*a*c - b^2)^3)
^(1/2) - 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(1/3))/12)*((3^(1/2)*1i)/2 +
1/2)*((b^7 + b^4*(-(4*a*c - b^2)^3)^(1/2) - 32*a^3*b*c^3 + 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2)
- 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(54*(a^4*b^6 - 64*a^7*c^3 - 12*a^5*b^4*c + 48*a^6*b^2*c^2)
))^(1/3) - log(36*a^9*c^6 + 9*a^7*b^4*c^4 - 45*a^8*b^2*c^5 + (2^(2/3)*(27*a^7*c^3*x*(b^6 - 8*a^3*c^3 + 18*a^2*
b^2*c^2 - 8*a*b^4*c) + (27*2^(1/3)*a^10*b*c^3*(3^(1/2)*1i - 1)*(4*a*c - b^2)^2*((b^4*(-(4*a*c - b^2)^3)^(1/2)
- b^7 + 32*a^3*b*c^3 - 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c -
b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(2/3))/4)*(3^(1/2)*1i + 1)*((b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 32*a^
3*b*c^3 - 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2
))/(a^4*(4*a*c - b^2)^3))^(1/3))/12)*((3^(1/2)*1i)/2 + 1/2)*(-(b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 32*a^3*b*c
^3 - 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/2) + 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(5
4*(a^4*b^6 - 64*a^7*c^3 - 12*a^5*b^4*c + 48*a^6*b^2*c^2)))^(1/3) + log(36*a^9*c^6 + 9*a^7*b^4*c^4 - 45*a^8*b^2
*c^5 - (2^(2/3)*(27*a^7*c^3*x*(b^6 - 8*a^3*c^3 + 18*a^2*b^2*c^2 - 8*a*b^4*c) - (27*2^(1/3)*a^10*b*c^3*(3^(1/2)
*1i + 1)*(4*a*c - b^2)^2*((b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 32*a^3*b*c^3 - 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4
*a*c - b^2)^3)^(1/2) + 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(2/3))/4)*(3^(1
/2)*1i - 1)*((b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 32*a^3*b*c^3 - 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4*a*c - b^2)^3
)^(1/2) + 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(1/3))/12)*((3^(1/2)*1i)/2 -
1/2)*(-(b^4*(-(4*a*c - b^2)^3)^(1/2) - b^7 + 32*a^3*b*c^3 - 32*a^2*b^3*c^2 + 2*a^2*c^2*(-(4*a*c - b^2)^3)^(1/
2) + 10*a*b^5*c - 4*a*b^2*c*(-(4*a*c - b^2)^3)^(1/2))/(54*(a^4*b^6 - 64*a^7*c^3 - 12*a^5*b^4*c + 48*a^6*b^2*c^
2)))^(1/3)