\(\int \frac {d+e x^3}{x^2 (a+b x^3+c x^6)} \, dx\) [18]

Optimal result

Integrand size = 25, antiderivative size = 653 \[ \int \frac {d+e x^3}{x^2 \left (a+b x^3+c x^6\right )} \, dx=-\frac {d}{a x}+\frac {\sqrt [3]{c} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} a \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{c} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} a \sqrt [3]{b+\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{c} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} a \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{c} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} a \sqrt [3]{b+\sqrt {b^2-4 a c}}}-\frac {\sqrt [3]{c} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} a \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\sqrt [3]{c} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} a \sqrt [3]{b+\sqrt {b^2-4 a c}}} \]

[Out]

-d/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))*(d+(-2*a*e+b*d)/(-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))*(d+(-2*a*e+b*d)/(-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))*(d+(-2*a*e+b*d)/(-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))*(d+(2*a*e-b*d)/(-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))*(d+(2*a*e-b*d)/(-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))*(d+(2*a*e-b*d)/(-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 = 0.68 (sec) , antiderivative size = 653, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1518, 1524, 298, 31, 648, 631, 210, 642} \[ \int \frac {d+e x^3}{x^2 \left (a+b x^3+c x^6\right )} \, dx=\frac {\sqrt [3]{c} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right ) \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right )}{2^{2/3} \sqrt {3} a \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{c} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right ) \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )}{2^{2/3} \sqrt {3} a \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {\sqrt [3]{c} \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt {b^2-4 a c}}+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} a \sqrt [3]{b-\sqrt {b^2-4 a c}}}-\frac {\sqrt [3]{c} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt {b^2-4 a c}+b}+\left (\sqrt {b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} a \sqrt [3]{\sqrt {b^2-4 a c}+b}}+\frac {\sqrt [3]{c} \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right ) \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} a \sqrt [3]{b-\sqrt {b^2-4 a c}}}+\frac {\sqrt [3]{c} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} a \sqrt [3]{\sqrt {b^2-4 a c}+b}}-\frac {d}{a x} \]

[In]

Int[(d + e*x^3)/(x^2*(a + b*x^3 + c*x^6)),x]

[Out]

-(d/(a*x)) + (c^(1/3)*(d + (b*d - 2*a*e)/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)*(d - (b*d - 2*a*e)/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)*(d + (b*d - 2*a*e)/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)*(d - (b*d - 2*a*e)/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)*(d + (b*d - 2*a*e)/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)*(d - (b*d -
2*a*e)/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 1518

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :>
Simp[d*(f*x)^(m + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(a*f*(m + 1))), x] + Dist[1/(a*f^n*(m + 1)), Int[(f*x)^
(m + n)*(a + b*x^n + c*x^(2*n))^p*Simp[a*e*(m + 1) - b*d*(m + n*(p + 1) + 1) - c*d*(m + 2*n*(p + 1) + 1)*x^n,
x], x], x] /; FreeQ[{a, b, c, d, e, f, 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 {d}{a x}-\frac {\int \frac {x \left (b d-a e+c d x^3\right )}{a+b x^3+c x^6} \, dx}{a} \\ & = -\frac {d}{a x}-\frac {\left (c \left (d-\frac {b d-2 a e}{\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 (d+\frac {b d-2 a e}{\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 {d}{a x}+\frac {\left (c^{2/3} \left (d-\frac {b d-2 a e}{\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 (d-\frac {b d-2 a e}{\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 (d+\frac {b d-2 a e}{\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 (d+\frac {b d-2 a e}{\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 {d}{a x}+\frac {\sqrt [3]{c} \left (d+\frac {b d-2 a e}{\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 (d-\frac {b d-2 a e}{\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 (d-\frac {b d-2 a e}{\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 (d-\frac {b d-2 a e}{\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 (c^{2/3} \left (d+\frac {b d-2 a e}{\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 (d+\frac {b d-2 a e}{\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 {d}{a x}+\frac {\sqrt [3]{c} \left (d+\frac {b d-2 a e}{\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 (d-\frac {b d-2 a e}{\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 (d+\frac {b d-2 a e}{\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 (d-\frac {b d-2 a e}{\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 (d-\frac {b d-2 a e}{\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 (d+\frac {b d-2 a e}{\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 {d}{a x}+\frac {\sqrt [3]{c} \left (d+\frac {b d-2 a e}{\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 (d-\frac {b d-2 a e}{\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 (d+\frac {b d-2 a e}{\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 (d-\frac {b d-2 a e}{\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 (d+\frac {b d-2 a e}{\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 (d-\frac {b d-2 a e}{\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 = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.13 \[ \int \frac {d+e x^3}{x^2 \left (a+b x^3+c x^6\right )} \, dx=-\frac {d}{a x}-\frac {\text {RootSum}\left [a+b \text {$\#$1}^3+c \text {$\#$1}^6\&,\frac {b d \log (x-\text {$\#$1})-a e \log (x-\text {$\#$1})+c d \log (x-\text {$\#$1}) \text {$\#$1}^3}{b \text {$\#$1}+2 c \text {$\#$1}^4}\&\right ]}{3 a} \]

[In]

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

[Out]

-(d/(a*x)) - RootSum[a + b*#1^3 + c*#1^6 & , (b*d*Log[x - #1] - a*e*Log[x - #1] + c*d*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 = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.11

method	result	size
default	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c +\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (-c d \,\textit {\_R}^{4}+\left (a e -b d \right ) \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} c +\textit {\_R}^{2} b}}{3 a}-\frac {d}{a x}\)	\(71\)
risch	\(-\frac {d}{a x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 a^{7} c^{3}-48 b^{2} c^{2} a^{6}+12 a^{5} b^{4} c -b^{6} a^{4}\right ) \textit {\_Z}^{6}+\left (-16 a^{5} c^{2} e^{3}+8 a^{4} b^{2} c \,e^{3}+48 a^{4} b \,c^{2} d \,e^{2}+48 a^{4} c^{3} d^{2} e -a^{3} b^{4} e^{3}-24 a^{3} b^{3} c d \,e^{2}-72 a^{3} b^{2} c^{2} d^{2} e -32 a^{3} b \,c^{3} d^{3}+3 a^{2} b^{5} d \,e^{2}+27 a^{2} b^{4} c \,d^{2} e +32 a^{2} b^{3} c^{2} d^{3}-3 a \,b^{6} d^{2} e -10 a \,b^{5} c \,d^{3}+b^{7} d^{3}\right ) \textit {\_Z}^{3}+a^{3} c \,e^{6}-3 a^{2} b c d \,e^{5}+3 a^{2} c^{2} d^{2} e^{4}+3 a \,b^{2} c \,d^{2} e^{4}-6 a b \,c^{2} d^{3} e^{3}+3 a \,c^{3} d^{4} e^{2}-b^{3} c \,d^{3} e^{3}+3 b^{2} c^{2} d^{4} e^{2}-3 b \,c^{3} d^{5} e +c^{4} d^{6}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (224 a^{7} c^{3}-176 b^{2} c^{2} a^{6}+46 a^{5} b^{4} c -4 b^{6} a^{4}\right ) \textit {\_R}^{6}+\left (-52 a^{5} c^{2} e^{3}+25 a^{4} b^{2} c \,e^{3}+144 a^{4} b \,c^{2} d \,e^{2}+156 a^{4} c^{3} d^{2} e -3 a^{3} b^{4} e^{3}-72 a^{3} b^{3} c d \,e^{2}-219 a^{3} b^{2} c^{2} d^{2} e -100 a^{3} b \,c^{3} d^{3}+9 a^{2} b^{5} d \,e^{2}+81 a^{2} b^{4} c \,d^{2} e +97 a^{2} b^{3} c^{2} d^{3}-9 a \,b^{6} d^{2} e -30 a \,b^{5} c \,d^{3}+3 b^{7} d^{3}\right ) \textit {\_R}^{3}+3 a^{3} c \,e^{6}-9 a^{2} b c d \,e^{5}+9 a^{2} c^{2} d^{2} e^{4}+9 a \,b^{2} c \,d^{2} e^{4}-18 a b \,c^{2} d^{3} e^{3}+9 a \,c^{3} d^{4} e^{2}-3 b^{3} c \,d^{3} e^{3}+9 b^{2} c^{2} d^{4} e^{2}-9 b \,c^{3} d^{5} e +3 c^{4} d^{6}\right ) x +\left (16 a^{6} b \,c^{2} e +16 a^{6} c^{3} d -8 a^{5} b^{3} c e -24 a^{5} b^{2} c^{2} d +a^{4} b^{5} e +9 a^{4} b^{4} c d -a^{3} b^{6} d \right ) \textit {\_R}^{5}+\left (-a^{4} b c \,e^{4}+4 a^{4} c^{2} d \,e^{3}+a^{3} b^{2} c d \,e^{3}-6 a^{3} b \,c^{2} d^{2} e^{2}+4 a^{3} c^{3} d^{3} e +a^{2} b^{2} c^{2} d^{3} e -a^{2} b \,c^{3} d^{4}\right ) \textit {\_R}^{2}\right )\right )}{3}\)	\(843\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11285 vs. \(2 (517) = 1034\).

Time = 37.04 (sec) , antiderivative size = 11285, normalized size of antiderivative = 17.28 \[ \int \frac {d+e x^3}{x^2 \left (a+b x^3+c x^6\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 36.13 (sec) , antiderivative size = 11174, normalized size of antiderivative = 17.11 \[ \int \frac {d+e x^3}{x^2 \left (a+b x^3+c x^6\right )} \, dx=\text {Too large to display} \]

[In]

int((d + e*x^3)/(x^2*(a + b*x^3 + c*x^6)),x)

[Out]

log((2^(1/3)*(-(b^7*d^3 - a^3*b^4*e^3 + b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b*c^3*d^3 -
a^3*b*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c^2*d^
3 + 2*a^2*c^2*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e - 4*a*b^2*c*d^3*(-(4*a*c - b^2)^3)
^(1/2) - 3*a*b^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*d*e^2
- 6*a^3*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a^3*b^2*c^2*d^2*e +
9*a^2*b*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(2/3)*((2^(2/3)*(27*a^7*c^3*x*(4*a*c - b^2)*(
b^4*d^2 - 2*a^3*c*e^2 + a^2*b^2*e^2 + 2*a^2*c^2*d^2 - 2*a*b^3*d*e - 4*a*b^2*c*d^2 + 6*a^2*b*c*d*e) - (27*2^(1/
3)*a^10*b*c^3*(4*a*c - b^2)^2*(-(b^7*d^3 - a^3*b^4*e^3 + b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 3
2*a^3*b*c^3*d^3 - a^3*b*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e +
32*a^2*b^3*c^2*d^3 + 2*a^2*c^2*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e - 4*a*b^2*c*d^3*(
-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 + 4
8*a^4*b*c^2*d*e^2 - 6*a^3*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a^3
*b^2*c^2*d^2*e + 9*a^2*b*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(2/3))/2)*(-(b^7*d^3 - a^3*b
^4*e^3 + b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b*c^3*d^3 - a^3*b*e^3*(-(4*a*c - b^2)^3)^(
1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c^2*d^3 + 2*a^2*c^2*d^3*(-(4*a*c - b^
2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e - 4*a*b^2*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*d^2*e*(-(4*a*c
- b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*d*e^2 - 6*a^3*c*d*e^2*(-(4*a*c - b^2
)^3)^(1/2) + 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a^3*b^2*c^2*d^2*e + 9*a^2*b*c*d^2*e*(-(4*a*c - b^2)
^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(1/3))/6 + 36*a^9*c^6*d^3 - 108*a^10*c^5*d*e^2 + 9*a^7*b^4*c^4*d^3 - 45*a^8*
b^2*c^5*d^3 + 108*a^9*b*c^5*d^2*e - 27*a^8*b^3*c^4*d^2*e + 27*a^9*b^2*c^4*d*e^2))/18 + a^7*c^4*e*x*(a*e^2 + c*
d^2 - b*d*e)^2)*((b^7*d^3 - a^3*b^4*e^3 + b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b*c^3*d^3
- a^3*b*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c^2*
d^3 + 2*a^2*c^2*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e - 4*a*b^2*c*d^3*(-(4*a*c - b^2)^
3)^(1/2) - 3*a*b^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*d*e
^2 - 6*a^3*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a^3*b^2*c^2*d^2*e
+ 9*a^2*b*c*d^2*e*(-(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((2^(1/3)*(-(b^7*d^3 - a^3*b^4*e^3 - b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b*c^3*d^
3 + a^3*b*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c^2
*d^3 - 2*a^2*c^2*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e + 4*a*b^2*c*d^3*(-(4*a*c - b^2)
^3)^(1/2) + 3*a*b^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*d*
e^2 + 6*a^3*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a^3*b^2*c^2*d^2*e
- 9*a^2*b*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(2/3)*((2^(2/3)*(27*a^7*c^3*x*(4*a*c - b^2
)*(b^4*d^2 - 2*a^3*c*e^2 + a^2*b^2*e^2 + 2*a^2*c^2*d^2 - 2*a*b^3*d*e - 4*a*b^2*c*d^2 + 6*a^2*b*c*d*e) - (27*2^
(1/3)*a^10*b*c^3*(4*a*c - b^2)^2*(-(b^7*d^3 - a^3*b^4*e^3 - b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3
- 32*a^3*b*c^3*d^3 + a^3*b*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e
+ 32*a^2*b^3*c^2*d^3 - 2*a^2*c^2*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e + 4*a*b^2*c*d^
3*(-(4*a*c - b^2)^3)^(1/2) + 3*a*b^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2
+ 48*a^4*b*c^2*d*e^2 + 6*a^3*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*
a^3*b^2*c^2*d^2*e - 9*a^2*b*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(2/3))/2)*(-(b^7*d^3 - a^
3*b^4*e^3 - b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b*c^3*d^3 + a^3*b*e^3*(-(4*a*c - b^2)^3
)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c^2*d^3 - 2*a^2*c^2*d^3*(-(4*a*c -
b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e + 4*a*b^2*c*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a*b^3*d^2*e*(-(4*
a*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*d*e^2 + 6*a^3*c*d*e^2*(-(4*a*c -
b^2)^3)^(1/2) - 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a^3*b^2*c^2*d^2*e - 9*a^2*b*c*d^2*e*(-(4*a*c - b
^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(1/3))/6 + 36*a^9*c^6*d^3 - 108*a^10*c^5*d*e^2 + 9*a^7*b^4*c^4*d^3 - 45*a
^8*b^2*c^5*d^3 + 108*a^9*b*c^5*d^2*e - 27*a^8*b^3*c^4*d^2*e + 27*a^9*b^2*c^4*d*e^2))/18 + a^7*c^4*e*x*(a*e^2 +
c*d^2 - b*d*e)^2)*((b^7*d^3 - a^3*b^4*e^3 - b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b*c^3*
d^3 + a^3*b*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c
^2*d^3 - 2*a^2*c^2*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e + 4*a*b^2*c*d^3*(-(4*a*c - b^
2)^3)^(1/2) + 3*a*b^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*
d*e^2 + 6*a^3*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a^3*b^2*c^2*d^2
*e - 9*a^2*b*c*d^2*e*(-(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((2^(1/3)*(3^(1/2)*1i - 1)*(-(b^7*d^3 - a^3*b^4*e^3 + b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e
^3 - 32*a^3*b*c^3*d^3 - a^3*b*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^
2*e + 32*a^2*b^3*c^2*d^3 + 2*a^2*c^2*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e - 4*a*b^2*c
*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e
^2 + 48*a^4*b*c^2*d*e^2 - 6*a^3*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) -
72*a^3*b^2*c^2*d^2*e + 9*a^2*b*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(2/3)*(36*a^9*c^6*d^3
- 108*a^10*c^5*d*e^2 + 9*a^7*b^4*c^4*d^3 - 45*a^8*b^2*c^5*d^3 - (2^(2/3)*(3^(1/2)*1i + 1)*(27*a^7*c^3*x*(4*a*c
- b^2)*(b^4*d^2 - 2*a^3*c*e^2 + a^2*b^2*e^2 + 2*a^2*c^2*d^2 - 2*a*b^3*d*e - 4*a*b^2*c*d^2 + 6*a^2*b*c*d*e) -
(27*2^(1/3)*a^10*b*c^3*(3^(1/2)*1i - 1)*(4*a*c - b^2)^2*(-(b^7*d^3 - a^3*b^4*e^3 + b^4*d^3*(-(4*a*c - b^2)^3)^
(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b*c^3*d^3 - a^3*b*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d
*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c^2*d^3 + 2*a^2*c^2*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b
^6*d^2*e - 4*a*b^2*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*
e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*d*e^2 - 6*a^3*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*d*e^2*(-(4*a*
c - b^2)^3)^(1/2) - 72*a^3*b^2*c^2*d^2*e + 9*a^2*b*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(2
/3))/4)*(-(b^7*d^3 - a^3*b^4*e^3 + b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b*c^3*d^3 - a^3*
b*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c^2*d^3 + 2
*a^2*c^2*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e - 4*a*b^2*c*d^3*(-(4*a*c - b^2)^3)^(1/2
) - 3*a*b^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*d*e^2 - 6*
a^3*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a^3*b^2*c^2*d^2*e + 9*a^2
*b*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(1/3))/12 + 108*a^9*b*c^5*d^2*e - 27*a^8*b^3*c^4*d
^2*e + 27*a^9*b^2*c^4*d*e^2))/36 + a^7*c^4*e*x*(a*e^2 + c*d^2 - b*d*e)^2)*((3^(1/2)*1i)/2 + 1/2)*((b^7*d^3 - a
^3*b^4*e^3 + b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b*c^3*d^3 - a^3*b*e^3*(-(4*a*c - b^2)^
3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c^2*d^3 + 2*a^2*c^2*d^3*(-(4*a*c
- b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e - 4*a*b^2*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*d^2*e*(-(4
*a*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*d*e^2 - 6*a^3*c*d*e^2*(-(4*a*c -
b^2)^3)^(1/2) + 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a^3*b^2*c^2*d^2*e + 9*a^2*b*c*d^2*e*(-(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((2^(1/3)*(3^(1/2)*1i -
1)*(-(b^7*d^3 - a^3*b^4*e^3 - b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b*c^3*d^3 + a^3*b*e^
3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c^2*d^3 - 2*a^2
*c^2*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e + 4*a*b^2*c*d^3*(-(4*a*c - b^2)^3)^(1/2) +
3*a*b^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*d*e^2 + 6*a^3*
c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a^3*b^2*c^2*d^2*e - 9*a^2*b*c
*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(2/3)*(36*a^9*c^6*d^3 - 108*a^10*c^5*d*e^2 + 9*a^7*b^4
*c^4*d^3 - 45*a^8*b^2*c^5*d^3 - (2^(2/3)*(3^(1/2)*1i + 1)*(27*a^7*c^3*x*(4*a*c - b^2)*(b^4*d^2 - 2*a^3*c*e^2 +
a^2*b^2*e^2 + 2*a^2*c^2*d^2 - 2*a*b^3*d*e - 4*a*b^2*c*d^2 + 6*a^2*b*c*d*e) - (27*2^(1/3)*a^10*b*c^3*(3^(1/2)*
1i - 1)*(4*a*c - b^2)^2*(-(b^7*d^3 - a^3*b^4*e^3 - b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*
b*c^3*d^3 + a^3*b*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2
*b^3*c^2*d^3 - 2*a^2*c^2*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e + 4*a*b^2*c*d^3*(-(4*a*
c - b^2)^3)^(1/2) + 3*a*b^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 + 48*a^4*
b*c^2*d*e^2 + 6*a^3*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a^3*b^2*c
^2*d^2*e - 9*a^2*b*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(2/3))/4)*(-(b^7*d^3 - a^3*b^4*e^3
- b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b*c^3*d^3 + a^3*b*e^3*(-(4*a*c - b^2)^3)^(1/2) +
8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c^2*d^3 - 2*a^2*c^2*d^3*(-(4*a*c - b^2)^3)^
(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e + 4*a*b^2*c*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a*b^3*d^2*e*(-(4*a*c - b^2
)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*d*e^2 + 6*a^3*c*d*e^2*(-(4*a*c - b^2)^3)^(
1/2) - 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a^3*b^2*c^2*d^2*e - 9*a^2*b*c*d^2*e*(-(4*a*c - b^2)^3)^(1
/2))/(a^4*(4*a*c - b^2)^3))^(1/3))/12 + 108*a^9*b*c^5*d^2*e - 27*a^8*b^3*c^4*d^2*e + 27*a^9*b^2*c^4*d*e^2))/36
+ a^7*c^4*e*x*(a*e^2 + c*d^2 - b*d*e)^2)*((3^(1/2)*1i)/2 + 1/2)*((b^7*d^3 - a^3*b^4*e^3 - b^4*d^3*(-(4*a*c -
b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b*c^3*d^3 + a^3*b*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a
^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c^2*d^3 - 2*a^2*c^2*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3
- 3*a*b^6*d^2*e + 4*a*b^2*c*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a*b^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a^2*b^
4*c*d^2*e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*d*e^2 + 6*a^3*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*d*e^2
*(-(4*a*c - b^2)^3)^(1/2) - 72*a^3*b^2*c^2*d^2*e - 9*a^2*b*c*d^2*e*(-(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(a^7*c^4*e*x*(a*e^2 + c*d^2 - b*d*e)^2 - (2^(1/3)*(3^(1
/2)*1i + 1)*(-(b^7*d^3 - a^3*b^4*e^3 + b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b*c^3*d^3 -
a^3*b*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c^2*d^3
+ 2*a^2*c^2*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e - 4*a*b^2*c*d^3*(-(4*a*c - b^2)^3)^
(1/2) - 3*a*b^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*d*e^2
- 6*a^3*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a^3*b^2*c^2*d^2*e + 9
*a^2*b*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(2/3)*(36*a^9*c^6*d^3 - 108*a^10*c^5*d*e^2 + 9
*a^7*b^4*c^4*d^3 - 45*a^8*b^2*c^5*d^3 + (2^(2/3)*(3^(1/2)*1i - 1)*(27*a^7*c^3*x*(4*a*c - b^2)*(b^4*d^2 - 2*a^3
*c*e^2 + a^2*b^2*e^2 + 2*a^2*c^2*d^2 - 2*a*b^3*d*e - 4*a*b^2*c*d^2 + 6*a^2*b*c*d*e) + (27*2^(1/3)*a^10*b*c^3*(
3^(1/2)*1i + 1)*(4*a*c - b^2)^2*(-(b^7*d^3 - a^3*b^4*e^3 + b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 -
32*a^3*b*c^3*d^3 - a^3*b*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e
+ 32*a^2*b^3*c^2*d^3 + 2*a^2*c^2*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e - 4*a*b^2*c*d^3
*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 +
48*a^4*b*c^2*d*e^2 - 6*a^3*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a
^3*b^2*c^2*d^2*e + 9*a^2*b*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(2/3))/4)*(-(b^7*d^3 - a^3
*b^4*e^3 + b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b*c^3*d^3 - a^3*b*e^3*(-(4*a*c - b^2)^3)
^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c^2*d^3 + 2*a^2*c^2*d^3*(-(4*a*c -
b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e - 4*a*b^2*c*d^3*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*d^2*e*(-(4*a
*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*d*e^2 - 6*a^3*c*d*e^2*(-(4*a*c - b
^2)^3)^(1/2) + 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a^3*b^2*c^2*d^2*e + 9*a^2*b*c*d^2*e*(-(4*a*c - b^
2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(1/3))/12 + 108*a^9*b*c^5*d^2*e - 27*a^8*b^3*c^4*d^2*e + 27*a^9*b^2*c^4*d*
e^2))/36)*((3^(1/2)*1i)/2 - 1/2)*((b^7*d^3 - a^3*b^4*e^3 + b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 -
32*a^3*b*c^3*d^3 - a^3*b*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e
+ 32*a^2*b^3*c^2*d^3 + 2*a^2*c^2*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e - 4*a*b^2*c*d^3
*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 +
48*a^4*b*c^2*d*e^2 - 6*a^3*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a
^3*b^2*c^2*d^2*e + 9*a^2*b*c*d^2*e*(-(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(a^7*c^4*e*x*(a*e^2 + c*d^2 - b*d*e)^2 - (2^(1/3)*(3^(1/2)*1i + 1)*(-(b^7*d^3 - a^3*b^4
*e^3 - b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b*c^3*d^3 + a^3*b*e^3*(-(4*a*c - b^2)^3)^(1/
2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c^2*d^3 - 2*a^2*c^2*d^3*(-(4*a*c - b^2)
^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e + 4*a*b^2*c*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a*b^3*d^2*e*(-(4*a*c -
b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*d*e^2 + 6*a^3*c*d*e^2*(-(4*a*c - b^2)^
3)^(1/2) - 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a^3*b^2*c^2*d^2*e - 9*a^2*b*c*d^2*e*(-(4*a*c - b^2)^3
)^(1/2))/(a^4*(4*a*c - b^2)^3))^(2/3)*(36*a^9*c^6*d^3 - 108*a^10*c^5*d*e^2 + 9*a^7*b^4*c^4*d^3 - 45*a^8*b^2*c^
5*d^3 + (2^(2/3)*(3^(1/2)*1i - 1)*(27*a^7*c^3*x*(4*a*c - b^2)*(b^4*d^2 - 2*a^3*c*e^2 + a^2*b^2*e^2 + 2*a^2*c^2
*d^2 - 2*a*b^3*d*e - 4*a*b^2*c*d^2 + 6*a^2*b*c*d*e) + (27*2^(1/3)*a^10*b*c^3*(3^(1/2)*1i + 1)*(4*a*c - b^2)^2*
(-(b^7*d^3 - a^3*b^4*e^3 - b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b*c^3*d^3 + a^3*b*e^3*(-
(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c^2*d^3 - 2*a^2*c^2
*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e + 4*a*b^2*c*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a*
b^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*d*e^2 + 6*a^3*c*d*
e^2*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a^3*b^2*c^2*d^2*e - 9*a^2*b*c*d^2
*e*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^3))^(2/3))/4)*(-(b^7*d^3 - a^3*b^4*e^3 - b^4*d^3*(-(4*a*c - b^
2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b*c^3*d^3 + a^3*b*e^3*(-(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2
*b^5*d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c^2*d^3 - 2*a^2*c^2*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3 -
3*a*b^6*d^2*e + 4*a*b^2*c*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a*b^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a^2*b^4*
c*d^2*e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*d*e^2 + 6*a^3*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*d*e^2*(
-(4*a*c - b^2)^3)^(1/2) - 72*a^3*b^2*c^2*d^2*e - 9*a^2*b*c*d^2*e*(-(4*a*c - b^2)^3)^(1/2))/(a^4*(4*a*c - b^2)^
3))^(1/3))/12 + 108*a^9*b*c^5*d^2*e - 27*a^8*b^3*c^4*d^2*e + 27*a^9*b^2*c^4*d*e^2))/36)*((3^(1/2)*1i)/2 - 1/2)
*((b^7*d^3 - a^3*b^4*e^3 - b^4*d^3*(-(4*a*c - b^2)^3)^(1/2) - 16*a^5*c^2*e^3 - 32*a^3*b*c^3*d^3 + a^3*b*e^3*(-
(4*a*c - b^2)^3)^(1/2) + 8*a^4*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 48*a^4*c^3*d^2*e + 32*a^2*b^3*c^2*d^3 - 2*a^2*c^2
*d^3*(-(4*a*c - b^2)^3)^(1/2) - 10*a*b^5*c*d^3 - 3*a*b^6*d^2*e + 4*a*b^2*c*d^3*(-(4*a*c - b^2)^3)^(1/2) + 3*a*
b^3*d^2*e*(-(4*a*c - b^2)^3)^(1/2) + 27*a^2*b^4*c*d^2*e - 24*a^3*b^3*c*d*e^2 + 48*a^4*b*c^2*d*e^2 + 6*a^3*c*d*
e^2*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*d*e^2*(-(4*a*c - b^2)^3)^(1/2) - 72*a^3*b^2*c^2*d^2*e - 9*a^2*b*c*d^2
*e*(-(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) - d/(a*x)