\(\int \frac {d+e x^3}{a+c x^6} \, dx\) [1]

Optimal result

Integrand size = 17, antiderivative size = 305 \[ \int \frac {d+e x^3}{a+c x^6} \, dx=\frac {d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}-\frac {\left (\sqrt {c} d+\sqrt {3} \sqrt {a} e\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} c^{2/3}}+\frac {\left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} c^{2/3}}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}-\frac {\left (\sqrt {3} \sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}+\frac {\left (\sqrt {3} \sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}} \]

[Out]

1/3*d*arctan(c^(1/6)*x/a^(1/6))/a^(5/6)/c^(1/6)-1/6*e*ln(a^(1/3)+c^(1/3)*x^2)/a^(1/3)/c^(2/3)+1/6*arctan(2*c^(
1/6)*x/a^(1/6)+3^(1/2))*(-e*3^(1/2)*a^(1/2)+d*c^(1/2))/a^(5/6)/c^(2/3)+1/6*arctan(2*c^(1/6)*x/a^(1/6)-3^(1/2))
*(e*3^(1/2)*a^(1/2)+d*c^(1/2))/a^(5/6)/c^(2/3)-1/12*ln(a^(1/3)+c^(1/3)*x^2-a^(1/6)*c^(1/6)*x*3^(1/2))*(-e*a^(1
/2)+d*3^(1/2)*c^(1/2))/a^(5/6)/c^(2/3)+1/12*ln(a^(1/3)+c^(1/3)*x^2+a^(1/6)*c^(1/6)*x*3^(1/2))*(e*a^(1/2)+d*3^(
1/2)*c^(1/2))/a^(5/6)/c^(2/3)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {1430, 649, 209, 266, 648, 631, 210, 642} \[ \int \frac {d+e x^3}{a+c x^6} \, dx=-\frac {\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right ) \left (\sqrt {3} \sqrt {a} e+\sqrt {c} d\right )}{6 a^{5/6} c^{2/3}}+\frac {\arctan \left (\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}+\sqrt {3}\right ) \left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right )}{6 a^{5/6} c^{2/3}}+\frac {d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}-\frac {\left (\sqrt {3} \sqrt {c} d-\sqrt {a} e\right ) \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}+\frac {\left (\sqrt {a} e+\sqrt {3} \sqrt {c} d\right ) \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}} \]

[In]

Int[(d + e*x^3)/(a + c*x^6),x]

[Out]

(d*ArcTan[(c^(1/6)*x)/a^(1/6)])/(3*a^(5/6)*c^(1/6)) - ((Sqrt[c]*d + Sqrt[3]*Sqrt[a]*e)*ArcTan[Sqrt[3] - (2*c^(
1/6)*x)/a^(1/6)])/(6*a^(5/6)*c^(2/3)) + ((Sqrt[c]*d - Sqrt[3]*Sqrt[a]*e)*ArcTan[Sqrt[3] + (2*c^(1/6)*x)/a^(1/6
)])/(6*a^(5/6)*c^(2/3)) - (e*Log[a^(1/3) + c^(1/3)*x^2])/(6*a^(1/3)*c^(2/3)) - ((Sqrt[3]*Sqrt[c]*d - Sqrt[a]*e
)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(5/6)*c^(2/3)) + ((Sqrt[3]*Sqrt[c]*d + Sqrt[a]
*e)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(5/6)*c^(2/3))

Rule 209

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

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 266

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

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 649

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

Rule 1430

Int[((d_) + (e_.)*(x_)^3)/((a_) + (c_.)*(x_)^6), x_Symbol] :> With[{q = Rt[c/a, 6]}, Dist[1/(3*a*q^2), Int[(q^
2*d - e*x)/(1 + q^2*x^2), x], x] + (Dist[1/(6*a*q^2), Int[(2*q^2*d - (Sqrt[3]*q^3*d - e)*x)/(1 - Sqrt[3]*q*x +
q^2*x^2), x], x] + Dist[1/(6*a*q^2), Int[(2*q^2*d + (Sqrt[3]*q^3*d + e)*x)/(1 + Sqrt[3]*q*x + q^2*x^2), x], x
])] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && PosQ[c/a]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\frac {2 \sqrt [3]{c} d}{\sqrt [3]{a}}-\left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}-e\right ) x}{1-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{6 a^{2/3} \sqrt [3]{c}}+\frac {\int \frac {\frac {2 \sqrt [3]{c} d}{\sqrt [3]{a}}+\left (\frac {\sqrt {3} \sqrt {c} d}{\sqrt {a}}+e\right ) x}{1+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{6 a^{2/3} \sqrt [3]{c}}+\frac {\int \frac {\frac {\sqrt [3]{c} d}{\sqrt [3]{a}}-e x}{1+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{3 a^{2/3} \sqrt [3]{c}} \\ & = \frac {d \int \frac {1}{1+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{3 a}-\frac {e \int \frac {x}{1+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{3 a^{2/3} \sqrt [3]{c}}-\frac {\left (\sqrt {3} \sqrt {c} d-\sqrt {a} e\right ) \int \frac {-\frac {\sqrt {3} \sqrt [6]{c}}{\sqrt [6]{a}}+\frac {2 \sqrt [3]{c} x}{\sqrt [3]{a}}}{1-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{12 a^{5/6} c^{2/3}}+\frac {\left (\sqrt {3} \sqrt {c} d+\sqrt {a} e\right ) \int \frac {\frac {\sqrt {3} \sqrt [6]{c}}{\sqrt [6]{a}}+\frac {2 \sqrt [3]{c} x}{\sqrt [3]{a}}}{1+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{12 a^{5/6} c^{2/3}}+\frac {\left (d-\frac {\sqrt {3} \sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{1+\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{12 a}+\frac {\left (d+\frac {\sqrt {3} \sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{1-\frac {\sqrt {3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{12 a} \\ & = \frac {d \tan ^{-1}\left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}-\frac {\left (\sqrt {3} \sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}+\frac {\left (\sqrt {3} \sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}-\frac {\left (\sqrt {3} \sqrt {c} d-3 \sqrt {a} e\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )}{18 a^{5/6} c^{2/3}}+\frac {\left (\sqrt {3} \sqrt {c} d+3 \sqrt {a} e\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )}{18 a^{5/6} c^{2/3}} \\ & = \frac {d \tan ^{-1}\left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}-\frac {\left (\sqrt {c} d+\sqrt {3} \sqrt {a} e\right ) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} c^{2/3}}+\frac {\left (\sqrt {c} d-\sqrt {3} \sqrt {a} e\right ) \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} c^{2/3}}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}-\frac {\left (\sqrt {3} \sqrt {c} d-\sqrt {a} e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}+\frac {\left (\sqrt {3} \sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.10 \[ \int \frac {d+e x^3}{a+c x^6} \, dx=\frac {d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}+\frac {\left (\sqrt [6]{a} \sqrt {c} d+\sqrt {3} a^{2/3} e\right ) \arctan \left (\frac {-\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a c^{2/3}}+\frac {\left (\sqrt [6]{a} \sqrt {c} d-\sqrt {3} a^{2/3} e\right ) \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a c^{2/3}}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}-\frac {\left (\sqrt {3} \sqrt [6]{a} \sqrt {c} d-a^{2/3} e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a c^{2/3}}-\frac {\left (-\sqrt {3} \sqrt [6]{a} \sqrt {c} d-a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a c^{2/3}} \]

[In]

Integrate[(d + e*x^3)/(a + c*x^6),x]

[Out]

(d*ArcTan[(c^(1/6)*x)/a^(1/6)])/(3*a^(5/6)*c^(1/6)) + ((a^(1/6)*Sqrt[c]*d + Sqrt[3]*a^(2/3)*e)*ArcTan[(-(Sqrt[
3]*a^(1/6)) + 2*c^(1/6)*x)/a^(1/6)])/(6*a*c^(2/3)) + ((a^(1/6)*Sqrt[c]*d - Sqrt[3]*a^(2/3)*e)*ArcTan[(Sqrt[3]*
a^(1/6) + 2*c^(1/6)*x)/a^(1/6)])/(6*a*c^(2/3)) - (e*Log[a^(1/3) + c^(1/3)*x^2])/(6*a^(1/3)*c^(2/3)) - ((Sqrt[3
]*a^(1/6)*Sqrt[c]*d - a^(2/3)*e)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a*c^(2/3)) - ((-(
Sqrt[3]*a^(1/6)*Sqrt[c]*d) - a^(2/3)*e)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a*c^(2/3))

Maple [C] (verified)

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

Time = 0.67 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.11

method	result	size
risch	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c +a \right )}{\sum }\frac {\left (\textit {\_R}^{3} e +d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}}{6 c}\)	\(34\)
default	\(\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \left (\frac {a}{c}\right )^{\frac {2}{3}} e}{12 a}-\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} d}{12 a}+\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, e}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) d}{6 a}+\frac {c \left (\frac {a}{c}\right )^{\frac {7}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, d}{12 a^{2}}+\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) e}{12 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{6 a}-\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, e}{6 a}-\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} e \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} d \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 a}\)	\(329\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1631 vs. \(2 (207) = 414\).

Time = 0.38 (sec) , antiderivative size = 1631, normalized size of antiderivative = 5.35 \[ \int \frac {d+e x^3}{a+c x^6} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/12*(sqrt(-3) + 1)*((a^2*c^2*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + 3*c*d^2*e - a*e^3)
/(a^2*c^2))^(1/3)*log(-(c^2*d^5 - 2*a*c*d^3*e^2 - 3*a^2*d*e^4)*x + 1/2*(a*c^2*d^4 - 3*a^2*c*d^2*e^2 + sqrt(-3)
*(a*c^2*d^4 - 3*a^2*c*d^2*e^2) + (sqrt(-3)*a^4*c^2*e + a^4*c^2*e)*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e
^4)/(a^5*c^3)))*((a^2*c^2*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + 3*c*d^2*e - a*e^3)/(a^2
*c^2))^(1/3)) + 1/12*(sqrt(-3) - 1)*((a^2*c^2*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + 3*c
*d^2*e - a*e^3)/(a^2*c^2))^(1/3)*log(-(c^2*d^5 - 2*a*c*d^3*e^2 - 3*a^2*d*e^4)*x + 1/2*(a*c^2*d^4 - 3*a^2*c*d^2
*e^2 - sqrt(-3)*(a*c^2*d^4 - 3*a^2*c*d^2*e^2) - (sqrt(-3)*a^4*c^2*e - a^4*c^2*e)*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^
2 + 9*a^2*d^2*e^4)/(a^5*c^3)))*((a^2*c^2*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + 3*c*d^2*
e - a*e^3)/(a^2*c^2))^(1/3)) - 1/12*(sqrt(-3) + 1)*(-(a^2*c^2*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/
(a^5*c^3)) - 3*c*d^2*e + a*e^3)/(a^2*c^2))^(1/3)*log(-(c^2*d^5 - 2*a*c*d^3*e^2 - 3*a^2*d*e^4)*x + 1/2*(a*c^2*d
^4 - 3*a^2*c*d^2*e^2 + sqrt(-3)*(a*c^2*d^4 - 3*a^2*c*d^2*e^2) - (sqrt(-3)*a^4*c^2*e + a^4*c^2*e)*sqrt(-(c^2*d^
6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)))*(-(a^2*c^2*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5
*c^3)) - 3*c*d^2*e + a*e^3)/(a^2*c^2))^(1/3)) + 1/12*(sqrt(-3) - 1)*(-(a^2*c^2*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2
+ 9*a^2*d^2*e^4)/(a^5*c^3)) - 3*c*d^2*e + a*e^3)/(a^2*c^2))^(1/3)*log(-(c^2*d^5 - 2*a*c*d^3*e^2 - 3*a^2*d*e^4)
*x + 1/2*(a*c^2*d^4 - 3*a^2*c*d^2*e^2 - sqrt(-3)*(a*c^2*d^4 - 3*a^2*c*d^2*e^2) + (sqrt(-3)*a^4*c^2*e - a^4*c^2
*e)*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)))*(-(a^2*c^2*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*
a^2*d^2*e^4)/(a^5*c^3)) - 3*c*d^2*e + a*e^3)/(a^2*c^2))^(1/3)) + 1/6*((a^2*c^2*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2
+ 9*a^2*d^2*e^4)/(a^5*c^3)) + 3*c*d^2*e - a*e^3)/(a^2*c^2))^(1/3)*log(-(c^2*d^5 - 2*a*c*d^3*e^2 - 3*a^2*d*e^4)
*x - (a^4*c^2*e*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + a*c^2*d^4 - 3*a^2*c*d^2*e^2)*((a^
2*c^2*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) + 3*c*d^2*e - a*e^3)/(a^2*c^2))^(1/3)) + 1/6*
(-(a^2*c^2*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) - 3*c*d^2*e + a*e^3)/(a^2*c^2))^(1/3)*lo
g(-(c^2*d^5 - 2*a*c*d^3*e^2 - 3*a^2*d*e^4)*x + (a^4*c^2*e*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5
*c^3)) - a*c^2*d^4 + 3*a^2*c*d^2*e^2)*(-(a^2*c^2*sqrt(-(c^2*d^6 - 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^5*c^3)) -
3*c*d^2*e + a*e^3)/(a^2*c^2))^(1/3))

Sympy [A] (verification not implemented)

Time = 9.41 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.54 \[ \int \frac {d+e x^3}{a+c x^6} \, dx=\operatorname {RootSum} {\left (46656 t^{6} a^{5} c^{4} + t^{3} \cdot \left (432 a^{4} c^{2} e^{3} - 1296 a^{3} c^{3} d^{2} e\right ) + a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}, \left ( t \mapsto t \log {\left (x + \frac {- 1296 t^{4} a^{4} c^{2} e - 6 t a^{3} e^{4} + 36 t a^{2} c d^{2} e^{2} - 6 t a c^{2} d^{4}}{3 a^{2} d e^{4} + 2 a c d^{3} e^{2} - c^{2} d^{5}} \right )} \right )\right )} \]

[In]

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

[Out]

RootSum(46656*_t**6*a**5*c**4 + _t**3*(432*a**4*c**2*e**3 - 1296*a**3*c**3*d**2*e) + a**3*e**6 + 3*a**2*c*d**2
*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6, Lambda(_t, _t*log(x + (-1296*_t**4*a**4*c**2*e - 6*_t*a**3*e**4 + 36*_
t*a**2*c*d**2*e**2 - 6*_t*a*c**2*d**4)/(3*a**2*d*e**4 + 2*a*c*d**3*e**2 - c**2*d**5))))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.92 \[ \int \frac {d+e x^3}{a+c x^6} \, dx=-\frac {e \log \left (c^{\frac {1}{3}} x^{2} + a^{\frac {1}{3}}\right )}{6 \, a^{\frac {1}{3}} c^{\frac {2}{3}}} + \frac {d \arctan \left (\frac {c^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{3 \, a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} + \frac {{\left (\sqrt {3} a^{\frac {1}{6}} \sqrt {c} d + a^{\frac {2}{3}} e\right )} \log \left (c^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{12 \, a c^{\frac {2}{3}}} - \frac {{\left (\sqrt {3} a^{\frac {1}{6}} \sqrt {c} d - a^{\frac {2}{3}} e\right )} \log \left (c^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{12 \, a c^{\frac {2}{3}}} - \frac {{\left (\sqrt {3} a^{\frac {5}{6}} c^{\frac {1}{6}} e - a^{\frac {1}{3}} c^{\frac {2}{3}} d\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{6 \, a c^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} + \frac {{\left (\sqrt {3} a^{\frac {5}{6}} c^{\frac {1}{6}} e + a^{\frac {1}{3}} c^{\frac {2}{3}} d\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{6 \, a c^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} \]

[In]

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

[Out]

-1/6*e*log(c^(1/3)*x^2 + a^(1/3))/(a^(1/3)*c^(2/3)) + 1/3*d*arctan(c^(1/3)*x/sqrt(a^(1/3)*c^(1/3)))/(a^(2/3)*s
qrt(a^(1/3)*c^(1/3))) + 1/12*(sqrt(3)*a^(1/6)*sqrt(c)*d + a^(2/3)*e)*log(c^(1/3)*x^2 + sqrt(3)*a^(1/6)*c^(1/6)
*x + a^(1/3))/(a*c^(2/3)) - 1/12*(sqrt(3)*a^(1/6)*sqrt(c)*d - a^(2/3)*e)*log(c^(1/3)*x^2 - sqrt(3)*a^(1/6)*c^(
1/6)*x + a^(1/3))/(a*c^(2/3)) - 1/6*(sqrt(3)*a^(5/6)*c^(1/6)*e - a^(1/3)*c^(2/3)*d)*arctan((2*c^(1/3)*x + sqrt
(3)*a^(1/6)*c^(1/6))/sqrt(a^(1/3)*c^(1/3)))/(a*c^(2/3)*sqrt(a^(1/3)*c^(1/3))) + 1/6*(sqrt(3)*a^(5/6)*c^(1/6)*e
+ a^(1/3)*c^(2/3)*d)*arctan((2*c^(1/3)*x - sqrt(3)*a^(1/6)*c^(1/6))/sqrt(a^(1/3)*c^(1/3)))/(a*c^(2/3)*sqrt(a^
(1/3)*c^(1/3)))

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.93 \[ \int \frac {d+e x^3}{a+c x^6} \, dx=-\frac {e {\left | c \right |} \log \left (x^{2} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 \, \left (a c^{5}\right )^{\frac {1}{3}}} + \frac {\left (a c^{5}\right )^{\frac {1}{6}} d \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 \, a c} + \frac {{\left (\left (a c^{5}\right )^{\frac {1}{6}} c^{3} d - \sqrt {3} \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a c^{4}} + \frac {{\left (\left (a c^{5}\right )^{\frac {1}{6}} c^{3} d + \sqrt {3} \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a c^{4}} + \frac {{\left (\sqrt {3} \left (a c^{5}\right )^{\frac {1}{6}} c^{3} d + \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a c^{4}} - \frac {{\left (\sqrt {3} \left (a c^{5}\right )^{\frac {1}{6}} c^{3} d - \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a c^{4}} \]

[In]

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

[Out]

-1/6*e*abs(c)*log(x^2 + (a/c)^(1/3))/(a*c^5)^(1/3) + 1/3*(a*c^5)^(1/6)*d*arctan(x/(a/c)^(1/6))/(a*c) + 1/6*((a
*c^5)^(1/6)*c^3*d - sqrt(3)*(a*c^5)^(2/3)*e)*arctan((2*x + sqrt(3)*(a/c)^(1/6))/(a/c)^(1/6))/(a*c^4) + 1/6*((a
*c^5)^(1/6)*c^3*d + sqrt(3)*(a*c^5)^(2/3)*e)*arctan((2*x - sqrt(3)*(a/c)^(1/6))/(a/c)^(1/6))/(a*c^4) + 1/12*(s
qrt(3)*(a*c^5)^(1/6)*c^3*d + (a*c^5)^(2/3)*e)*log(x^2 + sqrt(3)*x*(a/c)^(1/6) + (a/c)^(1/3))/(a*c^4) - 1/12*(s
qrt(3)*(a*c^5)^(1/6)*c^3*d - (a*c^5)^(2/3)*e)*log(x^2 - sqrt(3)*x*(a/c)^(1/6) + (a/c)^(1/3))/(a*c^4)

Mupad [B] (verification not implemented)

Time = 0.92 (sec) , antiderivative size = 1331, normalized size of antiderivative = 4.36 \[ \int \frac {d+e x^3}{a+c x^6} \, dx=\ln \left (a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3+c\,d^3\,\sqrt {-a^5\,c^5}-3\,a^3\,c^3\,d^2\,e-3\,a\,d\,e^2\,\sqrt {-a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}+e\,x\,\sqrt {-a^5\,c^5}+a^2\,c^3\,d\,x\right )\,{\left (-\frac {a^4\,c^2\,e^3+c\,d^3\,\sqrt {-a^5\,c^5}-3\,a^3\,c^3\,d^2\,e-3\,a\,d\,e^2\,\sqrt {-a^5\,c^5}}{216\,a^5\,c^4}\right )}^{1/3}+\ln \left (a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3-c\,d^3\,\sqrt {-a^5\,c^5}-3\,a^3\,c^3\,d^2\,e+3\,a\,d\,e^2\,\sqrt {-a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}-e\,x\,\sqrt {-a^5\,c^5}+a^2\,c^3\,d\,x\right )\,{\left (-\frac {a^4\,c^2\,e^3-c\,d^3\,\sqrt {-a^5\,c^5}-3\,a^3\,c^3\,d^2\,e+3\,a\,d\,e^2\,\sqrt {-a^5\,c^5}}{216\,a^5\,c^4}\right )}^{1/3}-\ln \left (a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3+c\,d^3\,\sqrt {-a^5\,c^5}-3\,a^3\,c^3\,d^2\,e-3\,a\,d\,e^2\,\sqrt {-a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}-2\,e\,x\,\sqrt {-a^5\,c^5}-2\,a^2\,c^3\,d\,x+\sqrt {3}\,a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3+c\,d^3\,\sqrt {-a^5\,c^5}-3\,a^3\,c^3\,d^2\,e-3\,a\,d\,e^2\,\sqrt {-a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a^4\,c^2\,e^3+c\,d^3\,\sqrt {-a^5\,c^5}-3\,a^3\,c^3\,d^2\,e-3\,a\,d\,e^2\,\sqrt {-a^5\,c^5}}{216\,a^5\,c^4}\right )}^{1/3}+\ln \left (e\,x\,\sqrt {-a^5\,c^5}-\frac {a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3+c\,d^3\,\sqrt {-a^5\,c^5}-3\,a^3\,c^3\,d^2\,e-3\,a\,d\,e^2\,\sqrt {-a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}}{2}+a^2\,c^3\,d\,x+\frac {\sqrt {3}\,a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3+c\,d^3\,\sqrt {-a^5\,c^5}-3\,a^3\,c^3\,d^2\,e-3\,a\,d\,e^2\,\sqrt {-a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a^4\,c^2\,e^3+c\,d^3\,\sqrt {-a^5\,c^5}-3\,a^3\,c^3\,d^2\,e-3\,a\,d\,e^2\,\sqrt {-a^5\,c^5}}{216\,a^5\,c^4}\right )}^{1/3}+\ln \left (a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3-c\,d^3\,\sqrt {-a^5\,c^5}-3\,a^3\,c^3\,d^2\,e+3\,a\,d\,e^2\,\sqrt {-a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}+2\,e\,x\,\sqrt {-a^5\,c^5}-2\,a^2\,c^3\,d\,x-\sqrt {3}\,a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3-c\,d^3\,\sqrt {-a^5\,c^5}-3\,a^3\,c^3\,d^2\,e+3\,a\,d\,e^2\,\sqrt {-a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a^4\,c^2\,e^3-c\,d^3\,\sqrt {-a^5\,c^5}-3\,a^3\,c^3\,d^2\,e+3\,a\,d\,e^2\,\sqrt {-a^5\,c^5}}{216\,a^5\,c^4}\right )}^{1/3}-\ln \left (a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3-c\,d^3\,\sqrt {-a^5\,c^5}-3\,a^3\,c^3\,d^2\,e+3\,a\,d\,e^2\,\sqrt {-a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}+2\,e\,x\,\sqrt {-a^5\,c^5}-2\,a^2\,c^3\,d\,x+\sqrt {3}\,a^3\,c^3\,{\left (-\frac {a^4\,c^2\,e^3-c\,d^3\,\sqrt {-a^5\,c^5}-3\,a^3\,c^3\,d^2\,e+3\,a\,d\,e^2\,\sqrt {-a^5\,c^5}}{a^5\,c^4}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a^4\,c^2\,e^3-c\,d^3\,\sqrt {-a^5\,c^5}-3\,a^3\,c^3\,d^2\,e+3\,a\,d\,e^2\,\sqrt {-a^5\,c^5}}{216\,a^5\,c^4}\right )}^{1/3} \]

[In]

int((d + e*x^3)/(a + c*x^6),x)

[Out]

log(a^3*c^3*(-(a^4*c^2*e^3 + c*d^3*(-a^5*c^5)^(1/2) - 3*a^3*c^3*d^2*e - 3*a*d*e^2*(-a^5*c^5)^(1/2))/(a^5*c^4))
^(1/3) + e*x*(-a^5*c^5)^(1/2) + a^2*c^3*d*x)*(-(a^4*c^2*e^3 + c*d^3*(-a^5*c^5)^(1/2) - 3*a^3*c^3*d^2*e - 3*a*d
*e^2*(-a^5*c^5)^(1/2))/(216*a^5*c^4))^(1/3) + log(a^3*c^3*(-(a^4*c^2*e^3 - c*d^3*(-a^5*c^5)^(1/2) - 3*a^3*c^3*
d^2*e + 3*a*d*e^2*(-a^5*c^5)^(1/2))/(a^5*c^4))^(1/3) - e*x*(-a^5*c^5)^(1/2) + a^2*c^3*d*x)*(-(a^4*c^2*e^3 - c*
d^3*(-a^5*c^5)^(1/2) - 3*a^3*c^3*d^2*e + 3*a*d*e^2*(-a^5*c^5)^(1/2))/(216*a^5*c^4))^(1/3) - log(a^3*c^3*(-(a^4
*c^2*e^3 + c*d^3*(-a^5*c^5)^(1/2) - 3*a^3*c^3*d^2*e - 3*a*d*e^2*(-a^5*c^5)^(1/2))/(a^5*c^4))^(1/3) - 2*e*x*(-a
^5*c^5)^(1/2) + 3^(1/2)*a^3*c^3*(-(a^4*c^2*e^3 + c*d^3*(-a^5*c^5)^(1/2) - 3*a^3*c^3*d^2*e - 3*a*d*e^2*(-a^5*c^
5)^(1/2))/(a^5*c^4))^(1/3)*1i - 2*a^2*c^3*d*x)*((3^(1/2)*1i)/2 + 1/2)*(-(a^4*c^2*e^3 + c*d^3*(-a^5*c^5)^(1/2)
- 3*a^3*c^3*d^2*e - 3*a*d*e^2*(-a^5*c^5)^(1/2))/(216*a^5*c^4))^(1/3) + log(e*x*(-a^5*c^5)^(1/2) - (a^3*c^3*(-(
a^4*c^2*e^3 + c*d^3*(-a^5*c^5)^(1/2) - 3*a^3*c^3*d^2*e - 3*a*d*e^2*(-a^5*c^5)^(1/2))/(a^5*c^4))^(1/3))/2 + (3^
(1/2)*a^3*c^3*(-(a^4*c^2*e^3 + c*d^3*(-a^5*c^5)^(1/2) - 3*a^3*c^3*d^2*e - 3*a*d*e^2*(-a^5*c^5)^(1/2))/(a^5*c^4
))^(1/3)*1i)/2 + a^2*c^3*d*x)*((3^(1/2)*1i)/2 - 1/2)*(-(a^4*c^2*e^3 + c*d^3*(-a^5*c^5)^(1/2) - 3*a^3*c^3*d^2*e
- 3*a*d*e^2*(-a^5*c^5)^(1/2))/(216*a^5*c^4))^(1/3) + log(a^3*c^3*(-(a^4*c^2*e^3 - c*d^3*(-a^5*c^5)^(1/2) - 3*
a^3*c^3*d^2*e + 3*a*d*e^2*(-a^5*c^5)^(1/2))/(a^5*c^4))^(1/3) + 2*e*x*(-a^5*c^5)^(1/2) - 3^(1/2)*a^3*c^3*(-(a^4
*c^2*e^3 - c*d^3*(-a^5*c^5)^(1/2) - 3*a^3*c^3*d^2*e + 3*a*d*e^2*(-a^5*c^5)^(1/2))/(a^5*c^4))^(1/3)*1i - 2*a^2*
c^3*d*x)*((3^(1/2)*1i)/2 - 1/2)*(-(a^4*c^2*e^3 - c*d^3*(-a^5*c^5)^(1/2) - 3*a^3*c^3*d^2*e + 3*a*d*e^2*(-a^5*c^
5)^(1/2))/(216*a^5*c^4))^(1/3) - log(a^3*c^3*(-(a^4*c^2*e^3 - c*d^3*(-a^5*c^5)^(1/2) - 3*a^3*c^3*d^2*e + 3*a*d
*e^2*(-a^5*c^5)^(1/2))/(a^5*c^4))^(1/3) + 2*e*x*(-a^5*c^5)^(1/2) + 3^(1/2)*a^3*c^3*(-(a^4*c^2*e^3 - c*d^3*(-a^
5*c^5)^(1/2) - 3*a^3*c^3*d^2*e + 3*a*d*e^2*(-a^5*c^5)^(1/2))/(a^5*c^4))^(1/3)*1i - 2*a^2*c^3*d*x)*((3^(1/2)*1i
)/2 + 1/2)*(-(a^4*c^2*e^3 - c*d^3*(-a^5*c^5)^(1/2) - 3*a^3*c^3*d^2*e + 3*a*d*e^2*(-a^5*c^5)^(1/2))/(216*a^5*c^
4))^(1/3)