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

Optimal result

Integrand size = 18, antiderivative size = 323 \[ \int \frac {d+e x^3}{a-c x^6} \, dx=-\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \arctan \left (\frac {\sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )}{2 \sqrt {3} a^{5/6} \sqrt [6]{c}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (\frac {\sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )}{2 \sqrt {3} a^{5/6} c^{2/3}}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{6 a^{5/6} c^{2/3}}+\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{6 a^{5/6} \sqrt [6]{c}}-\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \log \left (\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} \sqrt [6]{c}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1431, 206, 31, 648, 631, 210, 642} \[ \int \frac {d+e x^3}{a-c x^6} \, dx=\frac {\arctan \left (\frac {\sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right ) \left (\sqrt {a} e+\sqrt {c} d\right )}{2 \sqrt {3} a^{5/6} c^{2/3}}-\frac {\arctan \left (\frac {\sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right ) \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right )}{2 \sqrt {3} a^{5/6} \sqrt [6]{c}}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right ) \log \left (\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}-\frac {\left (\sqrt {a} e+\sqrt {c} d\right ) \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{6 a^{5/6} c^{2/3}}-\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \log \left (-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} \sqrt [6]{c}}+\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{6 a^{5/6} \sqrt [6]{c}} \]

[In]

Int[(d + e*x^3)/(a - c*x^6),x]

[Out]

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

Rule 31

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

Rule 206

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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 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 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 1431

Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{q = Rt[-a/c, 2]}, Dist[(d + e*q)/2,
Int[1/(a + c*q*x^n), x], x] + Dist[(d - e*q)/2, Int[1/(a - c*q*x^n), x], x]] /; FreeQ[{a, c, d, e, n}, x] &&
EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && NegQ[a*c] && IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{a+\sqrt {a} \sqrt {c} x^3} \, dx+\frac {1}{2} \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{a-\sqrt {a} \sqrt {c} x^3} \, dx \\ & = \frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x} \, dx}{6 a^{2/3}}+\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {2 \sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x}{a^{2/3}-\sqrt {a} \sqrt [6]{c} x+\sqrt [3]{a} \sqrt [3]{c} x^2} \, dx}{6 a^{2/3}}+\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x} \, dx}{6 a^{2/3}}+\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {2 \sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x}{a^{2/3}+\sqrt {a} \sqrt [6]{c} x+\sqrt [3]{a} \sqrt [3]{c} x^2} \, dx}{6 a^{2/3}} \\ & = -\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{6 a^{5/6} c^{2/3}}+\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{6 a^{5/6} \sqrt [6]{c}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \int \frac {\sqrt {a} \sqrt [6]{c}+2 \sqrt [3]{a} \sqrt [3]{c} x}{a^{2/3}+\sqrt {a} \sqrt [6]{c} x+\sqrt [3]{a} \sqrt [3]{c} x^2} \, dx}{12 a^{5/6} c^{2/3}}+\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{a^{2/3}-\sqrt {a} \sqrt [6]{c} x+\sqrt [3]{a} \sqrt [3]{c} x^2} \, dx}{4 \sqrt [3]{a}}-\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {-\sqrt {a} \sqrt [6]{c}+2 \sqrt [3]{a} \sqrt [3]{c} x}{a^{2/3}-\sqrt {a} \sqrt [6]{c} x+\sqrt [3]{a} \sqrt [3]{c} x^2} \, dx}{12 a^{5/6} \sqrt [6]{c}}+\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{a^{2/3}+\sqrt {a} \sqrt [6]{c} x+\sqrt [3]{a} \sqrt [3]{c} x^2} \, dx}{4 \sqrt [3]{a}} \\ & = -\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{6 a^{5/6} c^{2/3}}+\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{6 a^{5/6} \sqrt [6]{c}}-\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \log \left (\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} \sqrt [6]{c}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{2 a^{5/6} c^{2/3}}+\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{2 a^{5/6} \sqrt [6]{c}} \\ & = -\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \tan ^{-1}\left (\frac {\sqrt [6]{a}-2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )}{2 \sqrt {3} a^{5/6} \sqrt [6]{c}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt {3} \sqrt [6]{a}}\right )}{2 \sqrt {3} a^{5/6} c^{2/3}}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )}{6 a^{5/6} c^{2/3}}+\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )}{6 a^{5/6} \sqrt [6]{c}}-\frac {\left (d-\frac {\sqrt {a} e}{\sqrt {c}}\right ) \log \left (\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} \sqrt [6]{c}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \log \left (\sqrt [3]{a}+\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 = 337, normalized size of antiderivative = 1.04 \[ \int \frac {d+e x^3}{a-c x^6} \, dx=\frac {-2 \sqrt {3} \left (\sqrt {c} d-\sqrt {a} e\right ) \arctan \left (\frac {1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}}{\sqrt {3}}\right )+2 \sqrt {3} \left (\sqrt {c} d+\sqrt {a} e\right ) \arctan \left (\frac {1+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}}{\sqrt {3}}\right )-2 \sqrt {c} d \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )-2 \sqrt {a} e \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )+2 \sqrt {c} d \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )-2 \sqrt {a} e \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )-\sqrt {c} d \log \left (\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )+\sqrt {a} e \log \left (\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )+\sqrt {c} d \log \left (\sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )+\sqrt {a} e \log \left (\sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}} \]

[In]

Integrate[(d + e*x^3)/(a - c*x^6),x]

[Out]

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

Maple [C] (verified)

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

Time = 0.56 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.11

method	result	size
risch	\(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c -a \right )}{\sum }\frac {\left (\textit {\_R}^{3} e +d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}}{6 c}\)	\(36\)
default	\(-\frac {\ln \left (-x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right ) e}{6 c \left (\frac {a}{c}\right )^{\frac {1}{3}}}-\frac {\ln \left (-x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right ) d}{6 c \left (\frac {a}{c}\right )^{\frac {5}{6}}}+\frac {e \left (\frac {a}{c}\right )^{\frac {2}{3}} \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {e \left (\frac {a}{c}\right )^{\frac {2}{3}} \sqrt {3}\, \arctan \left (\frac {2 x \sqrt {3}}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}}{3}\right )}{6 a}+\frac {d \left (\frac {a}{c}\right )^{\frac {1}{6}} \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {d \left (\frac {a}{c}\right )^{\frac {1}{6}} \sqrt {3}\, \arctan \left (\frac {2 x \sqrt {3}}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}}{3}\right )}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} \ln \left (\left (\frac {a}{c}\right )^{\frac {1}{6}} x -x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) e}{12 a}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \ln \left (\left (\frac {a}{c}\right )^{\frac {1}{6}} x -x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) d}{12 a}-\frac {\left (\frac {a}{c}\right )^{\frac {2}{3}} \sqrt {3}\, e \arctan \left (-\frac {\sqrt {3}}{3}+\frac {2 x \sqrt {3}}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 a}+\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \sqrt {3}\, d \arctan \left (-\frac {\sqrt {3}}{3}+\frac {2 x \sqrt {3}}{3 \left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 a}-\frac {\ln \left (x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right ) e}{6 c \left (\frac {a}{c}\right )^{\frac {1}{3}}}+\frac {\ln \left (x +\left (\frac {a}{c}\right )^{\frac {1}{6}}\right ) d}{6 c \left (\frac {a}{c}\right )^{\frac {5}{6}}}\)	\(386\)

[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. 1613 vs. \(2 (223) = 446\).

Time = 0.37 (sec) , antiderivative size = 1613, normalized size of antiderivative = 4.99 \[ \int \frac {d+e x^3}{a-c x^6} \, dx=\text {Too large to display} \]

[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*sqr
t((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))

Sympy [A] (verification not implemented)

Time = 9.50 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.52 \[ \int \frac {d+e x^3}{a-c x^6} \, dx=- \operatorname {RootSum} {\left (46656 t^{6} a^{5} c^{4} + t^{3} \left (- 432 a^{4} c^{2} e^{3} - 1296 a^{3} c^{3} d^{2} e\right ) + a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} - c^{3} d^{6}, \left ( t \mapsto t \log {\left (x + \frac {- 1296 t^{4} a^{4} c^{2} e + 6 t a^{3} e^{4} + 36 t a^{2} c d^{2} e^{2} + 6 t a c^{2} d^{4}}{3 a^{2} d e^{4} - 2 a c d^{3} e^{2} - c^{2} d^{5}} \right )} \right )\right )} \]

[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 = 0.31 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.97 \[ \int \frac {d+e x^3}{a-c x^6} \, dx=\frac {\sqrt {3} {\left (\sqrt {c} d + \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}}\right )}{6 \, \sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} + \frac {\sqrt {3} {\left (\sqrt {c} d - \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}}\right )}{6 \, \sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} + \frac {{\left (\sqrt {c} d + \sqrt {a} e\right )} \log \left (x^{2} + x \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}} + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}\right )}{12 \, \sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} - \frac {{\left (\sqrt {c} d - \sqrt {a} e\right )} \log \left (x^{2} - x \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}} + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}\right )}{12 \, \sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} + \frac {{\left (\sqrt {c} d - \sqrt {a} e\right )} \log \left (x + \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}{6 \, \sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} - \frac {{\left (\sqrt {c} d + \sqrt {a} e\right )} \log \left (x - \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {1}{3}}\right )}{6 \, \sqrt {a} c \left (\frac {\sqrt {a}}{\sqrt {c}}\right )^{\frac {2}{3}}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.94 \[ \int \frac {d+e x^3}{a-c x^6} \, dx=\frac {e {\left | c \right |} \log \left (x^{2} + \left (-\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 \, \left (-a c^{5}\right )^{\frac {1}{3}}} + \frac {\left (-a c^{5}\right )^{\frac {1}{6}} d \arctan \left (\frac {x}{\left (-\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 \, a c} + \frac {{\left (\left (-a c^{5}\right )^{\frac {1}{6}} c^{3} d - \sqrt {3} \left (-a c^{5}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {2 \, x + \sqrt {3} \left (-\frac {a}{c}\right )^{\frac {1}{6}}}{\left (-\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a c^{4}} + \frac {{\left (\left (-a c^{5}\right )^{\frac {1}{6}} c^{3} d + \sqrt {3} \left (-a c^{5}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {2 \, x - \sqrt {3} \left (-\frac {a}{c}\right )^{\frac {1}{6}}}{\left (-\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a c^{4}} + \frac {{\left (\sqrt {3} \left (-a c^{5}\right )^{\frac {1}{6}} c^{3} d + \left (-a c^{5}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} + \sqrt {3} x \left (-\frac {a}{c}\right )^{\frac {1}{6}} + \left (-\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a c^{4}} - \frac {{\left (\sqrt {3} \left (-a c^{5}\right )^{\frac {1}{6}} c^{3} d - \left (-a c^{5}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} - \sqrt {3} x \left (-\frac {a}{c}\right )^{\frac {1}{6}} + \left (-\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a c^{4}} \]

[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*(sqrt(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*(sqrt(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 = 9.28 (sec) , antiderivative size = 1293, normalized size of antiderivative = 4.00 \[ \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)