\(\int \frac {d+\frac {e}{x^3}}{c+\frac {a}{x^6}} \, dx\) [38]

Optimal result

Integrand size = 17, antiderivative size = 311 \[ \int \frac {d+\frac {e}{x^3}}{c+\frac {a}{x^6}} \, dx=\frac {d x}{c}-\frac {\sqrt [6]{a} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 c^{7/6}}+\frac {\left (\sqrt {a} d-\sqrt {3} \sqrt {c} e\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 \sqrt [3]{a} c^{7/6}}-\frac {\left (\sqrt {a} d+\sqrt {3} \sqrt {c} e\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 \sqrt [3]{a} c^{7/6}}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}+\frac {\left (\sqrt {3} \sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}}-\frac {\left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}} \]

[Out]

d*x/c-1/3*a^(1/6)*d*arctan(c^(1/6)*x/a^(1/6))/c^(7/6)-1/6*e*ln(a^(1/3)+c^(1/3)*x^2)/a^(1/3)/c^(2/3)-1/12*ln(a^
(1/3)+c^(1/3)*x^2+a^(1/6)*c^(1/6)*x*3^(1/2))*(d*3^(1/2)*a^(1/2)-e*c^(1/2))/a^(1/3)/c^(7/6)+1/12*ln(a^(1/3)+c^(
1/3)*x^2-a^(1/6)*c^(1/6)*x*3^(1/2))*(d*3^(1/2)*a^(1/2)+e*c^(1/2))/a^(1/3)/c^(7/6)-1/6*arctan(2*c^(1/6)*x/a^(1/
6)-3^(1/2))*(d*a^(1/2)-e*3^(1/2)*c^(1/2))/a^(1/3)/c^(7/6)-1/6*arctan(2*c^(1/6)*x/a^(1/6)+3^(1/2))*(d*a^(1/2)+e
*3^(1/2)*c^(1/2))/a^(1/3)/c^(7/6)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {1408, 1517, 1430, 649, 209, 266, 648, 631, 210, 642} \[ \int \frac {d+\frac {e}{x^3}}{c+\frac {a}{x^6}} \, dx=\frac {\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right ) \left (\sqrt {a} d-\sqrt {3} \sqrt {c} e\right )}{6 \sqrt [3]{a} c^{7/6}}-\frac {\arctan \left (\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}+\sqrt {3}\right ) \left (\sqrt {a} d+\sqrt {3} \sqrt {c} e\right )}{6 \sqrt [3]{a} c^{7/6}}-\frac {\sqrt [6]{a} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 c^{7/6}}+\frac {\left (\sqrt {3} \sqrt {a} d+\sqrt {c} e\right ) \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}}-\frac {\left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}+\frac {d x}{c} \]

[In]

Int[(d + e/x^3)/(c + a/x^6),x]

[Out]

(d*x)/c - (a^(1/6)*d*ArcTan[(c^(1/6)*x)/a^(1/6)])/(3*c^(7/6)) + ((Sqrt[a]*d - Sqrt[3]*Sqrt[c]*e)*ArcTan[Sqrt[3
] - (2*c^(1/6)*x)/a^(1/6)])/(6*a^(1/3)*c^(7/6)) - ((Sqrt[a]*d + Sqrt[3]*Sqrt[c]*e)*ArcTan[Sqrt[3] + (2*c^(1/6)
*x)/a^(1/6)])/(6*a^(1/3)*c^(7/6)) - (e*Log[a^(1/3) + c^(1/3)*x^2])/(6*a^(1/3)*c^(2/3)) + ((Sqrt[3]*Sqrt[a]*d +
Sqrt[c]*e)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(1/3)*c^(7/6)) - ((Sqrt[3]*Sqrt[a]*d
- Sqrt[c]*e)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(12*a^(1/3)*c^(7/6))

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 1408

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[x^(n*(2*p + q))*(e + d/x
^n)^q*(c + a/x^(2*n))^p, x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && IntegersQ[p, q] && NegQ[n]

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]

Rule 1517

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3 \left (e+d x^3\right )}{a+c x^6} \, dx \\ & = \frac {d x}{c}-\frac {\int \frac {a d-c e x^3}{a+c x^6} \, dx}{c} \\ & = \frac {d x}{c}-\frac {\int \frac {2 a^{2/3} \sqrt [3]{c} d-\left (\sqrt {3} \sqrt {a} \sqrt {c} d+c 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} c^{4/3}}-\frac {\int \frac {2 a^{2/3} \sqrt [3]{c} d+\left (\sqrt {3} \sqrt {a} \sqrt {c} d-c 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} c^{4/3}}-\frac {\int \frac {a^{2/3} \sqrt [3]{c} d+c e x}{1+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{3 a^{2/3} c^{4/3}} \\ & = \frac {d x}{c}-\frac {d \int \frac {1}{1+\frac {\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{3 c}-\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 {a} d-\sqrt {c} 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 \sqrt [3]{a} c^{7/6}}+\frac {\left (\sqrt {3} \sqrt {a} d+\sqrt {c} 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 \sqrt [3]{a} c^{7/6}}-\frac {\left (d-\frac {\sqrt {3} \sqrt {c} e}{\sqrt {a}}\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 c}-\frac {\left (d+\frac {\sqrt {3} \sqrt {c} e}{\sqrt {a}}\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 c} \\ & = \frac {d x}{c}-\frac {\sqrt [6]{a} d \tan ^{-1}\left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 c^{7/6}}-\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 {a} d+\sqrt {c} e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}}-\frac {\left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}}-\frac {\left (\sqrt {3} \sqrt {a} d-3 \sqrt {c} 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 \sqrt [3]{a} c^{7/6}}+\frac {\left (\sqrt {3} \sqrt {a} d+3 \sqrt {c} 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 \sqrt [3]{a} c^{7/6}} \\ & = \frac {d x}{c}-\frac {\sqrt [6]{a} d \tan ^{-1}\left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 c^{7/6}}+\frac {\left (\sqrt {a} d-\sqrt {3} \sqrt {c} e\right ) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 \sqrt [3]{a} c^{7/6}}-\frac {\left (\sqrt {a} d+\sqrt {3} \sqrt {c} e\right ) \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 \sqrt [3]{a} c^{7/6}}-\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 {a} d+\sqrt {c} e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}}-\frac {\left (\sqrt {3} \sqrt {a} d-\sqrt {c} e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 \sqrt [3]{a} c^{7/6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.11 \[ \int \frac {d+\frac {e}{x^3}}{c+\frac {a}{x^6}} \, dx=\frac {d x}{c}-\frac {\sqrt [6]{a} d \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 c^{7/6}}+\frac {\left (-a^{7/6} \sqrt {c} d+\sqrt {3} a^{2/3} c e\right ) \arctan \left (\frac {-\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a c^{5/3}}+\frac {\left (-a^{7/6} \sqrt {c} d-\sqrt {3} a^{2/3} c e\right ) \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a c^{5/3}}-\frac {e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}-\frac {\left (-\sqrt {3} a^{7/6} \sqrt {c} d-a^{2/3} c e\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a c^{5/3}}-\frac {\left (\sqrt {3} a^{7/6} \sqrt {c} d-a^{2/3} c e\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a c^{5/3}} \]

[In]

Integrate[(d + e/x^3)/(c + a/x^6),x]

[Out]

(d*x)/c - (a^(1/6)*d*ArcTan[(c^(1/6)*x)/a^(1/6)])/(3*c^(7/6)) + ((-(a^(7/6)*Sqrt[c]*d) + Sqrt[3]*a^(2/3)*c*e)*
ArcTan[(-(Sqrt[3]*a^(1/6)) + 2*c^(1/6)*x)/a^(1/6)])/(6*a*c^(5/3)) + ((-(a^(7/6)*Sqrt[c]*d) - Sqrt[3]*a^(2/3)*c
*e)*ArcTan[(Sqrt[3]*a^(1/6) + 2*c^(1/6)*x)/a^(1/6)])/(6*a*c^(5/3)) - (e*Log[a^(1/3) + c^(1/3)*x^2])/(6*a^(1/3)
*c^(2/3)) - ((-(Sqrt[3]*a^(7/6)*Sqrt[c]*d) - a^(2/3)*c*e)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^
2])/(12*a*c^(5/3)) - ((Sqrt[3]*a^(7/6)*Sqrt[c]*d - a^(2/3)*c*e)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + c^(1
/3)*x^2])/(12*a*c^(5/3))

Maple [C] (verified)

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

Time = 0.16 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.14

method	result	size
risch	\(\frac {d x}{c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c +a \right )}{\sum }\frac {\left (\textit {\_R}^{3} c e -d a \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}}{6 c^{2}}\)	\(45\)
default	\(\frac {d x}{c}+\frac {\frac {c \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \left (\frac {a}{c}\right )^{\frac {2}{3}} e}{12 a}+\frac {\ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} d}{12}+\frac {c \left (\frac {a}{c}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) \sqrt {3}\, e}{6 a}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}-\sqrt {3}\right ) d}{6}-\frac {c \left (\frac {a}{c}\right )^{\frac {7}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) \sqrt {3}\, d}{12 a}+\frac {c \left (\frac {a}{c}\right )^{\frac {2}{3}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{c}\right )^{\frac {1}{6}} x +\left (\frac {a}{c}\right )^{\frac {1}{3}}\right ) e}{12 a}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) d}{6}-\frac {c \left (\frac {a}{c}\right )^{\frac {2}{3}} \arctan \left (\frac {2 x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}+\sqrt {3}\right ) \sqrt {3}\, e}{6 a}-\frac {c \left (\frac {a}{c}\right )^{\frac {2}{3}} e \ln \left (x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 a}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{6}} d \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3}}{c}\)	\(333\)

[In]

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

[Out]

d*x/c+1/6/c^2*sum((_R^3*c*e-a*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. 1608 vs. \(2 (213) = 426\).

Time = 0.39 (sec) , antiderivative size = 1608, normalized size of antiderivative = 5.17 \[ \int \frac {d+\frac {e}{x^3}}{c+\frac {a}{x^6}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/12*(2*c*((a*c^3*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*c^7)) + 3*a*d^2*e - c*e^3)/(a*c^3))^(1/3)
*log(-(a^2*d^5 - 2*a*c*d^3*e^2 - 3*c^2*d*e^4)*x + (a*c^5*e*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*
c^7)) + a^2*c*d^4 - 3*a*c^2*d^2*e^2)*((a*c^3*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*c^7)) + 3*a*d^
2*e - c*e^3)/(a*c^3))^(1/3)) - (sqrt(-3)*c + c)*((a*c^3*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*c^7
)) + 3*a*d^2*e - c*e^3)/(a*c^3))^(1/3)*log(-(a^2*d^5 - 2*a*c*d^3*e^2 - 3*c^2*d*e^4)*x - 1/2*(a^2*c*d^4 - 3*a*c
^2*d^2*e^2 + sqrt(-3)*(a^2*c*d^4 - 3*a*c^2*d^2*e^2) + (sqrt(-3)*a*c^5*e + a*c^5*e)*sqrt(-(a^2*d^6 - 6*a*c*d^4*
e^2 + 9*c^2*d^2*e^4)/(a*c^7)))*((a*c^3*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*c^7)) + 3*a*d^2*e -
c*e^3)/(a*c^3))^(1/3)) + (sqrt(-3)*c - c)*((a*c^3*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*c^7)) + 3
*a*d^2*e - c*e^3)/(a*c^3))^(1/3)*log(-(a^2*d^5 - 2*a*c*d^3*e^2 - 3*c^2*d*e^4)*x - 1/2*(a^2*c*d^4 - 3*a*c^2*d^2
*e^2 - sqrt(-3)*(a^2*c*d^4 - 3*a*c^2*d^2*e^2) - (sqrt(-3)*a*c^5*e - a*c^5*e)*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 +
9*c^2*d^2*e^4)/(a*c^7)))*((a*c^3*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*c^7)) + 3*a*d^2*e - c*e^3)
/(a*c^3))^(1/3)) + 2*c*(-(a*c^3*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*c^7)) - 3*a*d^2*e + c*e^3)/
(a*c^3))^(1/3)*log(-(a^2*d^5 - 2*a*c*d^3*e^2 - 3*c^2*d*e^4)*x - (a*c^5*e*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^
2*d^2*e^4)/(a*c^7)) - a^2*c*d^4 + 3*a*c^2*d^2*e^2)*(-(a*c^3*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a
*c^7)) - 3*a*d^2*e + c*e^3)/(a*c^3))^(1/3)) - (sqrt(-3)*c + c)*(-(a*c^3*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2
*d^2*e^4)/(a*c^7)) - 3*a*d^2*e + c*e^3)/(a*c^3))^(1/3)*log(-(a^2*d^5 - 2*a*c*d^3*e^2 - 3*c^2*d*e^4)*x - 1/2*(a
^2*c*d^4 - 3*a*c^2*d^2*e^2 + sqrt(-3)*(a^2*c*d^4 - 3*a*c^2*d^2*e^2) - (sqrt(-3)*a*c^5*e + a*c^5*e)*sqrt(-(a^2*
d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*c^7)))*(-(a*c^3*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*c^7
)) - 3*a*d^2*e + c*e^3)/(a*c^3))^(1/3)) + (sqrt(-3)*c - c)*(-(a*c^3*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2
*e^4)/(a*c^7)) - 3*a*d^2*e + c*e^3)/(a*c^3))^(1/3)*log(-(a^2*d^5 - 2*a*c*d^3*e^2 - 3*c^2*d*e^4)*x - 1/2*(a^2*c
*d^4 - 3*a*c^2*d^2*e^2 - sqrt(-3)*(a^2*c*d^4 - 3*a*c^2*d^2*e^2) + (sqrt(-3)*a*c^5*e - a*c^5*e)*sqrt(-(a^2*d^6
- 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*c^7)))*(-(a*c^3*sqrt(-(a^2*d^6 - 6*a*c*d^4*e^2 + 9*c^2*d^2*e^4)/(a*c^7)) -
3*a*d^2*e + c*e^3)/(a*c^3))^(1/3)) + 12*d*x)/c

Sympy [A] (verification not implemented)

Time = 5.67 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.54 \[ \int \frac {d+\frac {e}{x^3}}{c+\frac {a}{x^6}} \, dx=\operatorname {RootSum} {\left (46656 t^{6} a^{2} c^{7} + t^{3} \left (- 1296 a^{2} c^{4} d^{2} e + 432 a c^{5} e^{3}\right ) + a^{3} d^{6} + 3 a^{2} c d^{4} e^{2} + 3 a c^{2} d^{2} e^{4} + c^{3} e^{6}, \left ( t \mapsto t \log {\left (x + \frac {- 1296 t^{4} a c^{5} e - 6 t a^{2} c d^{4} + 36 t a c^{2} d^{2} e^{2} - 6 t c^{3} e^{4}}{a^{2} d^{5} - 2 a c d^{3} e^{2} - 3 c^{2} d e^{4}} \right )} \right )\right )} + \frac {d x}{c} \]

[In]

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

[Out]

RootSum(46656*_t**6*a**2*c**7 + _t**3*(-1296*a**2*c**4*d**2*e + 432*a*c**5*e**3) + a**3*d**6 + 3*a**2*c*d**4*e
**2 + 3*a*c**2*d**2*e**4 + c**3*e**6, Lambda(_t, _t*log(x + (-1296*_t**4*a*c**5*e - 6*_t*a**2*c*d**4 + 36*_t*a
*c**2*d**2*e**2 - 6*_t*c**3*e**4)/(a**2*d**5 - 2*a*c*d**3*e**2 - 3*c**2*d*e**4)))) + d*x/c

Maxima [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.95 \[ \int \frac {d+\frac {e}{x^3}}{c+\frac {a}{x^6}} \, dx=\frac {d x}{c} - \frac {\frac {2 \, c^{\frac {1}{3}} e \log \left (c^{\frac {1}{3}} x^{2} + a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}} + \frac {4 \, a^{\frac {1}{3}} d \arctan \left (\frac {c^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} + \frac {{\left (\sqrt {3} a^{\frac {7}{6}} \sqrt {c} d - a^{\frac {2}{3}} c e\right )} \log \left (c^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a c^{\frac {2}{3}}} - \frac {{\left (\sqrt {3} a^{\frac {7}{6}} \sqrt {c} d + a^{\frac {2}{3}} c e\right )} \log \left (c^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a c^{\frac {2}{3}}} + \frac {2 \, {\left (\sqrt {3} a^{\frac {5}{6}} c^{\frac {7}{6}} e + a^{\frac {4}{3}} c^{\frac {2}{3}} d\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a c^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}} - \frac {2 \, {\left (\sqrt {3} a^{\frac {5}{6}} c^{\frac {7}{6}} e - a^{\frac {4}{3}} c^{\frac {2}{3}} d\right )} \arctan \left (\frac {2 \, c^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} c^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}\right )}{a c^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} c^{\frac {1}{3}}}}}{12 \, c} \]

[In]

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

[Out]

d*x/c - 1/12*(2*c^(1/3)*e*log(c^(1/3)*x^2 + a^(1/3))/a^(1/3) + 4*a^(1/3)*d*arctan(c^(1/3)*x/sqrt(a^(1/3)*c^(1/
3)))/sqrt(a^(1/3)*c^(1/3)) + (sqrt(3)*a^(7/6)*sqrt(c)*d - a^(2/3)*c*e)*log(c^(1/3)*x^2 + sqrt(3)*a^(1/6)*c^(1/
6)*x + a^(1/3))/(a*c^(2/3)) - (sqrt(3)*a^(7/6)*sqrt(c)*d + a^(2/3)*c*e)*log(c^(1/3)*x^2 - sqrt(3)*a^(1/6)*c^(1
/6)*x + a^(1/3))/(a*c^(2/3)) + 2*(sqrt(3)*a^(5/6)*c^(7/6)*e + a^(4/3)*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))) - 2*(sqrt(3)*a^(5/6)*c^(7/6)*e - a^
(4/3)*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))))/c

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.93 \[ \int \frac {d+\frac {e}{x^3}}{c+\frac {a}{x^6}} \, dx=-\frac {e {\left | c \right |} \log \left (x^{2} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{6 \, \left (a c^{5}\right )^{\frac {1}{3}}} + \frac {d x}{c} - \frac {\left (a c^{5}\right )^{\frac {1}{6}} d \arctan \left (\frac {x}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{3 \, c^{2}} - \frac {{\left (\left (a c^{5}\right )^{\frac {1}{6}} a c^{2} d + \sqrt {3} \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a c^{4}} - \frac {{\left (\left (a c^{5}\right )^{\frac {1}{6}} a c^{2} d - \sqrt {3} \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{c}\right )^{\frac {1}{6}}}{\left (\frac {a}{c}\right )^{\frac {1}{6}}}\right )}{6 \, a c^{4}} - \frac {{\left (\sqrt {3} \left (a c^{5}\right )^{\frac {1}{6}} a c^{2} d - \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a c^{4}} + \frac {{\left (\sqrt {3} \left (a c^{5}\right )^{\frac {1}{6}} a c^{2} d + \left (a c^{5}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{c}\right )^{\frac {1}{6}} + \left (\frac {a}{c}\right )^{\frac {1}{3}}\right )}{12 \, a c^{4}} \]

[In]

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

[Out]

-1/6*e*abs(c)*log(x^2 + (a/c)^(1/3))/(a*c^5)^(1/3) + d*x/c - 1/3*(a*c^5)^(1/6)*d*arctan(x/(a/c)^(1/6))/c^2 - 1
/6*((a*c^5)^(1/6)*a*c^2*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)*a*c^2*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)*a*c^2*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)*a*c^2*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 = 9.37 (sec) , antiderivative size = 1308, normalized size of antiderivative = 4.21 \[ \int \frac {d+\frac {e}{x^3}}{c+\frac {a}{x^6}} \, dx=\ln \left (e\,x\,\sqrt {-a^3\,c^7}-a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3+a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e-3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}+a^2\,c^3\,d\,x\right )\,{\left (-\frac {a\,c^5\,e^3+a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e-3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{216\,a^2\,c^7}\right )}^{1/3}+\ln \left (e\,x\,\sqrt {-a^3\,c^7}+a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3-a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e+3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}-a^2\,c^3\,d\,x\right )\,{\left (-\frac {a\,c^5\,e^3-a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e+3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{216\,a^2\,c^7}\right )}^{1/3}+\ln \left (2\,e\,x\,\sqrt {-a^3\,c^7}+a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3+a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e-3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}+2\,a^2\,c^3\,d\,x-\sqrt {3}\,a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3+a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e-3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a\,c^5\,e^3+a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e-3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{216\,a^2\,c^7}\right )}^{1/3}-\ln \left (2\,e\,x\,\sqrt {-a^3\,c^7}+a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3+a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e-3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}+2\,a^2\,c^3\,d\,x+\sqrt {3}\,a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3+a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e-3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a\,c^5\,e^3+a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e-3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{216\,a^2\,c^7}\right )}^{1/3}-\ln \left (a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3-a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e+3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}-2\,e\,x\,\sqrt {-a^3\,c^7}+2\,a^2\,c^3\,d\,x+\sqrt {3}\,a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3-a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e+3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a\,c^5\,e^3-a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e+3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{216\,a^2\,c^7}\right )}^{1/3}+\ln \left (2\,e\,x\,\sqrt {-a^3\,c^7}-a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3-a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e+3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}-2\,a^2\,c^3\,d\,x+\sqrt {3}\,a^2\,c^4\,{\left (-\frac {a\,c^5\,e^3-a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e+3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{a^2\,c^7}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {a\,c^5\,e^3-a\,d^3\,\sqrt {-a^3\,c^7}-3\,a^2\,c^4\,d^2\,e+3\,c\,d\,e^2\,\sqrt {-a^3\,c^7}}{216\,a^2\,c^7}\right )}^{1/3}+\frac {d\,x}{c} \]

[In]

int((d + e/x^3)/(c + a/x^6),x)

[Out]

log(e*x*(-a^3*c^7)^(1/2) - a^2*c^4*(-(a*c^5*e^3 + a*d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e - 3*c*d*e^2*(-a^3*c
^7)^(1/2))/(a^2*c^7))^(1/3) + a^2*c^3*d*x)*(-(a*c^5*e^3 + a*d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e - 3*c*d*e^2
*(-a^3*c^7)^(1/2))/(216*a^2*c^7))^(1/3) + log(e*x*(-a^3*c^7)^(1/2) + a^2*c^4*(-(a*c^5*e^3 - a*d^3*(-a^3*c^7)^(
1/2) - 3*a^2*c^4*d^2*e + 3*c*d*e^2*(-a^3*c^7)^(1/2))/(a^2*c^7))^(1/3) - a^2*c^3*d*x)*(-(a*c^5*e^3 - a*d^3*(-a^
3*c^7)^(1/2) - 3*a^2*c^4*d^2*e + 3*c*d*e^2*(-a^3*c^7)^(1/2))/(216*a^2*c^7))^(1/3) + log(2*e*x*(-a^3*c^7)^(1/2)
+ a^2*c^4*(-(a*c^5*e^3 + a*d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e - 3*c*d*e^2*(-a^3*c^7)^(1/2))/(a^2*c^7))^(1
/3) - 3^(1/2)*a^2*c^4*(-(a*c^5*e^3 + a*d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e - 3*c*d*e^2*(-a^3*c^7)^(1/2))/(a
^2*c^7))^(1/3)*1i + 2*a^2*c^3*d*x)*((3^(1/2)*1i)/2 - 1/2)*(-(a*c^5*e^3 + a*d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^
2*e - 3*c*d*e^2*(-a^3*c^7)^(1/2))/(216*a^2*c^7))^(1/3) - log(2*e*x*(-a^3*c^7)^(1/2) + a^2*c^4*(-(a*c^5*e^3 + a
*d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e - 3*c*d*e^2*(-a^3*c^7)^(1/2))/(a^2*c^7))^(1/3) + 3^(1/2)*a^2*c^4*(-(a*
c^5*e^3 + a*d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e - 3*c*d*e^2*(-a^3*c^7)^(1/2))/(a^2*c^7))^(1/3)*1i + 2*a^2*c
^3*d*x)*((3^(1/2)*1i)/2 + 1/2)*(-(a*c^5*e^3 + a*d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e - 3*c*d*e^2*(-a^3*c^7)^
(1/2))/(216*a^2*c^7))^(1/3) - log(a^2*c^4*(-(a*c^5*e^3 - a*d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e + 3*c*d*e^2*
(-a^3*c^7)^(1/2))/(a^2*c^7))^(1/3) - 2*e*x*(-a^3*c^7)^(1/2) + 3^(1/2)*a^2*c^4*(-(a*c^5*e^3 - a*d^3*(-a^3*c^7)^
(1/2) - 3*a^2*c^4*d^2*e + 3*c*d*e^2*(-a^3*c^7)^(1/2))/(a^2*c^7))^(1/3)*1i + 2*a^2*c^3*d*x)*((3^(1/2)*1i)/2 + 1
/2)*(-(a*c^5*e^3 - a*d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e + 3*c*d*e^2*(-a^3*c^7)^(1/2))/(216*a^2*c^7))^(1/3)
+ log(2*e*x*(-a^3*c^7)^(1/2) - a^2*c^4*(-(a*c^5*e^3 - a*d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e + 3*c*d*e^2*(-
a^3*c^7)^(1/2))/(a^2*c^7))^(1/3) + 3^(1/2)*a^2*c^4*(-(a*c^5*e^3 - a*d^3*(-a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e + 3
*c*d*e^2*(-a^3*c^7)^(1/2))/(a^2*c^7))^(1/3)*1i - 2*a^2*c^3*d*x)*((3^(1/2)*1i)/2 - 1/2)*(-(a*c^5*e^3 - a*d^3*(-
a^3*c^7)^(1/2) - 3*a^2*c^4*d^2*e + 3*c*d*e^2*(-a^3*c^7)^(1/2))/(216*a^2*c^7))^(1/3) + (d*x)/c