3.4.47 \(\int \frac {x^6}{(a+b x^3)^3} \, dx\) [347]

Optimal. Leaf size=153 \[ -\frac {x^4}{6 b \left (a+b x^3\right )^2}-\frac {2 x}{9 b^2 \left (a+b x^3\right )}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{2/3} b^{7/3}}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{7/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{2/3} b^{7/3}} \]

[Out]

-1/6*x^4/b/(b*x^3+a)^2-2/9*x/b^2/(b*x^3+a)+2/27*ln(a^(1/3)+b^(1/3)*x)/a^(2/3)/b^(7/3)-1/27*ln(a^(2/3)-a^(1/3)*
b^(1/3)*x+b^(2/3)*x^2)/a^(2/3)/b^(7/3)-2/27*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(2/3)/b^(7/3)*
3^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {294, 206, 31, 648, 631, 210, 642} \begin {gather*} -\frac {2 \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{2/3} b^{7/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{2/3} b^{7/3}}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{7/3}}-\frac {2 x}{9 b^2 \left (a+b x^3\right )}-\frac {x^4}{6 b \left (a+b x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b*x^3)^3,x]

[Out]

-1/6*x^4/(b*(a + b*x^3)^2) - (2*x)/(9*b^2*(a + b*x^3)) - (2*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])
/(9*Sqrt[3]*a^(2/3)*b^(7/3)) + (2*Log[a^(1/3) + b^(1/3)*x])/(27*a^(2/3)*b^(7/3)) - Log[a^(2/3) - a^(1/3)*b^(1/
3)*x + b^(2/3)*x^2]/(27*a^(2/3)*b^(7/3))

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 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, 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]

Rubi steps

\begin {align*} \int \frac {x^6}{\left (a+b x^3\right )^3} \, dx &=-\frac {x^4}{6 b \left (a+b x^3\right )^2}+\frac {2 \int \frac {x^3}{\left (a+b x^3\right )^2} \, dx}{3 b}\\ &=-\frac {x^4}{6 b \left (a+b x^3\right )^2}-\frac {2 x}{9 b^2 \left (a+b x^3\right )}+\frac {2 \int \frac {1}{a+b x^3} \, dx}{9 b^2}\\ &=-\frac {x^4}{6 b \left (a+b x^3\right )^2}-\frac {2 x}{9 b^2 \left (a+b x^3\right )}+\frac {2 \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{2/3} b^2}+\frac {2 \int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{2/3} b^2}\\ &=-\frac {x^4}{6 b \left (a+b x^3\right )^2}-\frac {2 x}{9 b^2 \left (a+b x^3\right )}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{7/3}}-\frac {\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{2/3} b^{7/3}}+\frac {\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 \sqrt [3]{a} b^2}\\ &=-\frac {x^4}{6 b \left (a+b x^3\right )^2}-\frac {2 x}{9 b^2 \left (a+b x^3\right )}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{7/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{2/3} b^{7/3}}+\frac {2 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{2/3} b^{7/3}}\\ &=-\frac {x^4}{6 b \left (a+b x^3\right )^2}-\frac {2 x}{9 b^2 \left (a+b x^3\right )}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{2/3} b^{7/3}}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{7/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{2/3} b^{7/3}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 136, normalized size = 0.89 \begin {gather*} \frac {\frac {9 a \sqrt [3]{b} x}{\left (a+b x^3\right )^2}-\frac {21 \sqrt [3]{b} x}{a+b x^3}-\frac {4 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}+\frac {4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}-\frac {2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}}{54 b^{7/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b*x^3)^3,x]

[Out]

((9*a*b^(1/3)*x)/(a + b*x^3)^2 - (21*b^(1/3)*x)/(a + b*x^3) - (4*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sq
rt[3]])/a^(2/3) + (4*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) - (2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^
(2/3))/(54*b^(7/3))

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Maple [A]
time = 0.13, size = 123, normalized size = 0.80

method	result	size
risch	\(\frac {-\frac {7 x^{4}}{18 b}-\frac {2 a x}{9 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {2 \left (\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{27 b^{3}}\)	\(54\)
default	\(\frac {-\frac {7 x^{4}}{18 b}-\frac {2 a x}{9 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\frac {2 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{b^{2}}\)	\(123\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-7/18*x^4/b-2/9*a*x/b^2)/(b*x^3+a)^2+2/9/b^2*(1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/6/b/(a/b)^(2/3)*ln(x^2-(a
/b)^(1/3)*x+(a/b)^(2/3))+1/3/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)))

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Maxima [A]
time = 0.51, size = 137, normalized size = 0.90 \begin {gather*} -\frac {7 \, b x^{4} + 4 \, a x}{18 \, {\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(7*b*x^4 + 4*a*x)/(b^4*x^6 + 2*a*b^3*x^3 + a^2*b^2) + 2/27*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3)
)/(a/b)^(1/3))/(b^3*(a/b)^(2/3)) - 1/27*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^3*(a/b)^(2/3)) + 2/27*log(x
+ (a/b)^(1/3))/(b^3*(a/b)^(2/3))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (112) = 224\).
time = 0.36, size = 503, normalized size = 3.29 \begin {gather*} \left [-\frac {21 \, a^{2} b^{2} x^{4} + 12 \, a^{3} b x - 6 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) + 2 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) - 4 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{54 \, {\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )}}, -\frac {21 \, a^{2} b^{2} x^{4} + 12 \, a^{3} b x - 12 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) + 2 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) - 4 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{54 \, {\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/54*(21*a^2*b^2*x^4 + 12*a^3*b*x - 6*sqrt(1/3)*(a*b^3*x^6 + 2*a^2*b^2*x^3 + a^3*b)*sqrt(-(a^2*b)^(1/3)/b)*l
og((2*a*b*x^3 - 3*(a^2*b)^(1/3)*a*x - a^2 + 3*sqrt(1/3)*(2*a*b*x^2 + (a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt(-
(a^2*b)^(1/3)/b))/(b*x^3 + a)) + 2*(b^2*x^6 + 2*a*b*x^3 + a^2)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (
a^2*b)^(1/3)*a) - 4*(b^2*x^6 + 2*a*b*x^3 + a^2)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^2*b^5*x^6 + 2*a^3
*b^4*x^3 + a^4*b^3), -1/54*(21*a^2*b^2*x^4 + 12*a^3*b*x - 12*sqrt(1/3)*(a*b^3*x^6 + 2*a^2*b^2*x^3 + a^3*b)*sqr
t((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) + 2*(b^2*
x^6 + 2*a*b*x^3 + a^2)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) - 4*(b^2*x^6 + 2*a*b*x^3
+ a^2)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^2*b^5*x^6 + 2*a^3*b^4*x^3 + a^4*b^3)]

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Sympy [A]
time = 0.18, size = 68, normalized size = 0.44 \begin {gather*} \frac {- 4 a x - 7 b x^{4}}{18 a^{2} b^{2} + 36 a b^{3} x^{3} + 18 b^{4} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} a^{2} b^{7} - 8, \left ( t \mapsto t \log {\left (\frac {27 t a b^{2}}{2} + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-4*a*x - 7*b*x**4)/(18*a**2*b**2 + 36*a*b**3*x**3 + 18*b**4*x**6) + RootSum(19683*_t**3*a**2*b**7 - 8, Lambda
(_t, _t*log(27*_t*a*b**2/2 + x)))

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Giac [A]
time = 2.20, size = 140, normalized size = 0.92 \begin {gather*} -\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a b^{2}} + \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a b^{3}} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, a b^{3}} - \frac {7 \, b x^{4} + 4 \, a x}{18 \, {\left (b x^{3} + a\right )}^{2} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/27*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^2) + 2/27*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*(2*x +
(-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^3) + 1/27*(-a*b^2)^(1/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b^3) - 1
/18*(7*b*x^4 + 4*a*x)/((b*x^3 + a)^2*b^2)

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Mupad [B]
time = 0.31, size = 127, normalized size = 0.83 \begin {gather*} \frac {2\,\ln \left (x+\frac {a^{1/3}}{b^{1/3}}\right )}{27\,a^{2/3}\,b^{7/3}}-\frac {\frac {7\,x^4}{18\,b}+\frac {2\,a\,x}{9\,b^2}}{a^2+2\,a\,b\,x^3+b^2\,x^6}+\frac {\ln \left (x+\frac {a^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{1/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{27\,a^{2/3}\,b^{7/3}}-\frac {\ln \left (x-\frac {a^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,b^{1/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{27\,a^{2/3}\,b^{7/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(a + b*x^3)^3,x)

[Out]

(2*log(x + a^(1/3)/b^(1/3)))/(27*a^(2/3)*b^(7/3)) - ((7*x^4)/(18*b) + (2*a*x)/(9*b^2))/(a^2 + b^2*x^6 + 2*a*b*
x^3) + (log(x + (a^(1/3)*(3^(1/2)*1i - 1))/(2*b^(1/3)))*(3^(1/2)*1i - 1))/(27*a^(2/3)*b^(7/3)) - (log(x - (a^(
1/3)*(3^(1/2)*1i + 1))/(2*b^(1/3)))*(3^(1/2)*1i + 1))/(27*a^(2/3)*b^(7/3))

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