3.4.51 \(\int \frac {1}{(a+b x^3)^3} \, dx\) [351]

Optimal. Leaf size=151 \[ \frac {x}{6 a \left (a+b x^3\right )^2}+\frac {5 x}{18 a^2 \left (a+b x^3\right )}-\frac {5 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} \sqrt [3]{b}}+\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} \sqrt [3]{b}} \]

[Out]

1/6*x/a/(b*x^3+a)^2+5/18*x/a^2/(b*x^3+a)+5/27*ln(a^(1/3)+b^(1/3)*x)/a^(8/3)/b^(1/3)-5/54*ln(a^(2/3)-a^(1/3)*b^
(1/3)*x+b^(2/3)*x^2)/a^(8/3)/b^(1/3)-5/27*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(8/3)/b^(1/3)*3^
(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {205, 206, 31, 648, 631, 210, 642} \begin {gather*} -\frac {5 \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} \sqrt [3]{b}}-\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} \sqrt [3]{b}}+\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac {5 x}{18 a^2 \left (a+b x^3\right )}+\frac {x}{6 a \left (a+b x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(-3),x]

[Out]

x/(6*a*(a + b*x^3)^2) + (5*x)/(18*a^2*(a + b*x^3)) - (5*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*
Sqrt[3]*a^(8/3)*b^(1/3)) + (5*Log[a^(1/3) + b^(1/3)*x])/(27*a^(8/3)*b^(1/3)) - (5*Log[a^(2/3) - a^(1/3)*b^(1/3
)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^(1/3))

Rule 31

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

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b x^3\right )^3} \, dx &=\frac {x}{6 a \left (a+b x^3\right )^2}+\frac {5 \int \frac {1}{\left (a+b x^3\right )^2} \, dx}{6 a}\\ &=\frac {x}{6 a \left (a+b x^3\right )^2}+\frac {5 x}{18 a^2 \left (a+b x^3\right )}+\frac {5 \int \frac {1}{a+b x^3} \, dx}{9 a^2}\\ &=\frac {x}{6 a \left (a+b x^3\right )^2}+\frac {5 x}{18 a^2 \left (a+b x^3\right )}+\frac {5 \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{8/3}}+\frac {5 \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^{8/3}}\\ &=\frac {x}{6 a \left (a+b x^3\right )^2}+\frac {5 x}{18 a^2 \left (a+b x^3\right )}+\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac {5 \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{7/3}}-\frac {5 \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}{54 a^{8/3} \sqrt [3]{b}}\\ &=\frac {x}{6 a \left (a+b x^3\right )^2}+\frac {5 x}{18 a^2 \left (a+b x^3\right )}+\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} \sqrt [3]{b}}+\frac {5 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{8/3} \sqrt [3]{b}}\\ &=\frac {x}{6 a \left (a+b x^3\right )^2}+\frac {5 x}{18 a^2 \left (a+b x^3\right )}-\frac {5 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} \sqrt [3]{b}}+\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} \sqrt [3]{b}}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 135, normalized size = 0.89 \begin {gather*} \frac {\frac {9 a^{5/3} x}{\left (a+b x^3\right )^2}+\frac {15 a^{2/3} x}{a+b x^3}-\frac {10 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}}{54 a^{8/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^(-3),x]

[Out]

((9*a^(5/3)*x)/(a + b*x^3)^2 + (15*a^(2/3)*x)/(a + b*x^3) - (10*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqr
t[3]])/b^(1/3) + (10*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) - (5*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^
(1/3))/(54*a^(8/3))

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Maple [A]
time = 0.13, size = 133, normalized size = 0.88

method	result	size
risch	\(\frac {\frac {5 b \,x^{4}}{18 a^{2}}+\frac {4 x}{9 a}}{\left (b \,x^{3}+a \right )^{2}}+\frac {5 \left (\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{27 a^{2} b}\)	\(57\)
default	\(\frac {x}{6 a \left (b \,x^{3}+a \right )^{2}}+\frac {\frac {5 x}{18 a \left (b \,x^{3}+a \right )}+\frac {5 \left (\frac {2 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{6 a}}{a}\)	\(133\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^3+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/6*x/a/(b*x^3+a)^2+5/6/a*(1/3*x/a/(b*x^3+a)+2/3/a*(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 = 0.49, size = 145, normalized size = 0.96 \begin {gather*} \frac {5 \, b x^{4} + 8 \, a x}{18 \, {\left (a^{2} b^{2} x^{6} + 2 \, a^{3} b x^{3} + a^{4}\right )}} + \frac {5 \, \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 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {5 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {5 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

1/18*(5*b*x^4 + 8*a*x)/(a^2*b^2*x^6 + 2*a^3*b*x^3 + a^4) + 5/27*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))
/(a/b)^(1/3))/(a^2*b*(a/b)^(2/3)) - 5/54*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*b*(a/b)^(2/3)) + 5/27*log
(x + (a/b)^(1/3))/(a^2*b*(a/b)^(2/3))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (110) = 220\).
time = 0.36, size = 499, normalized size = 3.30 \begin {gather*} \left [\frac {15 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x + 15 \, \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 ) - 5 \, {\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 ) + 10 \, {\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^{4} b^{3} x^{6} + 2 \, a^{5} b^{2} x^{3} + a^{6} b\right )}}, \frac {15 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x + 30 \, \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 ) - 5 \, {\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 ) + 10 \, {\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^{4} b^{3} x^{6} + 2 \, a^{5} b^{2} x^{3} + a^{6} b\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

[1/54*(15*a^2*b^2*x^4 + 24*a^3*b*x + 15*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)) - 5*(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) + 10*(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^4*b^3*x^6 + 2*a^
5*b^2*x^3 + a^6*b), 1/54*(15*a^2*b^2*x^4 + 24*a^3*b*x + 30*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)*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) - 5*(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) + 10*(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^4*b^3*x^6 + 2*a^5*b^2*x^3 + a^6*b)]

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Sympy [A]
time = 0.18, size = 63, normalized size = 0.42 \begin {gather*} \frac {8 a x + 5 b x^{4}}{18 a^{4} + 36 a^{3} b x^{3} + 18 a^{2} b^{2} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} a^{8} b - 125, \left ( t \mapsto t \log {\left (\frac {27 t a^{3}}{5} + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**3+a)**3,x)

[Out]

(8*a*x + 5*b*x**4)/(18*a**4 + 36*a**3*b*x**3 + 18*a**2*b**2*x**6) + RootSum(19683*_t**3*a**8*b - 125, Lambda(_
t, _t*log(27*_t*a**3/5 + x)))

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Giac [A]
time = 2.25, size = 137, normalized size = 0.91 \begin {gather*} -\frac {5 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{3}} + \frac {5 \, \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^{3} b} + \frac {5 \, \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 )}{54 \, a^{3} b} + \frac {5 \, b x^{4} + 8 \, a x}{18 \, {\left (b x^{3} + a\right )}^{2} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^3,x, algorithm="giac")

[Out]

-5/27*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^3 + 5/27*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*(2*x + (-a/
b)^(1/3))/(-a/b)^(1/3))/(a^3*b) + 5/54*(-a*b^2)^(1/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b) + 1/18*
(5*b*x^4 + 8*a*x)/((b*x^3 + a)^2*a^2)

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Mupad [B]
time = 1.11, size = 142, normalized size = 0.94 \begin {gather*} \frac {\frac {4\,x}{9\,a}+\frac {5\,b\,x^4}{18\,a^2}}{a^2+2\,a\,b\,x^3+b^2\,x^6}+\frac {5\,\ln \left (b^{1/3}\,x+a^{1/3}\right )}{27\,a^{8/3}\,b^{1/3}}+\frac {\ln \left (\frac {5\,b^2\,x}{3\,a^2}+\frac {b^{5/3}\,\left (-5+\sqrt {3}\,5{}\mathrm {i}\right )}{6\,a^{5/3}}\right )\,\left (-5+\sqrt {3}\,5{}\mathrm {i}\right )}{54\,a^{8/3}\,b^{1/3}}-\frac {\ln \left (\frac {5\,b^2\,x}{3\,a^2}-\frac {b^{5/3}\,\left (5+\sqrt {3}\,5{}\mathrm {i}\right )}{6\,a^{5/3}}\right )\,\left (5+\sqrt {3}\,5{}\mathrm {i}\right )}{54\,a^{8/3}\,b^{1/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x^3)^3,x)

[Out]

((4*x)/(9*a) + (5*b*x^4)/(18*a^2))/(a^2 + b^2*x^6 + 2*a*b*x^3) + (5*log(b^(1/3)*x + a^(1/3)))/(27*a^(8/3)*b^(1
/3)) + (log((5*b^2*x)/(3*a^2) + (b^(5/3)*(3^(1/2)*5i - 5))/(6*a^(5/3)))*(3^(1/2)*5i - 5))/(54*a^(8/3)*b^(1/3))
- (log((5*b^2*x)/(3*a^2) - (b^(5/3)*(3^(1/2)*5i + 5))/(6*a^(5/3)))*(3^(1/2)*5i + 5))/(54*a^(8/3)*b^(1/3))

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