Integrand size = 13, antiderivative size = 142 \[ \int \frac {x^3}{\left (a+b x^6\right )^2} \, dx=\frac {x^4}{6 a \left (a+b x^6\right )}-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{6 \sqrt {3} a^{4/3} b^{2/3}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{18 a^{4/3} b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{36 a^{4/3} b^{2/3}} \]
1/6*x^4/a/(b*x^6+a)-1/18*ln(a^(1/3)+b^(1/3)*x^2)/a^(4/3)/b^(2/3)+1/36*ln(a ^(2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)/a^(4/3)/b^(2/3)-1/18*arctan(1/3*(a ^(1/3)-2*b^(1/3)*x^2)/a^(1/3)*3^(1/2))/a^(4/3)/b^(2/3)*3^(1/2)
Time = 0.11 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.37 \[ \int \frac {x^3}{\left (a+b x^6\right )^2} \, dx=\frac {\frac {6 \sqrt [3]{a} x^4}{a+b x^6}-\frac {2 \sqrt {3} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{b^{2/3}}-\frac {2 \sqrt {3} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{b^{2/3}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{b^{2/3}}+\frac {\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{b^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{b^{2/3}}}{36 a^{4/3}} \]
((6*a^(1/3)*x^4)/(a + b*x^6) - (2*Sqrt[3]*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a ^(1/6)])/b^(2/3) - (2*Sqrt[3]*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)])/b^( 2/3) - (2*Log[a^(1/3) + b^(1/3)*x^2])/b^(2/3) + Log[a^(1/3) - Sqrt[3]*a^(1 /6)*b^(1/6)*x + b^(1/3)*x^2]/b^(2/3) + Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/ 6)*x + b^(1/3)*x^2]/b^(2/3))/(36*a^(4/3))
Time = 0.31 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {807, 819, 821, 16, 1142, 25, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^3}{\left (a+b x^6\right )^2} \, dx\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {1}{2} \int \frac {x^2}{\left (b x^6+a\right )^2}dx^2\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {x^2}{b x^6+a}dx^2}{3 a}+\frac {x^4}{3 a \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\int \frac {\sqrt [3]{b} x^2+\sqrt [3]{a}}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{b} x^2+\sqrt [3]{a}}dx^2}{3 \sqrt [3]{a} \sqrt [3]{b}}}{3 a}+\frac {x^4}{3 a \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\int \frac {\sqrt [3]{b} x^2+\sqrt [3]{a}}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^4}{3 a \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x^2\right )}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^4}{3 a \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x^2\right )}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^4}{3 a \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^4}{3 a \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\frac {3 \int \frac {1}{-x^4-3}d\left (1-\frac {2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^4}{3 a \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^4}{3 a \left (a+b x^6\right )}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^4}{3 a \left (a+b x^6\right )}\right )\) |
(x^4/(3*a*(a + b*x^6)) + (-1/3*Log[a^(1/3) + b^(1/3)*x^2]/(a^(1/3)*b^(2/3) ) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x^2)/a^(1/3))/Sqrt[3]])/b^(1/3)) + Log[a^(2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4]/(2*b^(1/3)))/(3*a^(1/3)*b ^(1/3)))/(3*a))/2
3.14.36.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(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]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.58 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.35
| method | result | size |
| risch | \(\frac {x^{4}}{6 a \left (b \,x^{6}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b^{2} a^{4}+1\right )}{\sum }\textit {\_R} \ln \left (-a b \,x^{2} \textit {\_R} +1\right )\right )}{18}\) | \(50\) |
| default | \(\frac {x^{4}}{6 a \left (b \,x^{6}+a \right )}+\frac {-\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{4}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{2}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}}{6 a}\) | \(120\) |
Time = 0.28 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.93 \[ \int \frac {x^3}{\left (a+b x^6\right )^2} \, dx=\left [\frac {6 \, a b^{2} x^{4} + 3 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{6} + a^{2} b\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{6} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{2} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{4} + a b x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b x^{6} + a}\right ) + {\left (b x^{6} + a\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{4} + \left (-a b^{2}\right )^{\frac {1}{3}} b x^{2} + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (b x^{6} + a\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x^{2} - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{36 \, {\left (a^{2} b^{3} x^{6} + a^{3} b^{2}\right )}}, \frac {6 \, a b^{2} x^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{6} + a^{2} b\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) + {\left (b x^{6} + a\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{4} + \left (-a b^{2}\right )^{\frac {1}{3}} b x^{2} + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (b x^{6} + a\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x^{2} - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{36 \, {\left (a^{2} b^{3} x^{6} + a^{3} b^{2}\right )}}\right ] \]
[1/36*(6*a*b^2*x^4 + 3*sqrt(1/3)*(a*b^2*x^6 + a^2*b)*sqrt((-a*b^2)^(1/3)/a )*log((2*b^2*x^6 - 3*(-a*b^2)^(2/3)*x^2 - a*b + 3*sqrt(1/3)*(2*(-a*b^2)^(2 /3)*x^4 + a*b*x^2 + (-a*b^2)^(1/3)*a)*sqrt((-a*b^2)^(1/3)/a))/(b*x^6 + a)) + (b*x^6 + a)*(-a*b^2)^(2/3)*log(b^2*x^4 + (-a*b^2)^(1/3)*b*x^2 + (-a*b^2 )^(2/3)) - 2*(b*x^6 + a)*(-a*b^2)^(2/3)*log(b*x^2 - (-a*b^2)^(1/3)))/(a^2* b^3*x^6 + a^3*b^2), 1/36*(6*a*b^2*x^4 + 6*sqrt(1/3)*(a*b^2*x^6 + a^2*b)*sq rt(-(-a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2*b*x^2 + (-a*b^2)^(1/3))*sqrt(-(- a*b^2)^(1/3)/a)/b) + (b*x^6 + a)*(-a*b^2)^(2/3)*log(b^2*x^4 + (-a*b^2)^(1/ 3)*b*x^2 + (-a*b^2)^(2/3)) - 2*(b*x^6 + a)*(-a*b^2)^(2/3)*log(b*x^2 - (-a* b^2)^(1/3)))/(a^2*b^3*x^6 + a^3*b^2)]
Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.32 \[ \int \frac {x^3}{\left (a+b x^6\right )^2} \, dx=\frac {x^{4}}{6 a^{2} + 6 a b x^{6}} + \operatorname {RootSum} {\left (5832 t^{3} a^{4} b^{2} + 1, \left ( t \mapsto t \log {\left (324 t^{2} a^{3} b + x^{2} \right )} \right )\right )} \]
x**4/(6*a**2 + 6*a*b*x**6) + RootSum(5832*_t**3*a**4*b**2 + 1, Lambda(_t, _t*log(324*_t**2*a**3*b + x**2)))
Time = 0.28 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.92 \[ \int \frac {x^3}{\left (a+b x^6\right )^2} \, dx=\frac {x^{4}}{6 \, {\left (a b x^{6} + a^{2}\right )}} + \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{18 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{4} - x^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{36 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\log \left (x^{2} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{18 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
1/6*x^4/(a*b*x^6 + a^2) + 1/18*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 - (a/b)^( 1/3))/(a/b)^(1/3))/(a*b*(a/b)^(1/3)) + 1/36*log(x^4 - x^2*(a/b)^(1/3) + (a /b)^(2/3))/(a*b*(a/b)^(1/3)) - 1/18*log(x^2 + (a/b)^(1/3))/(a*b*(a/b)^(1/3 ))
Time = 0.28 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.95 \[ \int \frac {x^3}{\left (a+b x^6\right )^2} \, dx=\frac {x^{4}}{6 \, {\left (b x^{6} + a\right )} a} - \frac {\left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{2} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{18 \, a^{2}} - \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{18 \, a^{2} b^{2}} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{4} + x^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{36 \, a^{2} b^{2}} \]
1/6*x^4/((b*x^6 + a)*a) - 1/18*(-a/b)^(2/3)*log(abs(x^2 - (-a/b)^(1/3)))/a ^2 - 1/18*sqrt(3)*(-a*b^2)^(2/3)*arctan(1/3*sqrt(3)*(2*x^2 + (-a/b)^(1/3)) /(-a/b)^(1/3))/(a^2*b^2) + 1/36*(-a*b^2)^(2/3)*log(x^4 + x^2*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b^2)
Time = 5.81 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.03 \[ \int \frac {x^3}{\left (a+b x^6\right )^2} \, dx=\frac {x^4}{6\,a\,\left (b\,x^6+a\right )}+\frac {{\left (-1\right )}^{1/3}\,\ln \left (\frac {b^2}{81\,a^3}-\frac {{\left (-1\right )}^{1/3}\,b^{7/3}\,x^2}{81\,a^{10/3}}\right )}{18\,a^{4/3}\,b^{2/3}}-\frac {{\left (-1\right )}^{1/3}\,\ln \left ({\left (-1\right )}^{2/3}\,a^{1/3}-2\,b^{1/3}\,x^2+{\left (-1\right )}^{1/6}\,\sqrt {3}\,a^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{18\,a^{4/3}\,b^{2/3}}+\frac {{\left (-1\right )}^{1/3}\,\ln \left (2\,b^{1/3}\,x^2-{\left (-1\right )}^{2/3}\,a^{1/3}+{\left (-1\right )}^{1/6}\,\sqrt {3}\,a^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{18\,a^{4/3}\,b^{2/3}} \]
x^4/(6*a*(a + b*x^6)) + ((-1)^(1/3)*log(b^2/(81*a^3) - ((-1)^(1/3)*b^(7/3) *x^2)/(81*a^(10/3))))/(18*a^(4/3)*b^(2/3)) - ((-1)^(1/3)*log((-1)^(2/3)*a^ (1/3) - 2*b^(1/3)*x^2 + (-1)^(1/6)*3^(1/2)*a^(1/3))*((3^(1/2)*1i)/2 + 1/2) )/(18*a^(4/3)*b^(2/3)) + ((-1)^(1/3)*log(2*b^(1/3)*x^2 - (-1)^(2/3)*a^(1/3 ) + (-1)^(1/6)*3^(1/2)*a^(1/3))*((3^(1/2)*1i)/2 - 1/2))/(18*a^(4/3)*b^(2/3 ))