Integrand size = 13, antiderivative size = 215 \[ \int \frac {x^4}{a+b x^6} \, dx=\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 \sqrt [6]{a} b^{5/6}}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 \sqrt [6]{a} b^{5/6}}+\frac {\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} \sqrt [6]{a} b^{5/6}}-\frac {\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} \sqrt [6]{a} b^{5/6}} \]
1/3*arctan(b^(1/6)*x/a^(1/6))/a^(1/6)/b^(5/6)-1/6*arctan((-2*b^(1/6)*x+a^( 1/6)*3^(1/2))/a^(1/6))/a^(1/6)/b^(5/6)+1/6*arctan((2*b^(1/6)*x+a^(1/6)*3^( 1/2))/a^(1/6))/a^(1/6)/b^(5/6)+1/12*ln(a^(1/3)+b^(1/3)*x^2-a^(1/6)*b^(1/6) *x*3^(1/2))/a^(1/6)/b^(5/6)*3^(1/2)-1/12*ln(a^(1/3)+b^(1/3)*x^2+a^(1/6)*b^ (1/6)*x*3^(1/2))/a^(1/6)/b^(5/6)*3^(1/2)
Time = 0.02 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.72 \[ \int \frac {x^4}{a+b x^6} \, dx=\frac {4 \arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+\sqrt {3} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )-\sqrt {3} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{12 \sqrt [6]{a} b^{5/6}} \]
(4*ArcTan[(b^(1/6)*x)/a^(1/6)] - 2*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)] + 2*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)] + Sqrt[3]*Log[a^(1/3) - Sqrt[ 3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2] - Sqrt[3]*Log[a^(1/3) + Sqrt[3]*a^(1/6 )*b^(1/6)*x + b^(1/3)*x^2])/(12*a^(1/6)*b^(5/6))
Time = 0.42 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {824, 27, 218, 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.
| \(\displaystyle \int \frac {x^4}{a+b x^6} \, dx\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt [3]{b} x^2+\sqrt [3]{a}}dx}{3 b^{2/3}}+\frac {\int -\frac {\sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} x}{2 \left (\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}\right )}dx}{3 \sqrt [6]{a} b^{2/3}}+\frac {\int -\frac {\sqrt {3} \sqrt [6]{b} x+\sqrt [6]{a}}{2 \left (\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}\right )}dx}{3 \sqrt [6]{a} b^{2/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt [3]{b} x^2+\sqrt [3]{a}}dx}{3 b^{2/3}}-\frac {\int \frac {\sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 \sqrt [6]{a} b^{2/3}}-\frac {\int \frac {\sqrt {3} \sqrt [6]{b} x+\sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 \sqrt [6]{a} b^{2/3}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\int \frac {\sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 \sqrt [6]{a} b^{2/3}}-\frac {\int \frac {\sqrt {3} \sqrt [6]{b} x+\sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x\right )}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x\right )}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx-\frac {\int \frac {1}{-\left (1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )^2-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )}{\sqrt {3} \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx+\frac {\int \frac {1}{-\left (\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}+1\right )^2-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}+1\right )}{\sqrt {3} \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx+\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{2 \sqrt [6]{b}}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\) |
ArcTan[(b^(1/6)*x)/a^(1/6)]/(3*a^(1/6)*b^(5/6)) - (ArcTan[Sqrt[3]*(1 - (2* b^(1/6)*x)/(Sqrt[3]*a^(1/6)))]/b^(1/6) - (Sqrt[3]*Log[a^(1/3) - Sqrt[3]*a^ (1/6)*b^(1/6)*x + b^(1/3)*x^2])/(2*b^(1/6)))/(6*a^(1/6)*b^(2/3)) - (-(ArcT an[Sqrt[3]*(1 + (2*b^(1/6)*x)/(Sqrt[3]*a^(1/6)))]/b^(1/6)) + (Sqrt[3]*Log[ a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(2*b^(1/6)))/(6*a^(1/6 )*b^(2/3))
3.14.22.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] ; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m)) Int[1/(r^2 + s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGt Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
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 = 4.45 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.13
| method | result | size |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}}{6 b}\) | \(27\) |
| default | \(\frac {\arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}\) | \(159\) |
Time = 0.29 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.15 \[ \int \frac {x^4}{a+b x^6} \, dx=-\frac {1}{12} \, {\left (\sqrt {-3} - 1\right )} \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} a b^{4} + a b^{4}\right )} \left (-\frac {1}{a b^{5}}\right )^{\frac {5}{6}} + x\right ) + \frac {1}{12} \, {\left (\sqrt {-3} - 1\right )} \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} a b^{4} + a b^{4}\right )} \left (-\frac {1}{a b^{5}}\right )^{\frac {5}{6}} + x\right ) - \frac {1}{12} \, {\left (\sqrt {-3} + 1\right )} \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} a b^{4} - a b^{4}\right )} \left (-\frac {1}{a b^{5}}\right )^{\frac {5}{6}} + x\right ) + \frac {1}{12} \, {\left (\sqrt {-3} + 1\right )} \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} a b^{4} - a b^{4}\right )} \left (-\frac {1}{a b^{5}}\right )^{\frac {5}{6}} + x\right ) + \frac {1}{6} \, \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} \log \left (a b^{4} \left (-\frac {1}{a b^{5}}\right )^{\frac {5}{6}} + x\right ) - \frac {1}{6} \, \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} \log \left (-a b^{4} \left (-\frac {1}{a b^{5}}\right )^{\frac {5}{6}} + x\right ) \]
-1/12*(sqrt(-3) - 1)*(-1/(a*b^5))^(1/6)*log(1/2*(sqrt(-3)*a*b^4 + a*b^4)*( -1/(a*b^5))^(5/6) + x) + 1/12*(sqrt(-3) - 1)*(-1/(a*b^5))^(1/6)*log(-1/2*( sqrt(-3)*a*b^4 + a*b^4)*(-1/(a*b^5))^(5/6) + x) - 1/12*(sqrt(-3) + 1)*(-1/ (a*b^5))^(1/6)*log(1/2*(sqrt(-3)*a*b^4 - a*b^4)*(-1/(a*b^5))^(5/6) + x) + 1/12*(sqrt(-3) + 1)*(-1/(a*b^5))^(1/6)*log(-1/2*(sqrt(-3)*a*b^4 - a*b^4)*( -1/(a*b^5))^(5/6) + x) + 1/6*(-1/(a*b^5))^(1/6)*log(a*b^4*(-1/(a*b^5))^(5/ 6) + x) - 1/6*(-1/(a*b^5))^(1/6)*log(-a*b^4*(-1/(a*b^5))^(5/6) + x)
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.12 \[ \int \frac {x^4}{a+b x^6} \, dx=\operatorname {RootSum} {\left (46656 t^{6} a b^{5} + 1, \left ( t \mapsto t \log {\left (7776 t^{5} a b^{4} + x \right )} \right )\right )} \]
Time = 0.29 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.86 \[ \int \frac {x^4}{a+b x^6} \, dx=-\frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{12 \, a^{\frac {1}{6}} b^{\frac {5}{6}}} + \frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{12 \, a^{\frac {1}{6}} b^{\frac {5}{6}}} + \frac {\arctan \left (\frac {b^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{3 \, b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {\arctan \left (\frac {2 \, b^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{6 \, b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {\arctan \left (\frac {2 \, b^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{6 \, b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} \]
-1/12*sqrt(3)*log(b^(1/3)*x^2 + sqrt(3)*a^(1/6)*b^(1/6)*x + a^(1/3))/(a^(1 /6)*b^(5/6)) + 1/12*sqrt(3)*log(b^(1/3)*x^2 - sqrt(3)*a^(1/6)*b^(1/6)*x + a^(1/3))/(a^(1/6)*b^(5/6)) + 1/3*arctan(b^(1/3)*x/sqrt(a^(1/3)*b^(1/3)))/( b^(2/3)*sqrt(a^(1/3)*b^(1/3))) + 1/6*arctan((2*b^(1/3)*x + sqrt(3)*a^(1/6) *b^(1/6))/sqrt(a^(1/3)*b^(1/3)))/(b^(2/3)*sqrt(a^(1/3)*b^(1/3))) + 1/6*arc tan((2*b^(1/3)*x - sqrt(3)*a^(1/6)*b^(1/6))/sqrt(a^(1/3)*b^(1/3)))/(b^(2/3 )*sqrt(a^(1/3)*b^(1/3)))
Time = 0.44 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.88 \[ \int \frac {x^4}{a+b x^6} \, dx=-\frac {\sqrt {3} \left (a b^{5}\right )^{\frac {5}{6}} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 \, a b^{5}} + \frac {\sqrt {3} \left (a b^{5}\right )^{\frac {5}{6}} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 \, a b^{5}} + \frac {\left (a b^{5}\right )^{\frac {5}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 \, a b^{5}} + \frac {\left (a b^{5}\right )^{\frac {5}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 \, a b^{5}} + \frac {\left (a b^{5}\right )^{\frac {5}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 \, a b^{5}} \]
-1/12*sqrt(3)*(a*b^5)^(5/6)*log(x^2 + sqrt(3)*x*(a/b)^(1/6) + (a/b)^(1/3)) /(a*b^5) + 1/12*sqrt(3)*(a*b^5)^(5/6)*log(x^2 - sqrt(3)*x*(a/b)^(1/6) + (a /b)^(1/3))/(a*b^5) + 1/6*(a*b^5)^(5/6)*arctan((2*x + sqrt(3)*(a/b)^(1/6))/ (a/b)^(1/6))/(a*b^5) + 1/6*(a*b^5)^(5/6)*arctan((2*x - sqrt(3)*(a/b)^(1/6) )/(a/b)^(1/6))/(a*b^5) + 1/3*(a*b^5)^(5/6)*arctan(x/(a/b)^(1/6))/(a*b^5)
Time = 0.19 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.62 \[ \int \frac {x^4}{a+b x^6} \, dx=-\frac {\mathrm {atanh}\left (\frac {b^{1/6}\,x}{{\left (-a\right )}^{1/6}}\right )}{3\,{\left (-a\right )}^{1/6}\,b^{5/6}}-\frac {\mathrm {atanh}\left (\frac {2\,{\left (-a\right )}^{5/2}\,b^{3/2}\,x}{{\left (-a\right )}^{8/3}\,b^{4/3}-\sqrt {3}\,{\left (-a\right )}^{8/3}\,b^{4/3}\,1{}\mathrm {i}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-a\right )}^{1/6}\,b^{5/6}}+\frac {\mathrm {atanh}\left (\frac {2\,{\left (-a\right )}^{5/2}\,b^{3/2}\,x}{{\left (-a\right )}^{8/3}\,b^{4/3}+\sqrt {3}\,{\left (-a\right )}^{8/3}\,b^{4/3}\,1{}\mathrm {i}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-a\right )}^{1/6}\,b^{5/6}} \]
(atanh((2*(-a)^(5/2)*b^(3/2)*x)/((-a)^(8/3)*b^(4/3) + 3^(1/2)*(-a)^(8/3)*b ^(4/3)*1i))*(3^(1/2)*1i - 1))/(6*(-a)^(1/6)*b^(5/6)) - (atanh((2*(-a)^(5/2 )*b^(3/2)*x)/((-a)^(8/3)*b^(4/3) - 3^(1/2)*(-a)^(8/3)*b^(4/3)*1i))*(3^(1/2 )*1i + 1))/(6*(-a)^(1/6)*b^(5/6)) - atanh((b^(1/6)*x)/(-a)^(1/6))/(3*(-a)^ (1/6)*b^(5/6))