Integrand size = 13, antiderivative size = 193 \[ \int \frac {x^5}{a+b x^8} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{3/4}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} b^{3/4}}+\frac {\log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} \sqrt [4]{a} b^{3/4}}-\frac {\log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {b} x^4\right )}{8 \sqrt {2} \sqrt [4]{a} b^{3/4}} \]
1/8*arctan(-1+b^(1/4)*x^2*2^(1/2)/a^(1/4))/a^(1/4)/b^(3/4)*2^(1/2)+1/8*arc tan(1+b^(1/4)*x^2*2^(1/2)/a^(1/4))/a^(1/4)/b^(3/4)*2^(1/2)+1/16*ln(-a^(1/4 )*b^(1/4)*x^2*2^(1/2)+a^(1/2)+x^4*b^(1/2))/a^(1/4)/b^(3/4)*2^(1/2)-1/16*ln (a^(1/4)*b^(1/4)*x^2*2^(1/2)+a^(1/2)+x^4*b^(1/2))/a^(1/4)/b^(3/4)*2^(1/2)
Time = 0.08 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.45 \[ \int \frac {x^5}{a+b x^8} \, dx=-\frac {2 \arctan \left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right )+2 \arctan \left (\cot \left (\frac {\pi }{8}\right )+\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right )-2 \arctan \left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac {\pi }{8}\right )\right )+2 \arctan \left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac {\pi }{8}\right )\right )-\log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )-\log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right )}{8 \sqrt {2} \sqrt [4]{a} b^{3/4}} \]
-1/8*(2*ArcTan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(1/8)] + 2*ArcTan[Cot[P i/8] + (b^(1/8)*x*Csc[Pi/8])/a^(1/8)] - 2*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^( 1/8) - Tan[Pi/8]] + 2*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]] - Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] - Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] + Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]] + Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b ^(1/8)*x*Sin[Pi/8]])/(Sqrt[2]*a^(1/4)*b^(3/4))
Time = 0.41 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {807, 826, 1476, 1082, 217, 1479, 25, 27, 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.
| \(\displaystyle \int \frac {x^5}{a+b x^8} \, dx\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {1}{2} \int \frac {x^4}{b x^8+a}dx^2\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {\sqrt {b} x^4+\sqrt {a}}{b x^8+a}dx^2}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^4}{b x^8+a}dx^2}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\int \frac {1}{x^4-\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx^2}{2 \sqrt {b}}+\frac {\int \frac {1}{x^4+\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx^2}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^4}{b x^8+a}dx^2}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\int \frac {1}{-x^4-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x^4-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^4}{b x^8+a}dx^2}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x^4}{b x^8+a}dx^2}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x^2}{\sqrt [4]{b} \left (x^4-\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx^2}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x^2+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x^4+\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx^2}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x^2}{\sqrt [4]{b} \left (x^4-\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx^2}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x^2+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x^4+\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx^2}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x^2}{x^4-\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx^2}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} x^2+\sqrt [4]{a}}{x^4+\frac {\sqrt {2} \sqrt [4]{a} x^2}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx^2}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {b}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {a}+\sqrt {b} x^4\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt {a}+\sqrt {b} x^4\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )\) |
((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2*S qrt[b]) - (-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4]/( Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqr t[b]*x^4]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[b]))/2
3.15.53.3.1 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[(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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, 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_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 14.88 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.15
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b^{3} a +1\right )}{\sum }\textit {\_R} \ln \left (-b \,x^{2} \textit {\_R} +1\right )\right )}{8}\) | \(29\) |
| default | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{4}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x^{2} \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{4}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x^{2} \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(110\) |
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.68 \[ \int \frac {x^5}{a+b x^8} \, dx=\frac {1}{8} \, \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (a b^{2} \left (-\frac {1}{a b^{3}}\right )^{\frac {3}{4}} + x^{2}\right ) - \frac {1}{8} i \, \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (i \, a b^{2} \left (-\frac {1}{a b^{3}}\right )^{\frac {3}{4}} + x^{2}\right ) + \frac {1}{8} i \, \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (-i \, a b^{2} \left (-\frac {1}{a b^{3}}\right )^{\frac {3}{4}} + x^{2}\right ) - \frac {1}{8} \, \left (-\frac {1}{a b^{3}}\right )^{\frac {1}{4}} \log \left (-a b^{2} \left (-\frac {1}{a b^{3}}\right )^{\frac {3}{4}} + x^{2}\right ) \]
1/8*(-1/(a*b^3))^(1/4)*log(a*b^2*(-1/(a*b^3))^(3/4) + x^2) - 1/8*I*(-1/(a* b^3))^(1/4)*log(I*a*b^2*(-1/(a*b^3))^(3/4) + x^2) + 1/8*I*(-1/(a*b^3))^(1/ 4)*log(-I*a*b^2*(-1/(a*b^3))^(3/4) + x^2) - 1/8*(-1/(a*b^3))^(1/4)*log(-a* b^2*(-1/(a*b^3))^(3/4) + x^2)
Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.14 \[ \int \frac {x^5}{a+b x^8} \, dx=\operatorname {RootSum} {\left (4096 t^{4} a b^{3} + 1, \left ( t \mapsto t \log {\left (512 t^{3} a b^{2} + x^{2} \right )} \right )\right )} \]
Time = 0.28 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.92 \[ \int \frac {x^5}{a+b x^8} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{8 \, \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{8 \, \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} x^{4} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x^{2} + \sqrt {a}\right )}{16 \, a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} x^{4} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x^{2} + \sqrt {a}\right )}{16 \, a^{\frac {1}{4}} b^{\frac {3}{4}}} \]
1/8*sqrt(2)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4))/s qrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 1/8*sqrt(2)*arctan (1/2*sqrt(2)*(2*sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b )))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - 1/16*sqrt(2)*log(sqrt(b)*x^4 + sqrt( 2)*a^(1/4)*b^(1/4)*x^2 + sqrt(a))/(a^(1/4)*b^(3/4)) + 1/16*sqrt(2)*log(sqr t(b)*x^4 - sqrt(2)*a^(1/4)*b^(1/4)*x^2 + sqrt(a))/(a^(1/4)*b^(3/4))
Time = 0.28 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.03 \[ \int \frac {x^5}{a+b x^8} \, dx=\frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} x^{4} \arctan \left (\frac {\sqrt {2} {\left (2 \, x^{2} + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a b} + \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} x^{4} \arctan \left (\frac {\sqrt {2} {\left (2 \, x^{2} - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a b} + \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} x^{4} \log \left (x^{4} + \sqrt {2} x^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{16 \, a b} - \frac {\sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} x^{4} \log \left (x^{4} - \sqrt {2} x^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{16 \, a b} \]
1/8*sqrt(2)*(a*b^3)^(1/4)*x^4*arctan(1/2*sqrt(2)*(2*x^2 + sqrt(2)*(a/b)^(1 /4))/(a/b)^(1/4))/(a*b) + 1/8*sqrt(2)*(a*b^3)^(1/4)*x^4*arctan(1/2*sqrt(2) *(2*x^2 - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a*b) + 1/16*sqrt(2)*(a*b^3)^( 1/4)*x^4*log(x^4 + sqrt(2)*x^2*(a/b)^(1/4) + sqrt(a/b))/(a*b) - 1/16*sqrt( 2)*(a*b^3)^(1/4)*x^4*log(x^4 - sqrt(2)*x^2*(a/b)^(1/4) + sqrt(a/b))/(a*b)
Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.20 \[ \int \frac {x^5}{a+b x^8} \, dx=\frac {\mathrm {atan}\left (\frac {b^{1/4}\,x^2}{{\left (-a\right )}^{1/4}}\right )-\mathrm {atanh}\left (\frac {b^{1/4}\,x^2}{{\left (-a\right )}^{1/4}}\right )}{4\,{\left (-a\right )}^{1/4}\,b^{3/4}} \]