\(\int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} (-b-a x^4+2 x^8)} \, dx\) [1382]

Optimal result

Integrand size = 42, antiderivative size = 99 \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\frac {1}{4} \text {RootSum}\left [a^4+a^3 b-2 b^3-4 a^3 \text {$\#$1}^4-2 a^2 b \text {$\#$1}^4+6 a^2 \text {$\#$1}^8+a b \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log (x)+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1161\) vs. \(2(99)=198\).

Time = 1.86 (sec) , antiderivative size = 1161, normalized size of antiderivative = 11.73, number of steps used = 22, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2081, 6847, 6860, 1443, 385, 218, 214, 211} \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \arctan \left (\frac {\sqrt [4]{a \sqrt {a-\sqrt {a^2+8 b}}-2 b} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a \sqrt {a-\sqrt {a^2+8 b}}-2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \arctan \left (\frac {\sqrt [4]{\sqrt {a-\sqrt {a^2+8 b}} a+2 b} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{\sqrt {a-\sqrt {a^2+8 b}} a+2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \arctan \left (\frac {\sqrt [4]{a \sqrt {a+\sqrt {a^2+8 b}}-2 b} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a \sqrt {a+\sqrt {a^2+8 b}}-2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \arctan \left (\frac {\sqrt [4]{\sqrt {a+\sqrt {a^2+8 b}} a+2 b} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{\sqrt {a+\sqrt {a^2+8 b}} a+2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \text {arctanh}\left (\frac {\sqrt [4]{a \sqrt {a-\sqrt {a^2+8 b}}-2 b} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a \sqrt {a-\sqrt {a^2+8 b}}-2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {a-\sqrt {a^2+8 b}} a+2 b} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{\sqrt {a-\sqrt {a^2+8 b}} a+2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \text {arctanh}\left (\frac {\sqrt [4]{a \sqrt {a+\sqrt {a^2+8 b}}-2 b} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a \sqrt {a+\sqrt {a^2+8 b}}-2 b} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {a+\sqrt {a^2+8 b}} a+2 b} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{\sqrt {a+\sqrt {a^2+8 b}} a+2 b} \sqrt [4]{a x^4+b x^2}} \]

[In]

Int[(2*b + a*x^4)/((b*x^2 + a*x^4)^(1/4)*(-b - a*x^4 + 2*x^8)),x]

[Out]

-1/2*((a - Sqrt[a^2 + 8*b])^(1/8)*Sqrt[x]*(b + a*x^2)^(1/4)*ArcTan[((-2*b + a*Sqrt[a - Sqrt[a^2 + 8*b]])^(1/4)
*Sqrt[x])/((a - Sqrt[a^2 + 8*b])^(1/8)*(b + a*x^2)^(1/4))])/((-2*b + a*Sqrt[a - Sqrt[a^2 + 8*b]])^(1/4)*(b*x^2
+ a*x^4)^(1/4)) - ((a - Sqrt[a^2 + 8*b])^(1/8)*Sqrt[x]*(b + a*x^2)^(1/4)*ArcTan[((2*b + a*Sqrt[a - Sqrt[a^2 +
8*b]])^(1/4)*Sqrt[x])/((a - Sqrt[a^2 + 8*b])^(1/8)*(b + a*x^2)^(1/4))])/(2*(2*b + a*Sqrt[a - Sqrt[a^2 + 8*b]]
)^(1/4)*(b*x^2 + a*x^4)^(1/4)) - ((a + Sqrt[a^2 + 8*b])^(1/8)*Sqrt[x]*(b + a*x^2)^(1/4)*ArcTan[((-2*b + a*Sqrt
[a + Sqrt[a^2 + 8*b]])^(1/4)*Sqrt[x])/((a + Sqrt[a^2 + 8*b])^(1/8)*(b + a*x^2)^(1/4))])/(2*(-2*b + a*Sqrt[a +
Sqrt[a^2 + 8*b]])^(1/4)*(b*x^2 + a*x^4)^(1/4)) - ((a + Sqrt[a^2 + 8*b])^(1/8)*Sqrt[x]*(b + a*x^2)^(1/4)*ArcTan
[((2*b + a*Sqrt[a + Sqrt[a^2 + 8*b]])^(1/4)*Sqrt[x])/((a + Sqrt[a^2 + 8*b])^(1/8)*(b + a*x^2)^(1/4))])/(2*(2*b
+ a*Sqrt[a + Sqrt[a^2 + 8*b]])^(1/4)*(b*x^2 + a*x^4)^(1/4)) - ((a - Sqrt[a^2 + 8*b])^(1/8)*Sqrt[x]*(b + a*x^2
)^(1/4)*ArcTanh[((-2*b + a*Sqrt[a - Sqrt[a^2 + 8*b]])^(1/4)*Sqrt[x])/((a - Sqrt[a^2 + 8*b])^(1/8)*(b + a*x^2)^
(1/4))])/(2*(-2*b + a*Sqrt[a - Sqrt[a^2 + 8*b]])^(1/4)*(b*x^2 + a*x^4)^(1/4)) - ((a - Sqrt[a^2 + 8*b])^(1/8)*S
qrt[x]*(b + a*x^2)^(1/4)*ArcTanh[((2*b + a*Sqrt[a - Sqrt[a^2 + 8*b]])^(1/4)*Sqrt[x])/((a - Sqrt[a^2 + 8*b])^(1
/8)*(b + a*x^2)^(1/4))])/(2*(2*b + a*Sqrt[a - Sqrt[a^2 + 8*b]])^(1/4)*(b*x^2 + a*x^4)^(1/4)) - ((a + Sqrt[a^2
+ 8*b])^(1/8)*Sqrt[x]*(b + a*x^2)^(1/4)*ArcTanh[((-2*b + a*Sqrt[a + Sqrt[a^2 + 8*b]])^(1/4)*Sqrt[x])/((a + Sqr
t[a^2 + 8*b])^(1/8)*(b + a*x^2)^(1/4))])/(2*(-2*b + a*Sqrt[a + Sqrt[a^2 + 8*b]])^(1/4)*(b*x^2 + a*x^4)^(1/4))
- ((a + Sqrt[a^2 + 8*b])^(1/8)*Sqrt[x]*(b + a*x^2)^(1/4)*ArcTanh[((2*b + a*Sqrt[a + Sqrt[a^2 + 8*b]])^(1/4)*Sq
rt[x])/((a + Sqrt[a^2 + 8*b])^(1/8)*(b + a*x^2)^(1/4))])/(2*(2*b + a*Sqrt[a + Sqrt[a^2 + 8*b]])^(1/4)*(b*x^2 +
a*x^4)^(1/4))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 1443

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{r = Rt[(-a)*c, 2]}, Dist[-c/(2
*r), Int[(d + e*x^n)^q/(r - c*x^n), x], x] - Dist[c/(2*r), Int[(d + e*x^n)^q/(r + c*x^n), x], x]] /; FreeQ[{a,
c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[q]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {2 b+a x^4}{\sqrt {x} \sqrt [4]{b+a x^2} \left (-b-a x^4+2 x^8\right )} \, dx}{\sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {2 b+a x^8}{\sqrt [4]{b+a x^4} \left (-b-a x^8+2 x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \left (\frac {a+\sqrt {a^2+8 b}}{\sqrt [4]{b+a x^4} \left (-a-\sqrt {a^2+8 b}+4 x^8\right )}+\frac {a-\sqrt {a^2+8 b}}{\sqrt [4]{b+a x^4} \left (-a+\sqrt {a^2+8 b}+4 x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (2 \left (a-\sqrt {a^2+8 b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4} \left (-a+\sqrt {a^2+8 b}+4 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}+\frac {\left (2 \left (a+\sqrt {a^2+8 b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4} \left (-a-\sqrt {a^2+8 b}+4 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = -\frac {\left (2 \sqrt {a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {a-\sqrt {a^2+8 b}}-4 x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \sqrt {a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {a-\sqrt {a^2+8 b}}+4 x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \sqrt {a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {a+\sqrt {a^2+8 b}}-4 x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \sqrt {a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {a+\sqrt {a^2+8 b}}+4 x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = -\frac {\left (2 \sqrt {a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {a-\sqrt {a^2+8 b}}-\left (-4 b+2 a \sqrt {a-\sqrt {a^2+8 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \sqrt {a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {a-\sqrt {a^2+8 b}}-\left (4 b+2 a \sqrt {a-\sqrt {a^2+8 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \sqrt {a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {a+\sqrt {a^2+8 b}}-\left (-4 b+2 a \sqrt {a+\sqrt {a^2+8 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (2 \sqrt {a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {a+\sqrt {a^2+8 b}}-\left (4 b+2 a \sqrt {a+\sqrt {a^2+8 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = -\frac {\left (\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a-\sqrt {a^2+8 b}}-\sqrt {-2 b+a \sqrt {a-\sqrt {a^2+8 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a-\sqrt {a^2+8 b}}+\sqrt {-2 b+a \sqrt {a-\sqrt {a^2+8 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a-\sqrt {a^2+8 b}}-\sqrt {2 b+a \sqrt {a-\sqrt {a^2+8 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt [4]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a-\sqrt {a^2+8 b}}+\sqrt {2 b+a \sqrt {a-\sqrt {a^2+8 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+\sqrt {a^2+8 b}}-\sqrt {-2 b+a \sqrt {a+\sqrt {a^2+8 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+\sqrt {a^2+8 b}}+\sqrt {-2 b+a \sqrt {a+\sqrt {a^2+8 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+\sqrt {a^2+8 b}}-\sqrt {2 b+a \sqrt {a+\sqrt {a^2+8 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt [4]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+\sqrt {a^2+8 b}}+\sqrt {2 b+a \sqrt {a+\sqrt {a^2+8 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{b x^2+a x^4}} \\ & = -\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \arctan \left (\frac {\sqrt [4]{-2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{-2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \arctan \left (\frac {\sqrt [4]{2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \arctan \left (\frac {\sqrt [4]{-2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{-2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \arctan \left (\frac {\sqrt [4]{2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{-2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{-2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{2 b+a \sqrt {a-\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{-2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{-2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}}-\frac {\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt {x}}{\sqrt [8]{a+\sqrt {a^2+8 b}} \sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{2 b+a \sqrt {a+\sqrt {a^2+8 b}}} \sqrt [4]{b x^2+a x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 16.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.17 \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\frac {\sqrt [4]{a+\frac {b}{x^2}} x \text {RootSum}\left [a^4+a^3 b-2 b^3-4 a^3 \text {$\#$1}^4-2 a^2 b \text {$\#$1}^4+6 a^2 \text {$\#$1}^8+a b \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt [4]{a+\frac {b}{x^2}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{4 \sqrt [4]{x^2 \left (b+a x^2\right )}} \]

[In]

Integrate[(2*b + a*x^4)/((b*x^2 + a*x^4)^(1/4)*(-b - a*x^4 + 2*x^8)),x]

[Out]

((a + b/x^2)^(1/4)*x*RootSum[a^4 + a^3*b - 2*b^3 - 4*a^3*#1^4 - 2*a^2*b*#1^4 + 6*a^2*#1^8 + a*b*#1^8 - 4*a*#1^
12 + #1^16 & , Log[(a + b/x^2)^(1/4) - #1]/#1 & ])/(4*(x^2*(b + a*x^2))^(1/4))

Maple [N/A] (verified)

Time = 1.72 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-4 a \,\textit {\_Z}^{12}+\left (6 a^{2}+a b \right ) \textit {\_Z}^{8}+\left (-4 a^{3}-2 a^{2} b \right ) \textit {\_Z}^{4}+a^{4}+a^{3} b -2 b^{3}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right )}{4}\)	\(86\)

[In]

int((a*x^4+2*b)/(a*x^4+b*x^2)^(1/4)/(2*x^8-a*x^4-b),x,method=_RETURNVERBOSE)

[Out]

1/4*sum(ln((-_R*x+(x^2*(a*x^2+b))^(1/4))/x)/_R,_R=RootOf(_Z^16-4*a*_Z^12+(6*a^2+a*b)*_Z^8+(-4*a^3-2*a^2*b)*_Z^
4+a^4+a^3*b-2*b^3))

Fricas [F(-1)]

Timed out. \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\text {Timed out} \]

[In]

integrate((a*x^4+2*b)/(a*x^4+b*x^2)^(1/4)/(2*x^8-a*x^4-b),x, algorithm="fricas")

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 71.51 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.34 \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\int \frac {a x^{4} + 2 b}{\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (- a x^{4} - b + 2 x^{8}\right )}\, dx \]

[In]

integrate((a*x**4+2*b)/(a*x**4+b*x**2)**(1/4)/(2*x**8-a*x**4-b),x)

[Out]

Integral((a*x**4 + 2*b)/((x**2*(a*x**2 + b))**(1/4)*(-a*x**4 - b + 2*x**8)), x)

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.42 \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\int { \frac {a x^{4} + 2 \, b}{{\left (2 \, x^{8} - a x^{4} - b\right )} {\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate((a*x^4+2*b)/(a*x^4+b*x^2)^(1/4)/(2*x^8-a*x^4-b),x, algorithm="maxima")

[Out]

integrate((a*x^4 + 2*b)/((2*x^8 - a*x^4 - b)*(a*x^4 + b*x^2)^(1/4)), x)

Giac [N/A]

Not integrable

Time = 19.03 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.03 \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\int { \frac {a x^{4} + 2 \, b}{{\left (2 \, x^{8} - a x^{4} - b\right )} {\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate((a*x^4+2*b)/(a*x^4+b*x^2)^(1/4)/(2*x^8-a*x^4-b),x, algorithm="giac")

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 6.71 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.40 \[ \int \frac {2 b+a x^4}{\sqrt [4]{b x^2+a x^4} \left (-b-a x^4+2 x^8\right )} \, dx=\int -\frac {a\,x^4+2\,b}{{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (-2\,x^8+a\,x^4+b\right )} \,d x \]

[In]

int(-(2*b + a*x^4)/((a*x^4 + b*x^2)^(1/4)*(b + a*x^4 - 2*x^8)),x)

[Out]

int(-(2*b + a*x^4)/((a*x^4 + b*x^2)^(1/4)*(b + a*x^4 - 2*x^8)), x)