\(\int \frac {-b+a x^4}{\sqrt [4]{-b x^2+a x^4} (-b+2 a x^4+2 x^8)} \, dx\) [1454]

Optimal result

Integrand size = 43, antiderivative size = 102 \[ \int \frac {-b+a x^4}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+2 x^8\right )} \, dx=-\frac {1}{8} \text {RootSum}\left [a^4-2 a^3 b-2 b^3-4 a^3 \text {$\#$1}^4+4 a^2 b \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-2 a b \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log (x)+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1425\) vs. \(2(102)=204\).

Time = 1.31 (sec) , antiderivative size = 1425, normalized size of antiderivative = 13.97, number of steps used = 22, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {2081, 6847, 6860, 1443, 385, 218, 214, 211} \[ \int \frac {-b+a x^4}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+2 x^8\right )} \, dx=\frac {\sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}-2 b} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt [4]{a x^2-b}}\right )}{2\ 2^{7/8} \sqrt [4]{\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}-2 b} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{\sqrt {2} \sqrt {-a-\sqrt {a^2+2 b}} a+2 b} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt [4]{a x^2-b}}\right )}{2\ 2^{7/8} \sqrt [4]{\sqrt {2} \sqrt {-a-\sqrt {a^2+2 b}} a+2 b} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{\sqrt {a^2+2 b}-a} \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{\sqrt {2} a \sqrt {\sqrt {a^2+2 b}-a}-2 b} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{\sqrt {a^2+2 b}-a} \sqrt [4]{a x^2-b}}\right )}{2\ 2^{7/8} \sqrt [4]{\sqrt {2} a \sqrt {\sqrt {a^2+2 b}-a}-2 b} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{\sqrt {a^2+2 b}-a} \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{\sqrt {2} \sqrt {\sqrt {a^2+2 b}-a} a+2 b} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{\sqrt {a^2+2 b}-a} \sqrt [4]{a x^2-b}}\right )}{2\ 2^{7/8} \sqrt [4]{\sqrt {2} \sqrt {\sqrt {a^2+2 b}-a} a+2 b} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}-2 b} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt [4]{a x^2-b}}\right )}{2\ 2^{7/8} \sqrt [4]{\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}-2 b} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {2} \sqrt {-a-\sqrt {a^2+2 b}} a+2 b} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt [4]{a x^2-b}}\right )}{2\ 2^{7/8} \sqrt [4]{\sqrt {2} \sqrt {-a-\sqrt {a^2+2 b}} a+2 b} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{\sqrt {a^2+2 b}-a} \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {2} a \sqrt {\sqrt {a^2+2 b}-a}-2 b} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{\sqrt {a^2+2 b}-a} \sqrt [4]{a x^2-b}}\right )}{2\ 2^{7/8} \sqrt [4]{\sqrt {2} a \sqrt {\sqrt {a^2+2 b}-a}-2 b} \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{\sqrt {a^2+2 b}-a} \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {2} \sqrt {\sqrt {a^2+2 b}-a} a+2 b} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{\sqrt {a^2+2 b}-a} \sqrt [4]{a x^2-b}}\right )}{2\ 2^{7/8} \sqrt [4]{\sqrt {2} \sqrt {\sqrt {a^2+2 b}-a} a+2 b} \sqrt [4]{a x^4-b x^2}} \]

[In]

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

[Out]

((-a - Sqrt[a^2 + 2*b])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[((-2*b + Sqrt[2]*a*Sqrt[-a - Sqrt[a^2 + 2*b]])
^(1/4)*Sqrt[x])/(2^(1/8)*(-a - Sqrt[a^2 + 2*b])^(1/8)*(-b + a*x^2)^(1/4))])/(2*2^(7/8)*(-2*b + Sqrt[2]*a*Sqrt[
-a - Sqrt[a^2 + 2*b]])^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) + ((-a - Sqrt[a^2 + 2*b])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1
/4)*ArcTan[((2*b + Sqrt[2]*a*Sqrt[-a - Sqrt[a^2 + 2*b]])^(1/4)*Sqrt[x])/(2^(1/8)*(-a - Sqrt[a^2 + 2*b])^(1/8)*
(-b + a*x^2)^(1/4))])/(2*2^(7/8)*(2*b + Sqrt[2]*a*Sqrt[-a - Sqrt[a^2 + 2*b]])^(1/4)*(-(b*x^2) + a*x^4)^(1/4))
+ ((-a + Sqrt[a^2 + 2*b])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[((-2*b + Sqrt[2]*a*Sqrt[-a + Sqrt[a^2 + 2*b]
])^(1/4)*Sqrt[x])/(2^(1/8)*(-a + Sqrt[a^2 + 2*b])^(1/8)*(-b + a*x^2)^(1/4))])/(2*2^(7/8)*(-2*b + Sqrt[2]*a*Sqr
t[-a + Sqrt[a^2 + 2*b]])^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) + ((-a + Sqrt[a^2 + 2*b])^(1/8)*Sqrt[x]*(-b + a*x^2)^
(1/4)*ArcTan[((2*b + Sqrt[2]*a*Sqrt[-a + Sqrt[a^2 + 2*b]])^(1/4)*Sqrt[x])/(2^(1/8)*(-a + Sqrt[a^2 + 2*b])^(1/8
)*(-b + a*x^2)^(1/4))])/(2*2^(7/8)*(2*b + Sqrt[2]*a*Sqrt[-a + Sqrt[a^2 + 2*b]])^(1/4)*(-(b*x^2) + a*x^4)^(1/4)
) + ((-a - Sqrt[a^2 + 2*b])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTanh[((-2*b + Sqrt[2]*a*Sqrt[-a - Sqrt[a^2 + 2
*b]])^(1/4)*Sqrt[x])/(2^(1/8)*(-a - Sqrt[a^2 + 2*b])^(1/8)*(-b + a*x^2)^(1/4))])/(2*2^(7/8)*(-2*b + Sqrt[2]*a*
Sqrt[-a - Sqrt[a^2 + 2*b]])^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) + ((-a - Sqrt[a^2 + 2*b])^(1/8)*Sqrt[x]*(-b + a*x^
2)^(1/4)*ArcTanh[((2*b + Sqrt[2]*a*Sqrt[-a - Sqrt[a^2 + 2*b]])^(1/4)*Sqrt[x])/(2^(1/8)*(-a - Sqrt[a^2 + 2*b])^
(1/8)*(-b + a*x^2)^(1/4))])/(2*2^(7/8)*(2*b + Sqrt[2]*a*Sqrt[-a - Sqrt[a^2 + 2*b]])^(1/4)*(-(b*x^2) + a*x^4)^(
1/4)) + ((-a + Sqrt[a^2 + 2*b])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTanh[((-2*b + Sqrt[2]*a*Sqrt[-a + Sqrt[a^2
+ 2*b]])^(1/4)*Sqrt[x])/(2^(1/8)*(-a + Sqrt[a^2 + 2*b])^(1/8)*(-b + a*x^2)^(1/4))])/(2*2^(7/8)*(-2*b + Sqrt[2
]*a*Sqrt[-a + Sqrt[a^2 + 2*b]])^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) + ((-a + Sqrt[a^2 + 2*b])^(1/8)*Sqrt[x]*(-b +
a*x^2)^(1/4)*ArcTanh[((2*b + Sqrt[2]*a*Sqrt[-a + Sqrt[a^2 + 2*b]])^(1/4)*Sqrt[x])/(2^(1/8)*(-a + Sqrt[a^2 + 2*
b])^(1/8)*(-b + a*x^2)^(1/4))])/(2*2^(7/8)*(2*b + Sqrt[2]*a*Sqrt[-a + Sqrt[a^2 + 2*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 {-b+a x^4}{\sqrt {x} \sqrt [4]{-b+a x^2} \left (-b+2 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 {-b+a x^8}{\sqrt [4]{-b+a x^4} \left (-b+2 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+2 b}}{\sqrt [4]{-b+a x^4} \left (2 a-2 \sqrt {a^2+2 b}+4 x^8\right )}+\frac {a+\sqrt {a^2+2 b}}{\sqrt [4]{-b+a x^4} \left (2 a+2 \sqrt {a^2+2 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+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (2 a-2 \sqrt {a^2+2 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+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (2 a+2 \sqrt {a^2+2 b}+4 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}} \\ & = -\frac {\left (\sqrt {2} \left (a-\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {2} \sqrt {-a+\sqrt {a^2+2 b}}-4 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a+\sqrt {a^2+2 b}} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt {2} \left (a-\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {2} \sqrt {-a+\sqrt {a^2+2 b}}+4 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a+\sqrt {a^2+2 b}} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt {2} \left (a+\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {2} \sqrt {-a-\sqrt {a^2+2 b}}-4 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a-\sqrt {a^2+2 b}} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt {2} \left (a+\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (2 \sqrt {2} \sqrt {-a-\sqrt {a^2+2 b}}+4 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a-\sqrt {a^2+2 b}} \sqrt [4]{-b x^2+a x^4}} \\ & = -\frac {\left (\sqrt {2} \left (a-\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {2} \sqrt {-a+\sqrt {a^2+2 b}}-\left (-4 b+2 \sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {-a+\sqrt {a^2+2 b}} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt {2} \left (a-\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {2} \sqrt {-a+\sqrt {a^2+2 b}}-\left (4 b+2 \sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {-a+\sqrt {a^2+2 b}} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt {2} \left (a+\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {2} \sqrt {-a-\sqrt {a^2+2 b}}-\left (-4 b+2 \sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {-a-\sqrt {a^2+2 b}} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\sqrt {2} \left (a+\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{2 \sqrt {2} \sqrt {-a-\sqrt {a^2+2 b}}-\left (4 b+2 \sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {-a-\sqrt {a^2+2 b}} \sqrt [4]{-b x^2+a x^4}} \\ & = -\frac {\left (\left (a-\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{-a+\sqrt {a^2+2 b}}-\sqrt {-2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2\ 2^{3/4} \left (-a+\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\left (a-\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{-a+\sqrt {a^2+2 b}}+\sqrt {-2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2\ 2^{3/4} \left (-a+\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\left (a-\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{-a+\sqrt {a^2+2 b}}-\sqrt {2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2\ 2^{3/4} \left (-a+\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\left (a-\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{-a+\sqrt {a^2+2 b}}+\sqrt {2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2\ 2^{3/4} \left (-a+\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\left (a+\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{-a-\sqrt {a^2+2 b}}-\sqrt {-2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2\ 2^{3/4} \left (-a-\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\left (a+\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{-a-\sqrt {a^2+2 b}}+\sqrt {-2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2\ 2^{3/4} \left (-a-\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\left (a+\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{-a-\sqrt {a^2+2 b}}-\sqrt {2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2\ 2^{3/4} \left (-a-\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (\left (a+\sqrt {a^2+2 b}\right ) \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{-a-\sqrt {a^2+2 b}}+\sqrt {2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2\ 2^{3/4} \left (-a-\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{-2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt [4]{-b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [4]{-2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt [4]{-b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [4]{2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-a+\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{-2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a+\sqrt {a^2+2 b}} \sqrt [4]{-b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [4]{-2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-a+\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a+\sqrt {a^2+2 b}} \sqrt [4]{-b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [4]{2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{-2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt [4]{-b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [4]{-2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a-\sqrt {a^2+2 b}} \sqrt [4]{-b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [4]{2 b+\sqrt {2} a \sqrt {-a-\sqrt {a^2+2 b}}} \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-a+\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{-2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a+\sqrt {a^2+2 b}} \sqrt [4]{-b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [4]{-2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-a+\sqrt {a^2+2 b}} \sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} \sqrt {x}}{\sqrt [8]{2} \sqrt [8]{-a+\sqrt {a^2+2 b}} \sqrt [4]{-b+a x^2}}\right )}{2\ 2^{7/8} \sqrt [4]{2 b+\sqrt {2} a \sqrt {-a+\sqrt {a^2+2 b}}} \sqrt [4]{-b x^2+a x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.21 \[ \int \frac {-b+a x^4}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+2 x^8\right )} \, dx=-\frac {\sqrt [4]{-a+\frac {b}{x^2}} x \text {RootSum}\left [a^4-2 a^3 b-2 b^3+4 a^3 \text {$\#$1}^4-4 a^2 b \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-2 a b \text {$\#$1}^8+4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt [4]{-a+\frac {b}{x^2}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 \sqrt [4]{-b x^2+a x^4}} \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 0.39 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-4 a \,\textit {\_Z}^{12}+\left (6 a^{2}-2 a b \right ) \textit {\_Z}^{8}+\left (-4 a^{3}+4 a^{2} b \right ) \textit {\_Z}^{4}+a^{4}-2 a^{3} b -2 b^{3}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right )}{8}\)	\(90\)

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 69.97 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.33 \[ \int \frac {-b+a x^4}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+2 x^8\right )} \, dx=\int \frac {a x^{4} - b}{\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (2 a x^{4} - b + 2 x^{8}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.42 \[ \int \frac {-b+a x^4}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+2 x^8\right )} \, dx=\int { \frac {a x^{4} - b}{{\left (2 \, x^{8} + 2 \, a x^{4} - b\right )} {\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 19.13 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.03 \[ \int \frac {-b+a x^4}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+2 x^8\right )} \, dx=\int { \frac {a x^{4} - b}{{\left (2 \, x^{8} + 2 \, a x^{4} - b\right )} {\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.42 \[ \int \frac {-b+a x^4}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+2 x^8\right )} \, dx=\int -\frac {b-a\,x^4}{{\left (a\,x^4-b\,x^2\right )}^{1/4}\,\left (2\,x^8+2\,a\,x^4-b\right )} \,d x \]

[In]

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

[Out]

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