\(\int \frac {1}{\sqrt [4]{-b x^2+a x^4} (b+a x^8)} \, dx\) [1093]

Optimal result

Integrand size = 26, antiderivative size = 81 \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=-\frac {\text {RootSum}\left [a^4+a b^3-4 a^3 \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log (x)+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 b} \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1017\) vs. \(2(81)=162\).

Time = 1.71 (sec) , antiderivative size = 1017, normalized size of antiderivative = 12.56, number of steps used = 22, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2081, 6847, 6857, 1443, 385, 218, 212, 209, 214, 211} \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=\frac {\sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}}+\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} b \sqrt [4]{a x^4-b x^2}} \]

[In]

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

[Out]

(Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[((a - (-a)^(1/4)*b^(3/4))^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(4*(a - (-a)^
(1/4)*b^(3/4))^(1/4)*b*(-(b*x^2) + a*x^4)^(1/4)) + (Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[((a + (-a)^(1/4)*b^(3/4)
)^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(4*(a + (-a)^(1/4)*b^(3/4))^(1/4)*b*(-(b*x^2) + a*x^4)^(1/4)) + ((-Sqrt[
-a])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[((Sqrt[-Sqrt[-a]]*a - Sqrt[-a]*b^(3/4))^(1/4)*Sqrt[x])/((-Sqrt[-a
])^(1/8)*(-b + a*x^2)^(1/4))])/(4*(Sqrt[-Sqrt[-a]]*a - Sqrt[-a]*b^(3/4))^(1/4)*b*(-(b*x^2) + a*x^4)^(1/4)) + (
(-Sqrt[-a])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[((Sqrt[-Sqrt[-a]]*a + Sqrt[-a]*b^(3/4))^(1/4)*Sqrt[x])/((-
Sqrt[-a])^(1/8)*(-b + a*x^2)^(1/4))])/(4*(Sqrt[-Sqrt[-a]]*a + Sqrt[-a]*b^(3/4))^(1/4)*b*(-(b*x^2) + a*x^4)^(1/
4)) + (Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTanh[((a - (-a)^(1/4)*b^(3/4))^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(4*(a
- (-a)^(1/4)*b^(3/4))^(1/4)*b*(-(b*x^2) + a*x^4)^(1/4)) + (Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTanh[((a + (-a)^(1/4)
*b^(3/4))^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(4*(a + (-a)^(1/4)*b^(3/4))^(1/4)*b*(-(b*x^2) + a*x^4)^(1/4)) +
((-Sqrt[-a])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTanh[((Sqrt[-Sqrt[-a]]*a - Sqrt[-a]*b^(3/4))^(1/4)*Sqrt[x])/(
(-Sqrt[-a])^(1/8)*(-b + a*x^2)^(1/4))])/(4*(Sqrt[-Sqrt[-a]]*a - Sqrt[-a]*b^(3/4))^(1/4)*b*(-(b*x^2) + a*x^4)^(
1/4)) + ((-Sqrt[-a])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTanh[((Sqrt[-Sqrt[-a]]*a + Sqrt[-a]*b^(3/4))^(1/4)*Sq
rt[x])/((-Sqrt[-a])^(1/8)*(-b + a*x^2)^(1/4))])/(4*(Sqrt[-Sqrt[-a]]*a + Sqrt[-a]*b^(3/4))^(1/4)*b*(-(b*x^2) +
a*x^4)^(1/4))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 6857

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt [4]{-b+a x^2} \left (b+a 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 {1}{\sqrt [4]{-b+a x^4} \left (b+a 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 {1}{2 \sqrt {b} \sqrt [4]{-b+a x^4} \left (\sqrt {b}-\sqrt {-a} x^8\right )}+\frac {1}{2 \sqrt {b} \sqrt [4]{-b+a x^4} \left (\sqrt {b}+\sqrt {-a} x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (\sqrt {b}-\sqrt {-a} x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {b} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (\sqrt {b}+\sqrt {-a} x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {b} \sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\left (\sqrt {-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-\sqrt {-a}} \sqrt [4]{b}-\sqrt {-a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt {-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-\sqrt {-a}} \sqrt [4]{b}+\sqrt {-a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-a} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [4]{-a} \sqrt [4]{b}-\sqrt {-a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-a} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [4]{-a} \sqrt [4]{b}+\sqrt {-a} x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\left (\sqrt {-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-\sqrt {-a}} \sqrt [4]{b}-\left (\sqrt {-\sqrt {-a}} a \sqrt [4]{b}-\sqrt {-a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt {-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-\sqrt {-a}} \sqrt [4]{b}-\left (\sqrt {-\sqrt {-a}} a \sqrt [4]{b}+\sqrt {-a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-a} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a} \sqrt [4]{b}-\left (\sqrt [4]{-a} a \sqrt [4]{b}-\sqrt {-a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-a} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a} \sqrt [4]{b}-\left (\sqrt [4]{-a} a \sqrt [4]{b}+\sqrt {-a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{2 b^{3/4} \sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a-\sqrt [4]{-a} b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a-\sqrt [4]{-a} b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a+\sqrt [4]{-a} b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a+\sqrt [4]{-a} b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-\sqrt {-a}}-\sqrt {\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-\sqrt {-a}}+\sqrt {\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-\sqrt {-a}}-\sqrt {\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 b \sqrt [4]{-b x^2+a x^4}}+\frac {\left (\sqrt [4]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-\sqrt {-a}}+\sqrt {\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 b \sqrt [4]{-b x^2+a x^4}} \\ & = \frac {\sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} b \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} b \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} b \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2} \arctan \left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} b \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{a-\sqrt [4]{-a} b^{3/4}} b \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} \sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{a+\sqrt [4]{-a} b^{3/4}} b \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a-\sqrt {-a} b^{3/4}} b \sqrt [4]{-b x^2+a x^4}}+\frac {\sqrt [8]{-\sqrt {-a}} \sqrt {x} \sqrt [4]{-b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} \sqrt {x}}{\sqrt [8]{-\sqrt {-a}} \sqrt [4]{-b+a x^2}}\right )}{4 \sqrt [4]{\sqrt {-\sqrt {-a}} a+\sqrt {-a} b^{3/4}} b \sqrt [4]{-b x^2+a x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.48 \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=-\frac {\sqrt {x} \sqrt [4]{-b+a x^2} \text {RootSum}\left [a^4+a b^3-4 a^3 \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 b \sqrt [4]{-b x^2+a x^4}} \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 0.00 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-4 a \,\textit {\_Z}^{12}+6 a^{2} \textit {\_Z}^{8}-4 a^{3} \textit {\_Z}^{4}+a^{4}+a \,b^{3}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}}{8 b}\)	\(73\)

[In]

int(1/(a*x^4-b*x^2)^(1/4)/(a*x^8+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*_Z^12*a+6*_Z^8*a^2-4*_Z^4*a^3+a^4+a*b^3))/b

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 6.87 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.27 \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=\int \frac {1}{\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (a x^{8} + b\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.32 \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=\int { \frac {1}{{\left (a x^{8} + b\right )} {\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Not invertible Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.32 \[ \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx=\int \frac {1}{\left (a\,x^8+b\right )\,{\left (a\,x^4-b\,x^2\right )}^{1/4}} \,d x \]

[In]

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

[Out]

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