\(\int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} (-b+2 a x^4+x^8)} \, dx\) [2206]

Optimal result

Integrand size = 41, antiderivative size = 163 \[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{\sqrt [4]{a}}+\frac {1}{8} \text {RootSum}\left [a^4-2 a^3 b-b^3-4 a^3 \text {$\#$1}^4+4 a^2 b \text {$\#$1}^4+6 a^2 \text {$\#$1}^8-2 a b \text {$\#$1}^8-4 a \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {-\log (x)+\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1315\) vs. \(2(163)=326\).

Time = 1.28 (sec) , antiderivative size = 1315, normalized size of antiderivative = 8.07, number of steps used = 34, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.317, Rules used = {1607, 2081, 6860, 1284, 1531, 246, 218, 212, 209, 1443, 385, 214, 211} \[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\frac {\sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2-b}}\right )}{\sqrt [4]{a} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{a \sqrt {-a-\sqrt {a^2+b}}-b} \sqrt {x}}{\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{a \sqrt {-a-\sqrt {a^2+b}}-b} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{\sqrt {-a-\sqrt {a^2+b}} a+b} \sqrt {x}}{\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{\sqrt {-a-\sqrt {a^2+b}} a+b} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{\sqrt {a^2+b}-a} \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{a \sqrt {\sqrt {a^2+b}-a}-b} \sqrt {x}}{\sqrt [8]{\sqrt {a^2+b}-a} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{a \sqrt {\sqrt {a^2+b}-a}-b} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{\sqrt {a^2+b}-a} \sqrt {x} \sqrt [4]{a x^2-b} \arctan \left (\frac {\sqrt [4]{\sqrt {\sqrt {a^2+b}-a} a+b} \sqrt {x}}{\sqrt [8]{\sqrt {a^2+b}-a} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{\sqrt {\sqrt {a^2+b}-a} a+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 {x}}{\sqrt [4]{a x^2-b}}\right )}{\sqrt [4]{a} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{a \sqrt {-a-\sqrt {a^2+b}}-b} \sqrt {x}}{\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{a \sqrt {-a-\sqrt {a^2+b}}-b} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {-a-\sqrt {a^2+b}} a+b} \sqrt {x}}{\sqrt [8]{-a-\sqrt {a^2+b}} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{\sqrt {-a-\sqrt {a^2+b}} a+b} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{\sqrt {a^2+b}-a} \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{a \sqrt {\sqrt {a^2+b}-a}-b} \sqrt {x}}{\sqrt [8]{\sqrt {a^2+b}-a} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{a \sqrt {\sqrt {a^2+b}-a}-b} \sqrt [4]{a x^4-b x^2}}-\frac {\sqrt [8]{\sqrt {a^2+b}-a} \sqrt {x} \sqrt [4]{a x^2-b} \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {\sqrt {a^2+b}-a} a+b} \sqrt {x}}{\sqrt [8]{\sqrt {a^2+b}-a} \sqrt [4]{a x^2-b}}\right )}{4 \sqrt [4]{\sqrt {\sqrt {a^2+b}-a} a+b} \sqrt [4]{a x^4-b x^2}} \]

[In]

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

[Out]

(Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[(a^(1/4)*Sqrt[x])/(-b + a*x^2)^(1/4)])/(a^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) -
((-a - Sqrt[a^2 + b])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[((-b + a*Sqrt[-a - Sqrt[a^2 + b]])^(1/4)*Sqrt[x
])/((-a - Sqrt[a^2 + b])^(1/8)*(-b + a*x^2)^(1/4))])/(4*(-b + a*Sqrt[-a - Sqrt[a^2 + b]])^(1/4)*(-(b*x^2) + a*
x^4)^(1/4)) - ((-a - Sqrt[a^2 + b])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[((b + a*Sqrt[-a - Sqrt[a^2 + b]])^
(1/4)*Sqrt[x])/((-a - Sqrt[a^2 + b])^(1/8)*(-b + a*x^2)^(1/4))])/(4*(b + a*Sqrt[-a - Sqrt[a^2 + b]])^(1/4)*(-(
b*x^2) + a*x^4)^(1/4)) - ((-a + Sqrt[a^2 + b])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[((-b + a*Sqrt[-a + Sqrt
[a^2 + b]])^(1/4)*Sqrt[x])/((-a + Sqrt[a^2 + b])^(1/8)*(-b + a*x^2)^(1/4))])/(4*(-b + a*Sqrt[-a + Sqrt[a^2 + b
]])^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) - ((-a + Sqrt[a^2 + b])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[((b + a*Sq
rt[-a + Sqrt[a^2 + b]])^(1/4)*Sqrt[x])/((-a + Sqrt[a^2 + b])^(1/8)*(-b + a*x^2)^(1/4))])/(4*(b + a*Sqrt[-a + S
qrt[a^2 + b]])^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) + (Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTanh[(a^(1/4)*Sqrt[x])/(-b + a
*x^2)^(1/4)])/(a^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) - ((-a - Sqrt[a^2 + b])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcT
anh[((-b + a*Sqrt[-a - Sqrt[a^2 + b]])^(1/4)*Sqrt[x])/((-a - Sqrt[a^2 + b])^(1/8)*(-b + a*x^2)^(1/4))])/(4*(-b
+ a*Sqrt[-a - Sqrt[a^2 + b]])^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) - ((-a - Sqrt[a^2 + b])^(1/8)*Sqrt[x]*(-b + a*x
^2)^(1/4)*ArcTanh[((b + a*Sqrt[-a - Sqrt[a^2 + b]])^(1/4)*Sqrt[x])/((-a - Sqrt[a^2 + b])^(1/8)*(-b + a*x^2)^(1
/4))])/(4*(b + a*Sqrt[-a - Sqrt[a^2 + b]])^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) - ((-a + Sqrt[a^2 + b])^(1/8)*Sqrt[
x]*(-b + a*x^2)^(1/4)*ArcTanh[((-b + a*Sqrt[-a + Sqrt[a^2 + b]])^(1/4)*Sqrt[x])/((-a + Sqrt[a^2 + b])^(1/8)*(-
b + a*x^2)^(1/4))])/(4*(-b + a*Sqrt[-a + Sqrt[a^2 + b]])^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) - ((-a + Sqrt[a^2 + b
])^(1/8)*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTanh[((b + a*Sqrt[-a + Sqrt[a^2 + b]])^(1/4)*Sqrt[x])/((-a + Sqrt[a^2 +
b])^(1/8)*(-b + a*x^2)^(1/4))])/(4*(b + a*Sqrt[-a + Sqrt[a^2 + b]])^(1/4)*(-(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 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

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 1284

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{k = Denominat
or[m]}, Dist[k/f, Subst[Int[x^(k*(m + 1) - 1)*(d + e*(x^(2*k)/f))^q*(a + c*(x^(4*k)/f))^p, x], x, (f*x)^(1/k)]
, x]] /; FreeQ[{a, c, d, e, f, p, q}, x] && FractionQ[m] && IntegerQ[p]

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 1531

Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.)), x_Symbol] :> Dist[f^(2*n)/
c, Int[(f*x)^(m - 2*n)*(d + e*x^n)^q, x], x] - Dist[a*(f^(2*n)/c), Int[(f*x)^(m - 2*n)*((d + e*x^n)^q/(a + c*x
^(2*n))), x], x] /; FreeQ[{a, c, d, e, f, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] &&  !IntegerQ[q] && GtQ[m, 2*n
- 1]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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

Mathematica [F]

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

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 0.55 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.08

method	result	size
pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-4 a \,\textit {\_Z}^{12}+\left (6 a^{2}-2 a b \right ) \textit {\_Z}^{8}+\left (-4 a^{3}+4 a^{2} b \right ) \textit {\_Z}^{4}+a^{4}-2 a^{3} b -b^{3}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right ) a^{\frac {1}{4}}-8 \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+4 \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{2} \left (a \,x^{2}-b \right )\right )^{\frac {1}{4}}}\right )}{8 a^{\frac {1}{4}}}\)	\(176\)

[In]

int((x^8+a*x^4)/(a*x^4-b*x^2)^(1/4)/(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-b^3))*a^(1/4)-8*arctan(1/a^(1/4)/x*(x^2*(a*x^2-b))^(1/4))+4*ln((-a^(1/4)*x-(x^2*(a*x^2-b))^(1
/4))/(a^(1/4)*x-(x^2*(a*x^2-b))^(1/4))))/a^(1/4)

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.25 \[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\int { \frac {x^{8} + a x^{4}}{{\left (x^{8} + 2 \, a x^{4} - b\right )} {\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 19.83 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.02 \[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\int { \frac {x^{8} + a x^{4}}{{\left (x^{8} + 2 \, a x^{4} - b\right )} {\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.25 \[ \int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} \left (-b+2 a x^4+x^8\right )} \, dx=\int \frac {x^8+a\,x^4}{{\left (a\,x^4-b\,x^2\right )}^{1/4}\,\left (x^8+2\,a\,x^4-b\right )} \,d x \]

[In]

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

[Out]

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