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

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

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Rubi [B]  time = 3.34, antiderivative size = 1161, normalized size of antiderivative = 11.73, number of steps used = 22, number of rules used = 8, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2056, 6715, 6728, 1429, 377, 212, 208, 205} \begin {gather*} -\frac {\sqrt [8]{a-\sqrt {a^2+8 b}} \sqrt {x} \sqrt [4]{a x^2+b} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[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 205

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

Rule 208

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

Rule 212

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] &&
!GtQ[a/b, 0]

Rule 377

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 1429

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 2056

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 6715

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 6728

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*} \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 {\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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*}

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Mathematica [B]  time = 3.07, size = 782, normalized size = 7.90 \begin {gather*} \frac {x \sqrt [4]{a+\frac {b}{x^2}} \left (-\left (a^2 (2 a+b) \text {RootSum}\left [\text {$\#$1}^4-4 \text {$\#$1}^3 a+6 \text {$\#$1}^2 a^2+\text {$\#$1}^2 a b-4 \text {$\#$1} a^3-2 \text {$\#$1} a^2 b+a^4+a^3 b-2 b^3\&,\frac {\frac {\log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{a+\frac {b}{x^2}}\right )}{\sqrt [4]{\text {$\#$1}}}-\frac {\log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{a+\frac {b}{x^2}}\right )}{\sqrt [4]{\text {$\#$1}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\text {$\#$1}}}\right )}{\sqrt [4]{\text {$\#$1}}}}{2 \text {$\#$1}^3-6 \text {$\#$1}^2 a+6 \text {$\#$1} a^2+\text {$\#$1} a b-2 a^3-a^2 b}\&\right ]\right )+a (6 a+b) \text {RootSum}\left [\text {$\#$1}^4-4 \text {$\#$1}^3 a+6 \text {$\#$1}^2 a^2+\text {$\#$1}^2 a b-4 \text {$\#$1} a^3-2 \text {$\#$1} a^2 b+a^4+a^3 b-2 b^3\&,\frac {\text {$\#$1}^{3/4} \log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{a+\frac {b}{x^2}}\right )-\text {$\#$1}^{3/4} \log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{a+\frac {b}{x^2}}\right )+2 \text {$\#$1}^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\text {$\#$1}}}\right )}{2 \text {$\#$1}^3-6 \text {$\#$1}^2 a+6 \text {$\#$1} a^2+\text {$\#$1} a b-2 a^3-a^2 b}\&\right ]-6 a \text {RootSum}\left [\text {$\#$1}^4-4 \text {$\#$1}^3 a+6 \text {$\#$1}^2 a^2+\text {$\#$1}^2 a b-4 \text {$\#$1} a^3-2 \text {$\#$1} a^2 b+a^4+a^3 b-2 b^3\&,\frac {\text {$\#$1}^{7/4} \log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{a+\frac {b}{x^2}}\right )-\text {$\#$1}^{7/4} \log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{a+\frac {b}{x^2}}\right )+2 \text {$\#$1}^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\text {$\#$1}}}\right )}{2 \text {$\#$1}^3-6 \text {$\#$1}^2 a+6 \text {$\#$1} a^2+\text {$\#$1} a b-2 a^3-a^2 b}\&\right ]+2 \text {RootSum}\left [\text {$\#$1}^4-4 \text {$\#$1}^3 a+6 \text {$\#$1}^2 a^2+\text {$\#$1}^2 a b-4 \text {$\#$1} a^3-2 \text {$\#$1} a^2 b+a^4+a^3 b-2 b^3\&,\frac {\text {$\#$1}^{11/4} \log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{a+\frac {b}{x^2}}\right )-\text {$\#$1}^{11/4} \log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{a+\frac {b}{x^2}}\right )+2 \text {$\#$1}^{11/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\text {$\#$1}}}\right )}{2 \text {$\#$1}^3-6 \text {$\#$1}^2 a+6 \text {$\#$1} a^2+\text {$\#$1} a b-2 a^3-a^2 b}\&\right ]\right )}{4 \sqrt [4]{x^2 \left (a x^2+b\right )}} \end {gather*}

Antiderivative was successfully verified.

[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*(-(a^2*(2*a + b)*RootSum[a^4 + a^3*b - 2*b^3 - 4*a^3*#1 - 2*a^2*b*#1 + 6*a^2*#1^2 + a*b*#
1^2 - 4*a*#1^3 + #1^4 & , ((2*ArcTan[(a + b/x^2)^(1/4)/#1^(1/4)])/#1^(1/4) + Log[-(a + b/x^2)^(1/4) + #1^(1/4)
]/#1^(1/4) - Log[(a + b/x^2)^(1/4) + #1^(1/4)]/#1^(1/4))/(-2*a^3 - a^2*b + 6*a^2*#1 + a*b*#1 - 6*a*#1^2 + 2*#1
^3) & ]) + a*(6*a + b)*RootSum[a^4 + a^3*b - 2*b^3 - 4*a^3*#1 - 2*a^2*b*#1 + 6*a^2*#1^2 + a*b*#1^2 - 4*a*#1^3
+ #1^4 & , (2*ArcTan[(a + b/x^2)^(1/4)/#1^(1/4)]*#1^(3/4) + Log[-(a + b/x^2)^(1/4) + #1^(1/4)]*#1^(3/4) - Log[
(a + b/x^2)^(1/4) + #1^(1/4)]*#1^(3/4))/(-2*a^3 - a^2*b + 6*a^2*#1 + a*b*#1 - 6*a*#1^2 + 2*#1^3) & ] - 6*a*Roo
tSum[a^4 + a^3*b - 2*b^3 - 4*a^3*#1 - 2*a^2*b*#1 + 6*a^2*#1^2 + a*b*#1^2 - 4*a*#1^3 + #1^4 & , (2*ArcTan[(a +
b/x^2)^(1/4)/#1^(1/4)]*#1^(7/4) + Log[-(a + b/x^2)^(1/4) + #1^(1/4)]*#1^(7/4) - Log[(a + b/x^2)^(1/4) + #1^(1/
4)]*#1^(7/4))/(-2*a^3 - a^2*b + 6*a^2*#1 + a*b*#1 - 6*a*#1^2 + 2*#1^3) & ] + 2*RootSum[a^4 + a^3*b - 2*b^3 - 4
*a^3*#1 - 2*a^2*b*#1 + 6*a^2*#1^2 + a*b*#1^2 - 4*a*#1^3 + #1^4 & , (2*ArcTan[(a + b/x^2)^(1/4)/#1^(1/4)]*#1^(1
1/4) + Log[-(a + b/x^2)^(1/4) + #1^(1/4)]*#1^(11/4) - Log[(a + b/x^2)^(1/4) + #1^(1/4)]*#1^(11/4))/(-2*a^3 - a
^2*b + 6*a^2*#1 + a*b*#1 - 6*a*#1^2 + 2*#1^3) & ]))/(4*(x^2*(b + a*x^2))^(1/4))

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IntegrateAlgebraic [A]  time = 11.80, size = 99, normalized size = 1.00 \begin {gather*} \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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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[
x] + Log[(b*x^2 + a*x^4)^(1/4) - x*#1])/#1 & ]/4

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{4}+2 b}{\left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}} \left (2 x^{8}-a \,x^{4}-b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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