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

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

________________________________________________________________________________________

Rubi [B]  time = 1.96, antiderivative size = 1017, normalized size of antiderivative = 12.56, number of steps used = 22, number of rules used = 10, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2056, 6715, 6725, 1429, 377, 212, 206, 203, 208, 205} \begin {gather*} \frac {\sqrt {x} \sqrt [4]{a x^2-b} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[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 203

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

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 206

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

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 6725

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*} \int \frac {1}{\sqrt [4]{-b x^2+a x^4} \left (b+a x^8\right )} \, dx &=\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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 [B]  time = 0.84, size = 662, normalized size = 8.17 \begin {gather*} -\frac {x \sqrt [4]{\frac {b}{x^2}-a} \left (a^3 \text {RootSum}\left [\text {$\#$1}^4+4 \text {$\#$1}^3 a+6 \text {$\#$1}^2 a^2+4 \text {$\#$1} a^3+a^4+a b^3\&,\frac {\frac {\log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{\frac {b}{x^2}-a}\right )}{\sqrt [4]{\text {$\#$1}}}-\frac {\log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{\frac {b}{x^2}-a}\right )}{\sqrt [4]{\text {$\#$1}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [4]{\text {$\#$1}}}\right )}{\sqrt [4]{\text {$\#$1}}}}{\text {$\#$1}^3+3 \text {$\#$1}^2 a+3 \text {$\#$1} a^2+a^3}\&\right ]+3 a^2 \text {RootSum}\left [\text {$\#$1}^4+4 \text {$\#$1}^3 a+6 \text {$\#$1}^2 a^2+4 \text {$\#$1} a^3+a^4+a b^3\&,\frac {\text {$\#$1}^{3/4} \log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{\frac {b}{x^2}-a}\right )-\text {$\#$1}^{3/4} \log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{\frac {b}{x^2}-a}\right )+2 \text {$\#$1}^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [4]{\text {$\#$1}}}\right )}{\text {$\#$1}^3+3 \text {$\#$1}^2 a+3 \text {$\#$1} a^2+a^3}\&\right ]+3 a \text {RootSum}\left [\text {$\#$1}^4+4 \text {$\#$1}^3 a+6 \text {$\#$1}^2 a^2+4 \text {$\#$1} a^3+a^4+a b^3\&,\frac {\text {$\#$1}^{7/4} \log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{\frac {b}{x^2}-a}\right )-\text {$\#$1}^{7/4} \log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{\frac {b}{x^2}-a}\right )+2 \text {$\#$1}^{7/4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [4]{\text {$\#$1}}}\right )}{\text {$\#$1}^3+3 \text {$\#$1}^2 a+3 \text {$\#$1} a^2+a^3}\&\right ]+\text {RootSum}\left [\text {$\#$1}^4+4 \text {$\#$1}^3 a+6 \text {$\#$1}^2 a^2+4 \text {$\#$1} a^3+a^4+a b^3\&,\frac {\text {$\#$1}^{11/4} \log \left (\sqrt [4]{\text {$\#$1}}-\sqrt [4]{\frac {b}{x^2}-a}\right )-\text {$\#$1}^{11/4} \log \left (\sqrt [4]{\text {$\#$1}}+\sqrt [4]{\frac {b}{x^2}-a}\right )+2 \text {$\#$1}^{11/4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {b}{x^2}-a}}{\sqrt [4]{\text {$\#$1}}}\right )}{\text {$\#$1}^3+3 \text {$\#$1}^2 a+3 \text {$\#$1} a^2+a^3}\&\right ]\right )}{8 b \sqrt [4]{a x^4-b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*((-a + b/x^2)^(1/4)*x*(a^3*RootSum[a^4 + a*b^3 + 4*a^3*#1 + 6*a^2*#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))/(a^3 + 3*a^2*#1 + 3*a*#1^2 + #1^3) & ] + 3*a^2*RootSum[a^4 + a*b^3 + 4*a^3*#1 + 6*a^2*#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))/(a^3 + 3*a^2*#1 + 3*a*#1^2 + #1^3) & ] + 3*a*RootSum[
a^4 + a*b^3 + 4*a^3*#1 + 6*a^2*#1^2 + 4*a*#1^3 + #1^4 & , (2*ArcTan[(-a + b/x^2)^(1/4)/#1^(1/4)]*#1^(7/4) + Lo
g[-(-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))/(a^3 + 3*a^2*#1 + 3*
a*#1^2 + #1^3) & ] + RootSum[a^4 + a*b^3 + 4*a^3*#1 + 6*a^2*#1^2 + 4*a*#1^3 + #1^4 & , (2*ArcTan[(-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) - Log[(-a + b/x^2)^(1/4) + #1^(1/4)]
*#1^(11/4))/(a^3 + 3*a^2*#1 + 3*a*#1^2 + #1^3) & ]))/(b*(-(b*x^2) + a*x^4)^(1/4))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.00, size = 81, normalized size = 1.00 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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,x):;OUTPUT:Not
invertible Error: Bad Argument Value

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a x^{8} + 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(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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (a\,x^8+b\right )\,{\left (a\,x^4-b\,x^2\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (a x^{8} + b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________