3.23.6 \(\int \frac {a x^4+x^8}{\sqrt [4]{-b x^2+a x^4} (-b+2 a x^4+x^8)} \, dx\)
Optimal. Leaf size=163 \[ \frac {1}{8} \text {RootSum}\left [\text {$\#$1}^{16}-4 \text {$\#$1}^{12} a+6 \text {$\#$1}^8 a^2-2 \text {$\#$1}^8 a b-4 \text {$\#$1}^4 a^3+4 \text {$\#$1}^4 a^2 b+a^4-2 a^3 b-b^3\& ,\frac {\log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b x^2}}\right )}{\sqrt [4]{a}} \]
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Rubi [B] time = 2.17, antiderivative size = 1315, normalized size of antiderivative = 8.07,
number of steps used = 34, number of rules used = 13, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.317,
Rules used = {1593, 2056, 6728, 1270, 1517, 240, 212, 206, 203, 1429, 377, 208, 205} \begin {gather*} \frac {\sqrt {x} \sqrt [4]{a x^2-b} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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}} \end {gather*}
Antiderivative was successfully verified.
[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 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 240
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 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 1270
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 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 1517
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 1593
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 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 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 {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^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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \tan ^{-1}\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} \tanh ^{-1}\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\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] time = 2.73, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification is not applicable to the result.
[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]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.00, size = 163, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^2+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\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 ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(a*x^4 + x^8)/((-(b*x^2) + a*x^4)^(1/4)*(-b + 2*a*x^4 + x^8)),x]
[Out]
ArcTan[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]/a^(1/4) + ArcTanh[(a^(1/4)*x)/(-(b*x^2) + a*x^4)^(1/4)]/a^(1/4) +
RootSum[a^4 - 2*a^3*b - 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 & , (-L
og[x] + Log[(-(b*x^2) + a*x^4)^(1/4) - x*#1])/#1 & ]/8
________________________________________________________________________________________
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((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
________________________________________________________________________________________
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((x^8+a*x^4)/(a*x^4-b*x^2)^(1/4)/(x^8+2*a*x^4-b),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 2.8Unable to divide, perhaps due to rounding error%%%{-18014398509481984,[2,35,24,18]%%%}+%%%{108
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6,10]%%%}+%%%{-45097156608,[2,31,25,10]%%%}+%%%{-5767168,[2,31,25,6]%%%}+%%%{-64,[2,31,24,2]%%%}+%%%{576460752
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{369435906932736,[2,30,26,14]%%%}+%%%{246960619520,[2,30,26,10]%%%}+%%%{11010048,[2,30,26,6]%%%}+%%%{7340032,[
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76,[2,29,28,14]%%%}+%%%{-47244640256,[2,29,28,10]%%%}+%%%{-630503947831869440,[2,29,27,18]%%%}+%%%{-2093470139
285504,[2,29,27,14]%%%}+%%%{-488552529920,[2,29,27,10]%%%}+%%%{-8388608,[2,29,27,6]%%%}+%%%{-135291469824,[2,2
9,26,10]%%%}+%%%{-41943040,[2,29,26,6]%%%}+%%%{-832,[2,29,26,2]%%%}+%%%{-448,[2,29,25,2]%%%}+%%%{4611686018427
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498216206336,[2,28,28,10]%%%}+%%%{2097152,[2,28,28,6]%%%}+%%%{615726511554560,[2,28,27,14]%%%}+%%%{76665166233
6,[2,28,27,10]%%%}+%%%{81788928,[2,28,27,6]%%%}+%%%{768,[2,28,27,2]%%%}+%%%{22020096,[2,28,26,6]%%%}+%%%{2816,
[2,28,26,2]%%%}+%%%{-2594073385365405696,[2,27,31,18]%%%}+%%%{-18122484900538875904,[2,27,30,18]%%%}+%%%{-5277
65581332480,[2,27,30,14]%%%}+%%%{-10466365534009032704,[2,27,29,18]%%%}+%%%{-6781787720122368,[2,27,29,14]%%%}
+%%%{-369367187456,[2,27,29,10]%%%}+%%%{-630503947831869440,[2,27,28,18]%%%}+%%%{-3571213767016448,[2,27,28,14
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3040,[2,27,27,10]%%%}+%%%{-132120576,[2,27,27,6]%%%}+%%%{-6336,[2,27,27,2]%%%}+%%%{-1344,[2,27,26,2]%%%}+%%%{1
6140901064495857664,[2,26,31,18]%%%}+%%%{22698142121947299840,[2,26,30,18]%%%}+%%%{6429943999234048,[2,26,30,1
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+%%%{1337882312704,[2,26,28,10]%%%}+%%%{264241152,[2,26,28,6]%%%}+%%%{6016,[2,26,28,2]%%%}+%%%{36700160,[2,26,
27,6]%%%}+%%%{8960,[2,26,27,2]%%%}+%%%{-10520408729537478656,[2,25,32,18]%%%}+%%%{-34047213182920949760,[2,25,
31,18]%%%}+%%%{-1671257674219520,[2,25,31,14]%%%}+%%%{-9961962375743537152,[2,25,30,18]%%%}+%%%{-1175158027766
9888,[2,25,30,14]%%%}+%%%{-1270236577792,[2,25,30,10]%%%}+%%%{-378302368699121664,[2,25,29,18]%%%}+%%%{-369435
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,29,2]%%%}+%%%{-225485783040,[2,25,28,10]%%%}+%%%{-234881024,[2,25,28,6]%%%}+%%%{-21120,[2,25,28,2]%%%}+%%%{-2
240,[2,25,27,2]%%%}+%%%{32137686940915859456,[2,24,32,18]%%%}+%%%{23706948438478290944,[2,24,31,18]%%%}+%%%{12
516840370601984,[2,24,31,14]%%%}+%%%{554050781184,[2,24,31,10]%%%}+%%%{2269814212194729984,[2,24,30,18]%%%}+%%
%{7758154045587456,[2,24,30,14]%%%}+%%%{2641404887040,[2,24,30,10]%%%}+%%%{42467328,[2,24,30,6]%%%}+%%%{369435
906932736,[2,24,29,14]%%%}+%%%{1428076625920,[2,24,29,10]%%%}+%%%{484442112,[2,24,29,6]%%%}+%%%{20736,[2,24,29
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1,18]%%%}+%%%{-12068239626469376,[2,23,31,14]%%%}+%%%{-2520072060928,[2,23,31,10]%%%}+%%%{-126100789566373888,
[2,23,30,18]%%%}+%%%{-2339760743907328,[2,23,30,14]%%%}+%%%{-2780991324160,[2,23,30,10]%%%}+%%%{-348127232,[2,
23,30,6]%%%}+%%%{-7232,[2,23,30,2]%%%}+%%%{-135291469824,[2,23,29,10]%%%}+%%%{-256901120,[2,23,29,6]%%%}+%%%{-
40320,[2,23,29,2]%%%}+%%%{-2240,[2,23,28,2]%%%}+%%%{39307417547689689088,[2,22,33,18]%%%}+%%%{1462769158969937
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0,6]%%%}+%%%{41216,[2,22,30,2]%%%}+%%%{22020096,[2,22,29,6]%%%}+%%%{17920,[2,22,29,2]%%%}+%%%{-403702670597491
26144,[2,21,34,18]%%%}+%%%{-29129282389832368128,[2,21,33,18]%%%}+%%%{-2269391999729664,[2,21,33,14]%%%}+%%%{-
882705526964617216,[2,21,32,18]%%%}+%%%{-7019282231721984,[2,21,32,14]%%%}+%%%{-3180423282688,[2,21,32,10]%%%}
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8,[2,21,30,6]%%%}+%%%{-48384,[2,21,30,2]%%%}+%%%{-1344,[2,21,29,2]%%%}+%%%{29003181600265994240,[2,20,34,18]%%
%}+%%%{4539628424389459968,[2,20,33,18]%%%}+%%%{12683966138023936,[2,20,33,14]%%%}+%%%{1788853878784,[2,20,33,
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%%{396361728,[2,20,31,6]%%%}+%%%{51968,[2,20,31,2]%%%}+%%%{7340032,[2,20,30,6]%%%}+%%%{12544,[2,20,30,2]%%%}+%
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4176,[1,22,30,5]%%%}+%%%{1512,[1,22,30,1]%%%}+%%%{1342177280,[1,22,29,9]%%%}+%%%{13991936,[1,22,29,5]%%%}+%%%{
5824,[1,22,29,1]%%%}+%%%{56,[1,22,28,1]%%%}+%%%{-12977284455854669561856,[1,21,34,21]%%%}+%%%{5046283382468640
768,[1,21,34,17]%%%}+%%%{-3403424281599412273152,[1,21,33,21]%%%}+%%%{-5532672142224654336,[1,21,33,17]%%%}+%%
%{1973623371857920,[1,21,33,13]%%%}+%%%{-119903836479112085504,[1,21,32,21]%%%}+%%%{-1528972073492283392,[1,21
,32,17]%%%}+%%%{-1031341906853888,[1,21,32,13]%%%}+%%%{-7516192768,[1,21,32,9]%%%}+%%%{-20266198323167232,[1,2
1,31,17]%%%}+%%%{-460695372038144,[1,21,31,13]%%%}+%%%{-392721072128,[1,21,31,9]%%%}+%%%{22708224,[1,21,31,5]%
%%}+%%%{1848,[1,21,31,1]%%%}+%%%{-25232932864,[1,21,30,9]%%%}+%%%{-48168960,[1,21,30,5]%%%}+%%%{-6720,[1,21,30
,1]%%%}+%%%{-163840,[1,21,29,5]%%%}+%%%{-728,[1,21,29,1]%%%}+%%%{-22790952303068151021568,[1,20,35,21]%%%}+%%%
{12165627716611449290752,[1,20,34,21]%%%}+%%%{-488640559569698816,[1,20,34,17]%%%}+%%%{-175244068700240740352,
[1,20,33,21]%%%}+%%%{267964177828544512,[1,20,33,17]%%%}+%%%{-307863255777280,[1,20,33,13]%%%}+%%%{-2236067348
48,[1,20,33,9]%%%}+%%%{128352589380059136,[1,20,32,17]%%%}+%%%{485984139476992,[1,20,32,13]%%%}+%%%{-169114337
28,[1,20,32,9]%%%}+%%%{-31883264,[1,20,32,5]%%%}+%%%{9345848836096,[1,20,31,13]%%%}+%%%{102810779648,[1,20,31,
9]%%%}+%%%{40370176,[1,20,31,5]%%%}+%%%{56,[1,20,31,1]%%%}+%%%{3178496,[1,20,30,5]%%%}+%%%{2800,[1,20,30,1]%%%
}+%%%{8,[1,20,29,1]%%%}+%%%{-6686944726719712460800,[1,19,35,21]%%%}+%%%{5564197339616247808,[1,19,35,17]%%%}+
%%%{728646390911527288832,[1,19,34,21]%%%}+%%%{-1344324488770093056,[1,19,34,17]%%%}+%%%{1793303464902656,[1,1
9,34,13]%%%}+%%%{87820192733724672,[1,19,33,17]%%%}+%%%{184717953466368,[1,19,33,13]%%%}+%%%{33822867456,[1,19
,33,9]%%%}+%%%{-56075093016576,[1,19,32,13]%%%}+%%%{-73551314944,[1,19,32,9]%%%}+%%%{34177024,[1,19,32,5]%%%}+
%%%{2408,[1,19,32,1]%%%}+%%%{-2281701376,[1,19,31,9]%%%}+%%%{-13303808,[1,19,31,5]%%%}+%%%{-3248,[1,19,31,1]%%
%}+%%%{-168,[1,19,30,1]%%%}+%%%{-17754991170945443430400,[1,18,36,21]%%%}+%%%{3384977537525702721536,[1,18,35,
21]%%%}+%%%{-258956978573803520,[1,18,35,17]%%%}+%%%{-147573952589676412928,[1,18,34,21]%%%}+%%%{-898468125660
413952,[1,18,34,17]%%%}+%%%{-363938348793856,[1,18,34,13]%%%}+%%%{-219848638464,[1,18,34,9]%%%}+%%%{2251799813
685248,[1,18,33,17]%%%}+%%%{-15393162788864,[1,18,33,13]%%%}+%%%{-94757715968,[1,18,33,9]%%%}+%%%{-24084480,[1
,18,33,5]%%%}+%%%{13421772800,[1,18,32,9]%%%}+%%%{7929856,[1,18,32,5]%%%}+%%%{-952,[1,18,32,1]%%%}+%%%{294912,
[1,18,31,5]%%%}+%%%{816,[1,18,31,1]%%%}+%%%{-3200510096788607205376,[1,17,36,21]%%%}+%%%{4334714641344102400,[
1,17,36,17]%%%}+%%%{562625694248141324288,[1,17,35,21]%%%}+%%%{-141863388262170624,[1,17,35,17]%%%}+%%%{106982
4813826048,[1,17,35,13]%%%}+%%%{92323792361095168,[1,17,34,17]%%%}+%%%{256186209271808,[1,17,34,13]%%%}+%%%{16
374562816,[1,17,34,9]%%%}+%%%{-1649267441664,[1,17,33,13]%%%}+%%%{4294967296,[1,17,33,9]%%%}+%%%{23953408,[1,1
7,33,5]%%%}+%%%{2072,[1,17,33,1]%%%}+%%%{-1900544,[1,17,32,5]%%%}+%%%{-944,[1,17,32,1]%%%}+%%%{-16,[1,17,31,1]
%%%}+%%%{-9767550987029207580672,[1,16,37,21]%%%}+%%%{673306158690398633984,[1,16,36,21]%%%}+%%%{3175037737296
19968,[1,16,36,17]%%%}+%%%{-18446744073709551616,[1,16,35,21]%%%}+%%%{-385057768140177408,[1,16,35,17]%%%}+%%%
{-131941395333120,[1,16,35,13]%%%}+%%%{-148444807168,[1,16,35,9]%%%}+%%%{-23639499997184,[1,16,34,13]%%%}+%%%{
-51271172096,[1,16,34,9]%%%}+%%%{-10780672,[1,16,34,5]%%%}+%%%{536870912,[1,16,33,9]%%%}+%%%{-196608,[1,16,33,
5]%%%}+%%%{-888,[1,16,33,1]%%%}+%%%{128,[1,16,32,1]%%%}+%%%{-1429622665712490250240,[1,15,37,21]%%%}+%%%{23846
56002692677632,[1,15,37,17]%%%}+%%%{119903836479112085504,[1,15,36,21]%%%}+%%%{-38280596832649216,[1,15,36,17]
%%%}+%%%{391426139488256,[1,15,36,13]%%%}+%%%{11258999068426240,[1,15,35,17]%%%}+%%%{83562883710976,[1,15,35,1
3]%%%}+%%%{-6979321856,[1,15,35,9]%%%}+%%%{3892314112,[1,15,34,9]%%%}+%%%{9338880,[1,15,34,5]%%%}+%%%{1176,[1,
15,34,1]%%%}+%%%{-98304,[1,15,33,5]%%%}+%%%{-144,[1,15,33,1]%%%}+%%%{-3717018930852474650624,[1,14,38,21]%%%}+
%%%{101457092405402533888,[1,14,37,21]%%%}+%%%{412079365904400384,[1,14,37,17]%%%}+%%%{-67553994410557440,[1,1
4,36,17]%%%}+%%%{-4398046511104,[1,14,36,13]%%%}+%%%{-67914170368,[1,14,36,9]%%%}+%%%{-2748779069440,[1,14,35,
13]%%%}+%%%{-12079595520,[1,14,35,9]%%%}+%%%{-2129920,[1,14,35,5]%%%}+%%%{-327680,[1,14,34,5]%%%}+%%%{-392,[1,
14,34,1]%%%}+%%%{8,[1,14,33,1]%%%}+%%%{-488838717953303117824,[1,13,38,21]%%%}+%%%{907475324915154944,[1,13,38
,17]%%%}+%%%{9223372036854775808,[1,13,37,21]%%%}+%%%{-24769797950537728,[1,13,37,17]%%%}+%%%{67070209294336,[
1,13,37,13]%%%}+%%%{11544872091648,[1,13,36,13]%%%}+%%%{-9126805504,[1,13,36,9]%%%}+%%%{402653184,[1,13,35,9]%
%%}+%%%{1933312,[1,13,35,5]%%%}+%%%{424,[1,13,35,1]%%%}+%%%{-8,[1,13,34,1]%%%}+%%%{-931560575722332356608,[1,1
2,39,21]%%%}+%%%{9223372036854775808,[1,12,38,21]%%%}+%%%{200410183417987072,[1,12,38,17]%%%}+%%%{-45035996273
70496,[1,12,37,17]%%%}+%%%{8796093022208,[1,12,37,13]%%%}+%%%{-20132659200,[1,12,37,9]%%%}+%%%{-1073741824,[1,
12,36,9]%%%}+%%%{294912,[1,12,36,5]%%%}+%%%{-32768,[1,12,35,5]%%%}+%%%{-88,[1,12,35,1]%%%}+%%%{-10145709240540
2533888,[1,11,39,21]%%%}+%%%{227431781182210048,[1,11,39,17]%%%}+%%%{-4503599627370496,[1,11,38,17]%%%}+%%%{-5
497558138880,[1,11,38,13]%%%}+%%%{549755813888,[1,11,37,13]%%%}+%%%{-3221225472,[1,11,37,9]%%%}+%%%{163840,[1,
11,36,5]%%%}+%%%{88,[1,11,36,1]%%%}+%%%{-138350580552821637120,[1,10,40,21]%%%}+%%%{47287796087390208,[1,10,39
,17]%%%}+%%%{1649267441664,[1,10,38,13]%%%}+%%%{-3489660928,[1,10,38,9]%%%}+%%%{229376,[1,10,37,5]%%%}+%%%{-8,
[1,10,36,1]%%%}+%%%{-9223372036854775808,[1,9,40,21]%%%}+%%%{33776997205278720,[1,9,40,17]%%%}+%%%{-4398046511
104,[1,9,39,13]%%%}+%%%{-402653184,[1,9,38,9]%%%}+%%%{8,[1,9,37,1]%%%}+%%%{-9223372036854775808,[1,8,41,21]%%%
}+%%%{4503599627370496,[1,8,40,17]%%%}+%%%{-268435456,[1,8,39,9]%%%}+%%%{32768,[1,8,38,5]%%%}+%%%{225179981368
5248,[1,7,41,17]%%%}+%%%{-549755813888,[1,7,40,13]%%%} / %%%{134217728,[0,17,10,9]%%%}+%%%{-402653184,[0,16,11
,9]%%%}+%%%{-65536,[0,16,10,5]%%%}+%%%{536870912,[0,15,12,9]%%%}+%%%{402653184,[0,15,11,9]%%%}+%%%{163840,[0,1
5,11,5]%%%}+%%%{8,[0,15,10,1]%%%}+%%%{-536870912,[0,14,13,9]%%%}+%%%{-1207959552,[0,14,12,9]%%%}+%%%{-65536,[0
,14,12,5]%%%}+%%%{-196608,[0,14,11,5]%%%}+%%%{-24,[0,14,11,1]%%%}+%%%{1610612736,[0,13,13,9]%%%}+%%%{402653184
,[0,13,12,9]%%%}+%%%{524288,[0,13,12,5]%%%}+%%%{16,[0,13,12,1]%%%}+%%%{24,[0,13,11,1]%%%}+%%%{-2147483648,[0,1
2,14,9]%%%}+%%%{-1207959552,[0,12,13,9]%%%}+%%%{-163840,[0,12,13,5]%%%}+%%%{-196608,[0,12,12,5]%%%}+%%%{-80,[0
,12,12,1]%%%}+%%%{1476395008,[0,11,14,9]%%%}+%%%{134217728,[0,11,13,9]%%%}+%%%{589824,[0,11,13,5]%%%}+%%%{56,[
0,11,13,1]%%%}+%%%{24,[0,11,12,1]%%%}+%%%{-3355443200,[0,10,15,9]%%%}+%%%{-402653184,[0,10,14,9]%%%}+%%%{-6553
6,[0,10,14,5]%%%}+%%%{-65536,[0,10,13,5]%%%}+%%%{-96,[0,10,13,1]%%%}+%%%{134217728,[0,9,15,9]%%%}+%%%{262144,[
0,9,14,5]%%%}+%%%{72,[0,9,14,1]%%%}+%%%{8,[0,9,13,1]%%%}+%%%{-2550136832,[0,8,16,9]%%%}+%%%{131072,[0,8,15,5]%
%%}+%%%{-48,[0,8,14,1]%%%}+%%%{-402653184,[0,7,16,9]%%%}+%%%{32768,[0,7,15,5]%%%}+%%%{40,[0,7,15,1]%%%}+%%%{-9
39524096,[0,6,17,9]%%%}+%%%{131072,[0,6,16,5]%%%}+%%%{-8,[0,6,15,1]%%%}+%%%{-134217728,[0,5,17,9]%%%}+%%%{8,[0
,5,16,1]%%%}+%%%{-134217728,[0,4,18,9]%%%}+%%%{32768,[0,4,17,5]%%%} Error: Bad Argument Value
________________________________________________________________________________________
maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {x^{8}+a \,x^{4}}{\left (a \,x^{4}-b \,x^{2}\right )^{\frac {1}{4}} \left (x^{8}+2 a \,x^{4}-b \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^8+a*x^4)/(a*x^4-b*x^2)^(1/4)/(x^8+2*a*x^4-b),x)
[Out]
int((x^8+a*x^4)/(a*x^4-b*x^2)^(1/4)/(x^8+2*a*x^4-b),x)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
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((x**8+a*x**4)/(a*x**4-b*x**2)**(1/4)/(x**8+2*a*x**4-b),x)
[Out]
Timed out
________________________________________________________________________________________