3.29.64 \(\int \frac {x^8}{\sqrt {-b^4+a^4 x^4} (-b^{16}+a^{16} x^{16})} \, dx\)

Optimal. Leaf size=303 \[ -\frac {x}{8 a^8 b^8 \sqrt {a^4 x^4-b^4}}-\frac {\tan ^{-1}\left (\frac {-\frac {a^3 x^4}{2 b}+\frac {b^3}{2 a}+a b x^2}{x \sqrt {a^4 x^4-b^4}}\right )}{32 a^9 b^9}+\frac {\tanh ^{-1}\left (\frac {-\frac {a^3 x^4}{2 b}+\frac {b^3}{2 a}-a b x^2}{x \sqrt {a^4 x^4-b^4}}\right )}{32 a^9 b^9}-\frac {\tanh ^{-1}\left (\frac {-\frac {a^3 x^4}{2^{3/4} b}+\frac {b^3}{2^{3/4} a}-\frac {a b x^2}{\sqrt [4]{2}}}{x \sqrt {a^4 x^4-b^4}}\right )}{8\ 2^{3/4} a^9 b^9}-\frac {\tan ^{-1}\left (\frac {2^{3/4} a b x \sqrt {a^4 x^4-b^4}}{-a^4 x^4+\sqrt {2} a^2 b^2 x^2+b^4}\right )}{8\ 2^{3/4} a^9 b^9} \]

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Rubi [C]  time = 1.25, antiderivative size = 522, normalized size of antiderivative = 1.72, number of steps used = 40, number of rules used = 19, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.528, Rules used = {6725, 1152, 414, 21, 423, 427, 426, 424, 253, 224, 221, 409, 1211, 1699, 203, 206, 1429, 1219, 1218} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {a^4 x^4-b^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a^9 b^7 \sqrt {a^4 x^4-b^4}}-\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {a^4 x^4-b^4}}-\frac {x \left (a^2 x^2+b^2\right )}{16 a^8 b^{10} \sqrt {a^4 x^4-b^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {a^6}{\left (-a^8\right )^{3/4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {a^4 x^4-b^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [4]{-a^8}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {a^4 x^4-b^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {a^4 x^4-b^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {a^4 x^4-b^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8/(Sqrt[-b^4 + a^4*x^4]*(-b^16 + a^16*x^16)),x]

[Out]

-1/16*(x*(b^2 - a^2*x^2))/(a^8*b^10*Sqrt[-b^4 + a^4*x^4]) - (x*(b^2 + a^2*x^2))/(16*a^8*b^10*Sqrt[-b^4 + a^4*x
^4]) - ArcTan[(Sqrt[2]*(-a^4)^(1/4)*b*x)/Sqrt[-b^4 + a^4*x^4]]/(16*Sqrt[2]*(-a^4)^(9/4)*b^9) - ArcTanh[(Sqrt[2
]*(-a^4)^(1/4)*b*x)/Sqrt[-b^4 + a^4*x^4]]/(16*Sqrt[2]*(-a^4)^(9/4)*b^9) - (Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[A
rcSin[(a*x)/b], -1])/(4*a^9*b^7*Sqrt[-b^4 + a^4*x^4]) + (Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[a^6/(-a^8)^(3/4),
ArcSin[(a*x)/b], -1])/(8*a^9*b^7*Sqrt[-b^4 + a^4*x^4]) + (Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[(-a^8)^(1/4)/a^2,
ArcSin[(a*x)/b], -1])/(8*a^9*b^7*Sqrt[-b^4 + a^4*x^4]) + (Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[-(Sqrt[-Sqrt[-a^
8]]/a^2), ArcSin[(a*x)/b], -1])/(8*a^9*b^7*Sqrt[-b^4 + a^4*x^4]) + (Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[Sqrt[-S
qrt[-a^8]]/a^2, ArcSin[(a*x)/b], -1])/(8*a^9*b^7*Sqrt[-b^4 + a^4*x^4])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 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 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + (b*x^4)/a]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + (b*x^4)
/a], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 0]

Rule 253

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_)*((a2_.) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[((a1 + b1*x^n)^FracPa
rt[p]*(a2 + b2*x^n)^FracPart[p])/(a1*a2 + b1*b2*x^(2*n))^FracPart[p], Int[(a1*a2 + b1*b2*x^(2*n))^p, x], x] /;
FreeQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] &&  !IntegerQ[p]

Rule 409

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1
- Rt[-(d/c), 2]*x^2)), x], x] + Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-(d/c), 2]*x^2)), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 423

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[b/d, Int[Sqrt[c + d*x^2]/Sqrt[a + b
*x^2], x], x] - Dist[(b*c - a*d)/d, Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x]
&& PosQ[d/c] && NegQ[b/a]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 426

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[a + b*x^2]/Sqrt[1 + (b*x^2)/a]
, Int[Sqrt[1 + (b*x^2)/a]/Sqrt[c + d*x^2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &&  !GtQ
[a, 0]

Rule 427

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 + (d*x^2)/c]/Sqrt[c + d*x^2]
, Int[Sqrt[a + b*x^2]/Sqrt[1 + (d*x^2)/c], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]

Rule 1152

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + c*x^4)^FracPart[p]/((d + e*x
^2)^FracPart[p]*(a/d + (c*x^2)/e)^FracPart[p]), Int[(d + e*x^2)^(p + q)*(a/d + (c*x^2)/e)^p, x], x] /; FreeQ[{
a, c, d, e, p, q}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p]

Rule 1211

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[1/(2*d), Int[1/Sqrt[a + c*x^4], x],
x] + Dist[1/(2*d), Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d
^2 + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0]

Rule 1218

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-(c/a), 4]}, Simp[(1*Ellipt
icPi[-(e/(d*q^2)), ArcSin[q*x], -1])/(d*Sqrt[a]*q), x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rule 1219

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[Sqrt[1 + (c*x^4)/a]/Sqrt[a + c*x^4]
, Int[1/((d + e*x^2)*Sqrt[1 + (c*x^4)/a]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 0]

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 1699

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[A, Subst[Int[1/
(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ
[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]

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 {x^8}{\sqrt {-b^4+a^4 x^4} \left (-b^{16}+a^{16} x^{16}\right )} \, dx &=\int \left (\frac {1}{8 a^8 b^6 \left (-b^2+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}-\frac {1}{8 a^8 b^6 \left (b^2+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}-\frac {1}{4 a^8 b^4 \sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )}+\frac {1}{2 a^8 \sqrt {-b^4+a^4 x^4} \left (b^8+a^8 x^8\right )}\right ) \, dx\\ &=\frac {\int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (b^8+a^8 x^8\right )} \, dx}{2 a^8}+\frac {\int \frac {1}{\left (-b^2+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^6}-\frac {\int \frac {1}{\left (b^2+a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^6}-\frac {\int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (b^4+a^4 x^4\right )} \, dx}{4 a^8 b^4}\\ &=-\frac {\int \frac {1}{\left (1-\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}-\frac {\int \frac {1}{\left (1+\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}-\frac {\int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (\sqrt {-a^8} b^4-a^8 x^4\right )} \, dx}{4 \sqrt {-a^8} b^4}-\frac {\int \frac {1}{\sqrt {-b^4+a^4 x^4} \left (\sqrt {-a^8} b^4+a^8 x^4\right )} \, dx}{4 \sqrt {-a^8} b^4}-\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {1}{\sqrt {-b^2+a^2 x^2} \left (b^2+a^2 x^2\right )^{3/2}} \, dx}{8 a^8 b^6 \sqrt {-b^4+a^4 x^4}}+\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {1}{\left (-b^2+a^2 x^2\right )^{3/2} \sqrt {b^2+a^2 x^2}} \, dx}{8 a^8 b^6 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-2 \frac {\int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx}{16 a^8 b^8}-\frac {\int \frac {1-\frac {\sqrt {-a^4} x^2}{b^2}}{\left (1+\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{16 a^8 b^8}-\frac {\int \frac {1+\frac {\sqrt {-a^4} x^2}{b^2}}{\left (1-\frac {\sqrt {-a^4} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{16 a^8 b^8}+\frac {\int \frac {1}{\left (1-\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}+\frac {\int \frac {1}{\left (1+\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}+\frac {\int \frac {1}{\left (1-\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}+\frac {\int \frac {1}{\left (1+\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}+\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {-a^2 b^2+a^4 x^2}{\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}} \, dx}{16 a^{10} b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {a^2 b^2+a^4 x^2}{\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}} \, dx}{16 a^{10} b^{10} \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {\operatorname {Subst}\left (\int \frac {1}{1-2 \sqrt {-a^4} b^2 x^2} \, dx,x,\frac {x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 a^8 b^8}-\frac {\operatorname {Subst}\left (\int \frac {1}{1+2 \sqrt {-a^4} b^2 x^2} \, dx,x,\frac {x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 a^8 b^8}+\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {\sqrt {-b^2+a^2 x^2}}{\sqrt {b^2+a^2 x^2}} \, dx}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {\sqrt {b^2+a^2 x^2}}{\sqrt {-b^2+a^2 x^2}} \, dx}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-2 \frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{16 a^8 b^8 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1-\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{8 a^8 b^8 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1+\frac {\sqrt [4]{-a^8} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{8 a^8 b^8 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1-\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{8 a^8 b^8 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\left (1+\frac {\sqrt {-\sqrt {-a^8}} x^2}{b^2}\right ) \sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{8 a^8 b^8 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {a^6}{\left (-a^8\right )^{3/4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [4]{-a^8}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {\sqrt {b^2+a^2 x^2}}{\sqrt {-b^2+a^2 x^2}} \, dx}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}\right ) \int \frac {1}{\sqrt {-b^2+a^2 x^2} \sqrt {b^2+a^2 x^2}} \, dx}{8 a^8 b^8 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\sqrt {b^2+a^2 x^2} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {b^2+a^2 x^2}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {a^6}{\left (-a^8\right )^{3/4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [4]{-a^8}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}-\frac {\int \frac {1}{\sqrt {-b^4+a^4 x^4}} \, dx}{8 a^8 b^8}+\frac {\left (\sqrt {b^2+a^2 x^2} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {b^2+a^2 x^2}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {\left (\left (b^2+a^2 x^2\right ) \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {1+\frac {a^2 x^2}{b^2}}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{16 a^8 b^{10} \sqrt {1+\frac {a^2 x^2}{b^2}} \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\left (b^2+a^2 x^2\right ) \sqrt {1-\frac {a^2 x^2}{b^2}} E\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{16 a^9 b^9 \sqrt {1+\frac {a^2 x^2}{b^2}} \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {a^6}{\left (-a^8\right )^{3/4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [4]{-a^8}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\left (\left (b^2+a^2 x^2\right ) \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \int \frac {\sqrt {1+\frac {a^2 x^2}{b^2}}}{\sqrt {1-\frac {a^2 x^2}{b^2}}} \, dx}{16 a^8 b^{10} \sqrt {1+\frac {a^2 x^2}{b^2}} \sqrt {-b^4+a^4 x^4}}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \int \frac {1}{\sqrt {1-\frac {a^4 x^4}{b^4}}} \, dx}{8 a^8 b^8 \sqrt {-b^4+a^4 x^4}}\\ &=-\frac {x \left (b^2-a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {x \left (b^2+a^2 x^2\right )}{16 a^8 b^{10} \sqrt {-b^4+a^4 x^4}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^4} b x}{\sqrt {-b^4+a^4 x^4}}\right )}{16 \sqrt {2} \left (-a^4\right )^{9/4} b^9}-\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{4 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {a^6}{\left (-a^8\right )^{3/4}};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt [4]{-a^8}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}+\frac {\sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\frac {\sqrt {-\sqrt {-a^8}}}{a^2};\left .\sin ^{-1}\left (\frac {a x}{b}\right )\right |-1\right )}{8 a^9 b^7 \sqrt {-b^4+a^4 x^4}}\\ \end {align*}

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Mathematica [C]  time = 1.40, size = 379, normalized size = 1.25 \begin {gather*} \frac {x \left (-\sqrt {-\frac {a^2}{b^2}}\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} F\left (\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-i;\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )+i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (i;\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-\sqrt [4]{-1};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (\sqrt [4]{-1};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left (-(-1)^{3/4};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )-i \sqrt {1-\frac {a^4 x^4}{b^4}} \Pi \left ((-1)^{3/4};\left .i \sinh ^{-1}\left (\sqrt {-\frac {a^2}{b^2}} x\right )\right |-1\right )}{8 a^8 b^8 \sqrt {-\frac {a^2}{b^2}} \sqrt {a^4 x^4-b^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8/(Sqrt[-b^4 + a^4*x^4]*(-b^16 + a^16*x^16)),x]

[Out]

(-(Sqrt[-(a^2/b^2)]*x) + I*Sqrt[1 - (a^4*x^4)/b^4]*EllipticF[I*ArcSinh[Sqrt[-(a^2/b^2)]*x], -1] + I*Sqrt[1 - (
a^4*x^4)/b^4]*EllipticPi[-I, I*ArcSinh[Sqrt[-(a^2/b^2)]*x], -1] + I*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[I, I*Ar
cSinh[Sqrt[-(a^2/b^2)]*x], -1] - I*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[-(-1)^(1/4), I*ArcSinh[Sqrt[-(a^2/b^2)]*
x], -1] - I*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[(-1)^(1/4), I*ArcSinh[Sqrt[-(a^2/b^2)]*x], -1] - I*Sqrt[1 - (a^
4*x^4)/b^4]*EllipticPi[-(-1)^(3/4), I*ArcSinh[Sqrt[-(a^2/b^2)]*x], -1] - I*Sqrt[1 - (a^4*x^4)/b^4]*EllipticPi[
(-1)^(3/4), I*ArcSinh[Sqrt[-(a^2/b^2)]*x], -1])/(8*a^8*Sqrt[-(a^2/b^2)]*b^8*Sqrt[-b^4 + a^4*x^4])

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IntegrateAlgebraic [C]  time = 2.78, size = 612, normalized size = 2.02 \begin {gather*} -\frac {x}{8 a^8 b^8 \sqrt {-b^4+a^4 x^4}}-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) \tan ^{-1}\left (\frac {(1+i) a b x}{i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}}\right )}{a^9 b^9}+\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) \tanh ^{-1}\left (\frac {b^4-(1+i) a b^3 x-(1-i) a^3 b x^3-a^4 x^4+\left (-i b^2-(1-i) a b x-a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}{i b^4-(1-i) a b^3 x+(1+i) a^3 b x^3-i a^4 x^4+\left (b^2+(1+i) a b x-i a^2 x^2\right ) \sqrt {-b^4+a^4 x^4}}\right )}{a^9 b^9}+\frac {\text {RootSum}\left [16 a^8 b^8+32 i a^6 b^6 \text {$\#$1}^2+8 a^4 b^4 \text {$\#$1}^4-8 i a^2 b^2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-8 a^6 b^6 \log (x)+8 a^6 b^6 \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right )-4 i a^4 b^4 \log (x) \text {$\#$1}^2+4 i a^4 b^4 \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2-2 a^2 b^2 \log (x) \text {$\#$1}^4+2 a^2 b^2 \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-i \log (x) \text {$\#$1}^6+i \log \left (i b^2+a^2 x^2+\sqrt {-b^4+a^4 x^4}-x \text {$\#$1}\right ) \text {$\#$1}^6}{8 a^6 b^6 \text {$\#$1}-4 i a^4 b^4 \text {$\#$1}^3-6 a^2 b^2 \text {$\#$1}^5-i \text {$\#$1}^7}\&\right ]}{16 a^8 b^8} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^8/(Sqrt[-b^4 + a^4*x^4]*(-b^16 + a^16*x^16)),x]

[Out]

-1/8*x/(a^8*b^8*Sqrt[-b^4 + a^4*x^4]) - ((1/16 - I/16)*ArcTan[((1 + I)*a*b*x)/(I*b^2 + a^2*x^2 + Sqrt[-b^4 + a
^4*x^4])])/(a^9*b^9) + ((1/32 - I/32)*ArcTanh[(b^4 - (1 + I)*a*b^3*x - (1 - I)*a^3*b*x^3 - a^4*x^4 + ((-I)*b^2
- (1 - I)*a*b*x - a^2*x^2)*Sqrt[-b^4 + a^4*x^4])/(I*b^4 - (1 - I)*a*b^3*x + (1 + I)*a^3*b*x^3 - I*a^4*x^4 + (
b^2 + (1 + I)*a*b*x - I*a^2*x^2)*Sqrt[-b^4 + a^4*x^4])])/(a^9*b^9) + RootSum[16*a^8*b^8 + (32*I)*a^6*b^6*#1^2
+ 8*a^4*b^4*#1^4 - (8*I)*a^2*b^2*#1^6 + #1^8 & , (-8*a^6*b^6*Log[x] + 8*a^6*b^6*Log[I*b^2 + a^2*x^2 + Sqrt[-b^
4 + a^4*x^4] - x*#1] - (4*I)*a^4*b^4*Log[x]*#1^2 + (4*I)*a^4*b^4*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] -
x*#1]*#1^2 - 2*a^2*b^2*Log[x]*#1^4 + 2*a^2*b^2*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1^4 - I*Log
[x]*#1^6 + I*Log[I*b^2 + a^2*x^2 + Sqrt[-b^4 + a^4*x^4] - x*#1]*#1^6)/(8*a^6*b^6*#1 - (4*I)*a^4*b^4*#1^3 - 6*a
^2*b^2*#1^5 - I*#1^7) & ]/(16*a^8*b^8)

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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(x^8/(a^4*x^4-b^4)^(1/2)/(a^16*x^16-b^16),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 {x^{8}}{{\left (a^{16} x^{16} - b^{16}\right )} \sqrt {a^{4} x^{4} - b^{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(a^4*x^4-b^4)^(1/2)/(a^16*x^16-b^16),x, algorithm="giac")

[Out]

integrate(x^8/((a^16*x^16 - b^16)*sqrt(a^4*x^4 - b^4)), x)

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maple [B]  time = 0.26, size = 568, normalized size = 1.87

method	result	size
elliptic	\(\frac {\left (\frac {\sqrt {2}\, \ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\left (a^{4} b^{4}\right )^{\frac {1}{4}} \sqrt {a^{4} x^{4}-b^{4}}}{x}+\sqrt {a^{4} b^{4}}}\right )}{64 a^{8} b^{8} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}+1\right )}{32 a^{8} b^{8} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}}{\left (a^{4} b^{4}\right )^{\frac {1}{4}} x}-1\right )}{32 a^{8} b^{8} \left (a^{4} b^{4}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, x}{8 a^{8} b^{8} \sqrt {a^{4} x^{4}-b^{4}}}-\frac {\ln \left (\frac {\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}-\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}{\frac {a^{4} x^{4}-b^{4}}{2 x^{2}}+\frac {\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, \sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{2 x}+\frac {\sqrt {2}\, \sqrt {a^{4} b^{4}}}{2}}\right )}{16 a^{8} b^{8} \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}-\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}+1\right )}{8 a^{8} b^{8} \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}-\frac {\arctan \left (\frac {\sqrt {a^{4} x^{4}-b^{4}}\, \sqrt {2}}{\sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}\, x}-1\right )}{8 a^{8} b^{8} \sqrt {\sqrt {2}\, \sqrt {a^{4} b^{4}}}}\right ) \sqrt {2}}{2}\)	\(568\)
default	\(-\frac {\frac {a^{4} x^{3}-a^{3} b \,x^{2}+a^{2} b^{2} x -a \,b^{3}}{2 a^{2} b^{3} \sqrt {\left (x +\frac {b}{a}\right ) \left (a^{4} x^{3}-a^{3} b \,x^{2}+a^{2} b^{2} x -a \,b^{3}\right )}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}-\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}}{16 b^{7} a^{8}}+\frac {-\frac {a^{4} x^{3}+a^{3} b \,x^{2}+a^{2} b^{2} x +a \,b^{3}}{2 a^{2} b^{3} \sqrt {\left (x -\frac {b}{a}\right ) \left (a^{4} x^{3}+a^{3} b \,x^{2}+a^{2} b^{2} x +a \,b^{3}\right )}}-\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}}{16 b^{7} a^{8}}+\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{8} a^{8}+b^{8}\right )}{\sum }\frac {-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{6} a^{4}+b^{4} x^{2}\right ) a^{4}}{b^{4} \sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4} a^{4}-b^{4}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4} a^{4}-b^{4}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{7} a^{8} \sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{6} a^{6}}{b^{6}}, \frac {\sqrt {\frac {a^{2}}{b^{2}}}}{\sqrt {-\frac {a^{2}}{b^{2}}}}\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, b^{8} \sqrt {a^{4} x^{4}-b^{4}}}}{\underline {\hspace {1.25 ex}}\alpha ^{7}}}{32 a^{16}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}+b^{4}\right )}{\sum }\frac {-\frac {\sqrt {2}\, \arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+x^{2}\right ) a^{4}}{\sqrt {-2 b^{4}}\, \sqrt {a^{4} x^{4}-b^{4}}}\right )}{\sqrt {-b^{4}}}+\frac {4 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{4} \sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticPi \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}}{b^{2}}, \frac {\sqrt {\frac {a^{2}}{b^{2}}}}{\sqrt {-\frac {a^{2}}{b^{2}}}}\right )}{\sqrt {-\frac {a^{2}}{b^{2}}}\, b^{4} \sqrt {a^{4} x^{4}-b^{4}}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{64 a^{12} b^{4}}-\frac {-\frac {\left (a^{4} x^{2}-a^{2} b^{2}\right ) x}{2 b^{4} a^{2} \sqrt {\left (x^{2}+\frac {b^{2}}{a^{2}}\right ) \left (a^{4} x^{2}-a^{2} b^{2}\right )}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )}{2 b^{2} \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}+\frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}}+1}\, \sqrt {1-\frac {a^{2} x^{2}}{b^{2}}}\, \left (\EllipticF \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )-\EllipticE \left (x \sqrt {-\frac {a^{2}}{b^{2}}}, i\right )\right )}{2 b^{2} \sqrt {-\frac {a^{2}}{b^{2}}}\, \sqrt {a^{4} x^{4}-b^{4}}}}{8 a^{8} b^{6}}\)	\(1130\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(a^4*x^4-b^4)^(1/2)/(a^16*x^16-b^16),x,method=_RETURNVERBOSE)

[Out]

1/2*(1/64/a^8/b^8/(a^4*b^4)^(1/4)*2^(1/2)*ln((1/2*(a^4*x^4-b^4)/x^2-(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+(a^4
*b^4)^(1/2))/(1/2*(a^4*x^4-b^4)/x^2+(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+(a^4*b^4)^(1/2)))+1/32/a^8/b^8/(a^4*
b^4)^(1/4)*2^(1/2)*arctan(1/(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x+1)+1/32/a^8/b^8/(a^4*b^4)^(1/4)*2^(1/2)*arct
an(1/(a^4*b^4)^(1/4)*(a^4*x^4-b^4)^(1/2)/x-1)-1/8/a^8/b^8/(a^4*x^4-b^4)^(1/2)*2^(1/2)*x-1/16/a^8/b^8/(2^(1/2)*
(a^4*b^4)^(1/2))^(1/2)*ln((1/2*(a^4*x^4-b^4)/x^2-1/2*(2^(1/2)*(a^4*b^4)^(1/2))^(1/2)*(a^4*x^4-b^4)^(1/2)*2^(1/
2)/x+1/2*2^(1/2)*(a^4*b^4)^(1/2))/(1/2*(a^4*x^4-b^4)/x^2+1/2*(2^(1/2)*(a^4*b^4)^(1/2))^(1/2)*(a^4*x^4-b^4)^(1/
2)*2^(1/2)/x+1/2*2^(1/2)*(a^4*b^4)^(1/2)))-1/8/a^8/b^8/(2^(1/2)*(a^4*b^4)^(1/2))^(1/2)*arctan(1/(2^(1/2)*(a^4*
b^4)^(1/2))^(1/2)*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x+1)-1/8/a^8/b^8/(2^(1/2)*(a^4*b^4)^(1/2))^(1/2)*arctan(1/(2^(1/
2)*(a^4*b^4)^(1/2))^(1/2)*(a^4*x^4-b^4)^(1/2)*2^(1/2)/x-1))*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8}}{{\left (a^{16} x^{16} - b^{16}\right )} \sqrt {a^{4} x^{4} - b^{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(a^4*x^4-b^4)^(1/2)/(a^16*x^16-b^16),x, algorithm="maxima")

[Out]

integrate(x^8/((a^16*x^16 - b^16)*sqrt(a^4*x^4 - b^4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {x^8}{\sqrt {a^4\,x^4-b^4}\,\left (b^{16}-a^{16}\,x^{16}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^8/((a^4*x^4 - b^4)^(1/2)*(b^16 - a^16*x^16)),x)

[Out]

-int(x^8/((a^4*x^4 - b^4)^(1/2)*(b^16 - a^16*x^16)), 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(x**8/(a**4*x**4-b**4)**(1/2)/(a**16*x**16-b**16),x)

[Out]

Timed out

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