3.29.82 \(\int \frac {-b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} (b^6+a^6 x^6)} \, dx\)

Optimal. Leaf size=311 \[ -\frac {\tan ^{-1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3-b^2 x}}{a^2 x^2-2 a b x-b^2}\right )}{6 \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3-b^2 x}}{a^2 x^2-a b x-b^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\frac {a^{3/2} x^2}{2 \sqrt {b}}-\frac {b^{3/2}}{2 \sqrt {a}}+\sqrt {a} \sqrt {b} x}{\sqrt {a^2 x^3-b^2 x}}\right )}{6 \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\frac {a^{3/2} x^2}{\sqrt {2} \sqrt {b}}-\frac {b^{3/2}}{\sqrt {2} \sqrt {a}}+\frac {\sqrt {a} \sqrt {b} x}{\sqrt {2}}}{\sqrt {a^2 x^3-b^2 x}}\right )}{3 \sqrt {a} \sqrt {b}} \]

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Rubi [C]  time = 8.75, antiderivative size = 1295, normalized size of antiderivative = 4.16, number of steps used = 209, number of rules used = 20, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2056, 1586, 6715, 6725, 1729, 1209, 1201, 224, 221, 1200, 1199, 424, 1219, 1218, 1248, 735, 844, 217, 206, 725}

result too large to display

Antiderivative was successfully verified.

[In]

Int[(-b^6 + a^6*x^6)/(Sqrt[-(b^2*x) + a^2*x^3]*(b^6 + a^6*x^6)),x]

[Out]

-1/3*((-1)^(2/3)*(a - (-1)^(2/3)*(-a^6)^(1/6))*((-1)^(1/3)*a^4 - (-1)^(2/3)*a^2*(-a^6)^(1/3) - (-a^6)^(2/3))*S
qrt[b]*Sqrt[x]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticF[ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(a^(11/2)*Sqrt[-(b^2*
x) + a^2*x^3]) - ((-1)^(2/3)*(a + (-1)^(2/3)*(-a^6)^(1/6))*((-1)^(1/3)*a^4 - (-1)^(2/3)*a^2*(-a^6)^(1/3) - (-a
^6)^(2/3))*Sqrt[b]*Sqrt[x]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticF[ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(3*a^(11/
2)*Sqrt[-(b^2*x) + a^2*x^3]) - ((-1)^(1/3)*(a - (-1)^(1/3)*(-a^6)^(1/6))*((-1)^(2/3)*a^4 + ((-1)^(1/3)*a^8)/(-
a^6)^(2/3) + (-a^6)^(2/3))*Sqrt[b]*Sqrt[x]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticF[ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]]
, -1])/(3*a^(11/2)*Sqrt[-(b^2*x) + a^2*x^3]) - ((-1)^(1/3)*(a + (-1)^(1/3)*(-a^6)^(1/6))*((-1)^(2/3)*a^4 + ((-
1)^(1/3)*a^8)/(-a^6)^(2/3) + (-a^6)^(2/3))*Sqrt[b]*Sqrt[x]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticF[ArcSin[(Sqrt[a]*S
qrt[x])/Sqrt[b]], -1])/(3*a^(11/2)*Sqrt[-(b^2*x) + a^2*x^3]) + ((a - (-a^6)^(1/6))*(a^4 + a^2*(-a^6)^(1/3) + (
-a^6)^(2/3))*Sqrt[b]*Sqrt[x]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticF[ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(3*a^(1
1/2)*Sqrt[-(b^2*x) + a^2*x^3]) + ((a + (-a^6)^(1/6))*(a^4 + a^2*(-a^6)^(1/3) + (-a^6)^(2/3))*Sqrt[b]*Sqrt[x]*S
qrt[1 - (a^2*x^2)/b^2]*EllipticF[ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(3*a^(11/2)*Sqrt[-(b^2*x) + a^2*x^3])
- (2*Sqrt[b]*Sqrt[x]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticPi[a^5/(-a^6)^(5/6), ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]],
-1])/(3*Sqrt[a]*Sqrt[-(b^2*x) + a^2*x^3]) - (2*Sqrt[b]*Sqrt[x]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticPi[((-1)^(1/3)*
a^5)/(-a^6)^(5/6), ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(3*Sqrt[a]*Sqrt[-(b^2*x) + a^2*x^3]) - (2*Sqrt[b]*S
qrt[x]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticPi[((-1)^(2/3)*a^5)/(-a^6)^(5/6), ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1
])/(3*Sqrt[a]*Sqrt[-(b^2*x) + a^2*x^3]) - (2*Sqrt[b]*Sqrt[x]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticPi[(-a^6)^(1/6)/a
, ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(3*Sqrt[a]*Sqrt[-(b^2*x) + a^2*x^3]) - (2*Sqrt[b]*Sqrt[x]*Sqrt[1 - (
a^2*x^2)/b^2]*EllipticPi[((-1)^(1/3)*(-a^6)^(1/6))/a, ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(3*Sqrt[a]*Sqrt[
-(b^2*x) + a^2*x^3]) - (2*Sqrt[b]*Sqrt[x]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticPi[((-1)^(2/3)*(-a^6)^(1/6))/a, ArcS
in[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(3*Sqrt[a]*Sqrt[-(b^2*x) + a^2*x^3])

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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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 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 725

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

Rule 735

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(
e*(m + 2*p + 1)), x] + Dist[(2*p)/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[a*e - c*d*x, x]*(a + c*x^2)^(p - 1),
x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !Ration
alQ[m] || LtQ[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1199

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

Rule 1200

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

Rule 1201

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

Rule 1209

Int[((a_) + (c_.)*(x_)^4)^(p_)/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Dist[(e^2)^(-1), Int[(c*d - c*e*x^2)*(a +
c*x^4)^(p - 1), x], x] + Dist[(c*d^2 + a*e^2)/e^2, Int[(a + c*x^4)^(p - 1)/(d + e*x^2), x], x] /; FreeQ[{a, c,
d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p + 1/2, 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 1248

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 1729

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

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 {-b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \int \frac {-b^6+a^6 x^6}{\sqrt {x} \sqrt {-b^2+a^2 x^2} \left (b^6+a^6 x^6\right )} \, dx}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \int \frac {\sqrt {-b^2+a^2 x^2} \left (b^4+a^2 b^2 x^2+a^4 x^4\right )}{\sqrt {x} \left (b^6+a^6 x^6\right )} \, dx}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4} \left (b^4+a^2 b^2 x^4+a^4 x^8\right )}{b^6+a^6 x^{12}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {\left (b^{9/2}+\frac {a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+\frac {a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-\sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}+\frac {a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+\frac {a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-i \sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}+\frac {a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+\frac {a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+i \sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}+\frac {a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+\frac {a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+\sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}+\frac {(-1)^{2/3} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}-\frac {\sqrt [3]{-1} a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-\sqrt [6]{-1} \sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}+\frac {(-1)^{2/3} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}-\frac {\sqrt [3]{-1} a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+\sqrt [6]{-1} \sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}-\frac {\sqrt [3]{-1} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+\frac {(-1)^{2/3} a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-\sqrt [3]{-1} \sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}-\frac {\sqrt [3]{-1} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+\frac {(-1)^{2/3} a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+\sqrt [3]{-1} \sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}+\frac {(-1)^{2/3} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}-\frac {\sqrt [3]{-1} a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-(-1)^{2/3} \sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}+\frac {(-1)^{2/3} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}-\frac {\sqrt [3]{-1} a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+(-1)^{2/3} \sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}-\frac {\sqrt [3]{-1} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+\frac {(-1)^{2/3} a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-(-1)^{5/6} \sqrt [12]{-a^6} x\right )}+\frac {\left (b^{9/2}-\frac {\sqrt [3]{-1} a^4 b^{9/2}}{\left (-a^6\right )^{2/3}}+\frac {(-1)^{2/3} a^2 b^{9/2}}{\sqrt [3]{-a^6}}\right ) \sqrt {-b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+(-1)^{5/6} \sqrt [12]{-a^6} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}-\sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}-i \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}+i \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}+\sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}-\sqrt [3]{-1} \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b^{3/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}+\sqrt [3]{-1} \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b^{3/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}-(-1)^{5/6} \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b^{3/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}+(-1)^{5/6} \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b^{3/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}-\sqrt [6]{-1} \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b^{3/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}+\sqrt [6]{-1} \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b^{3/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}-(-1)^{2/3} \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b^{3/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{\sqrt {b}+(-1)^{2/3} \sqrt [12]{-a^6} x} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b^{3/2} \sqrt {-b^2 x+a^2 x^3}}\\ &=2 \frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{b-\sqrt [6]{-a^6} x^2} \, dx,x,\sqrt {x}\right )}{6 b \sqrt {-b^2 x+a^2 x^3}}+2 \frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{b+\sqrt [6]{-a^6} x^2} \, dx,x,\sqrt {x}\right )}{6 b \sqrt {-b^2 x+a^2 x^3}}-2 \frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{b-(-1)^{2/3} \sqrt [6]{-a^6} x^2} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b \sqrt {-b^2 x+a^2 x^3}}-2 \frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{b+(-1)^{2/3} \sqrt [6]{-a^6} x^2} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b \sqrt {-b^2 x+a^2 x^3}}+2 \frac {\left (\left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{b-\sqrt [3]{-1} \sqrt [6]{-a^6} x^2} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b \sqrt {-b^2 x+a^2 x^3}}+2 \frac {\left (\left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-b^2+a^2 x^4}}{b+\sqrt [3]{-1} \sqrt [6]{-a^6} x^2} \, dx,x,\sqrt {x}\right )}{6 \left (-a^6\right )^{2/3} b \sqrt {-b^2 x+a^2 x^3}}\\ &=2 \left (-\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {a^2 b+a^2 \sqrt [6]{-a^6} x^2}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [3]{-a^6} b \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \left (a^4+\left (-a^6\right )^{2/3}\right ) b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b-\sqrt [6]{-a^6} x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^4 \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {a^2 b-a^2 \sqrt [6]{-a^6} x^2}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [3]{-a^6} b \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \left (a^4+\left (-a^6\right )^{2/3}\right ) b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+\sqrt [6]{-a^6} x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^4 \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (\frac {\left ((-1)^{2/3} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {a^2 b+(-1)^{2/3} a^2 \sqrt [6]{-a^6} x^2}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^6 b \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \left (a^4+(-1)^{2/3} \left (-a^6\right )^{2/3}\right ) b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b-(-1)^{2/3} \sqrt [6]{-a^6} x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^4 \left (-a^6\right )^{2/3} \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (\frac {\left ((-1)^{2/3} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {a^2 b-(-1)^{2/3} a^2 \sqrt [6]{-a^6} x^2}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^6 b \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \left (a^4+(-1)^{2/3} \left (-a^6\right )^{2/3}\right ) b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+(-1)^{2/3} \sqrt [6]{-a^6} x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^4 \left (-a^6\right )^{2/3} \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\left (\sqrt [3]{-1} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {a^2 b+\sqrt [3]{-1} a^2 \sqrt [6]{-a^6} x^2}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^6 b \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [3]{-1} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \left (a^2 b^2-(-1)^{2/3} \sqrt [3]{-a^6} b^2\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b-\sqrt [3]{-1} \sqrt [6]{-a^6} x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^6 b \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\left (\sqrt [3]{-1} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {a^2 b-\sqrt [3]{-1} a^2 \sqrt [6]{-a^6} x^2}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^6 b \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [3]{-1} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \left (a^2 b^2-(-1)^{2/3} \sqrt [3]{-a^6} b^2\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+\sqrt [3]{-1} \sqrt [6]{-a^6} x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^6 b \sqrt {-b^2 x+a^2 x^3}}\right )\\ &=2 \left (-\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [6]{-a^6} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \left (a-\sqrt [6]{-a^6}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [3]{-a^6} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \left (a^4+\left (-a^6\right )^{2/3}\right ) b \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b-\sqrt [6]{-a^6} x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^4 \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [6]{-a^6} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \left (a+\sqrt [6]{-a^6}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [3]{-a^6} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \left (a^4+\left (-a^6\right )^{2/3}\right ) b \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+\sqrt [6]{-a^6} x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^4 \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (-\frac {\left (\sqrt [3]{-1} \sqrt [6]{-a^6} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left ((-1)^{2/3} \left (a-(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \left (a^4+(-1)^{2/3} \left (-a^6\right )^{2/3}\right ) b \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b-(-1)^{2/3} \sqrt [6]{-a^6} x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^4 \left (-a^6\right )^{2/3} \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (\frac {\left (\sqrt [3]{-1} \sqrt [6]{-a^6} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left ((-1)^{2/3} \left (a+(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \left (a^4+(-1)^{2/3} \left (-a^6\right )^{2/3}\right ) b \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+(-1)^{2/3} \sqrt [6]{-a^6} x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^4 \left (-a^6\right )^{2/3} \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\left ((-1)^{2/3} \sqrt [6]{-a^6} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [3]{-1} \left (a-\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [3]{-1} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \left (a^2 b^2-(-1)^{2/3} \sqrt [3]{-a^6} b^2\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b-\sqrt [3]{-1} \sqrt [6]{-a^6} x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^6 b \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (\frac {\left ((-1)^{2/3} \sqrt [6]{-a^6} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [3]{-1} \left (a+\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [3]{-1} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \left (a^2 b^2-(-1)^{2/3} \sqrt [3]{-a^6} b^2\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+\sqrt [3]{-1} \sqrt [6]{-a^6} x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^6 b \sqrt {-b^2 x+a^2 x^3}}\right )\\ &=2 \left (-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [6]{-a^6}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [6]{-a^6} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \left (a-\sqrt [6]{-a^6}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [3]{-a^6} \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {a^5}{\left (-a^6\right )^{5/6}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [6]{-a^6} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \left (a+\sqrt [6]{-a^6}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [3]{-a^6} \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {(-1)^{2/3} \sqrt [6]{-a^6}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [3]{-1} \sqrt [6]{-a^6} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left ((-1)^{2/3} \left (a-(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {(-1)^{2/3} a^5}{\left (-a^6\right )^{5/6}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [3]{-1} \sqrt [6]{-a^6} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left ((-1)^{2/3} \left (a+(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [3]{-1} \sqrt [6]{-a^6}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left ((-1)^{2/3} \sqrt [6]{-a^6} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [3]{-1} \left (a-\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [3]{-1} a^5}{\left (-a^6\right )^{5/6}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left ((-1)^{2/3} \sqrt [6]{-a^6} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {a x^2}{b}}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [3]{-1} \left (a+\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}\right )\\ &=2 \left (\frac {\left (a-\sqrt [6]{-a^6}\right ) \left (a^4+a^2 \sqrt [3]{-a^6}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [6]{-a^6}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {a x^2}{b}}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [6]{-a^6} \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (\frac {\left (a+\sqrt [6]{-a^6}\right ) \left (a^4+a^2 \sqrt [3]{-a^6}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {a^5}{\left (-a^6\right )^{5/6}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (a \left (1+\frac {a^4}{\left (-a^6\right )^{2/3}}+\frac {a^2}{\sqrt [3]{-a^6}}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {a x^2}{b}}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{6 \sqrt [6]{-a^6} \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (\frac {(-1)^{2/3} \left (a-(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {(-1)^{2/3} \sqrt [6]{-a^6}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [3]{-1} \sqrt [6]{-a^6} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {a x^2}{b}}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (\frac {(-1)^{2/3} \left (a+(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {(-1)^{2/3} a^5}{\left (-a^6\right )^{5/6}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [3]{-1} \sqrt [6]{-a^6} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {a x^2}{b}}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\sqrt [3]{-1} \left (a-\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [3]{-1} \sqrt [6]{-a^6}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left ((-1)^{2/3} \sqrt [6]{-a^6} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {a x^2}{b}}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\sqrt [3]{-1} \left (a+\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [3]{-1} a^5}{\left (-a^6\right )^{5/6}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left ((-1)^{2/3} \sqrt [6]{-a^6} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {a x^2}{b}}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{6 a^5 \sqrt {-b^2 x+a^2 x^3}}\right )\\ &=2 \left (\frac {\sqrt {a} \left (a^4+a^2 \sqrt [3]{-a^6}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 \left (-a^6\right )^{5/6} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (a+\sqrt [6]{-a^6}\right ) \left (a^4+a^2 \sqrt [3]{-a^6}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {a^5}{\left (-a^6\right )^{5/6}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (\frac {(-1)^{2/3} \sqrt [6]{-a^6} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt [3]{-1} \left (a+\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [3]{-1} a^5}{\left (-a^6\right )^{5/6}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (-\frac {\sqrt [3]{-1} \sqrt {a} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 \left (-a^6\right )^{5/6} \sqrt {-b^2 x+a^2 x^3}}+\frac {(-1)^{2/3} \left (a+(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {(-1)^{2/3} a^5}{\left (-a^6\right )^{5/6}};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {\sqrt {a} \left (a^4+a^2 \sqrt [3]{-a^6}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 \left (-a^6\right )^{5/6} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (a-\sqrt [6]{-a^6}\right ) \left (a^4+a^2 \sqrt [3]{-a^6}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [6]{-a^6}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}\right )+2 \left (-\frac {(-1)^{2/3} \sqrt [6]{-a^6} \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt [3]{-1} \left (a-\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left ((-1)^{2/3} a^4+\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [3]{-1} \sqrt [6]{-a^6}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}\right )-2 \left (\frac {\sqrt [3]{-1} \sqrt {a} \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 \left (-a^6\right )^{5/6} \sqrt {-b^2 x+a^2 x^3}}+\frac {(-1)^{2/3} \left (a-(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} a^2 \sqrt [3]{-a^6}-\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{6 a^{11/2} \sqrt {-b^2 x+a^2 x^3}}+\frac {\sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {(-1)^{2/3} \sqrt [6]{-a^6}}{a};\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}\right )\\ \end {align*}

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Mathematica [C]  time = 3.06, size = 294, normalized size = 0.95 \begin {gather*} -\frac {2 i x^{3/2} \sqrt {1-\frac {b^2}{a^2 x^2}} \left (3 F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-i;\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (i;\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-\frac {i}{2}-\frac {\sqrt {3}}{2};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {i}{2}-\frac {\sqrt {3}}{2};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {1}{2} \left (-i+\sqrt {3}\right );\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {1}{2} \left (i+\sqrt {3}\right );\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )\right )}{3 \sqrt {-\frac {b}{a}} \sqrt {a^2 x^3-b^2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-b^6 + a^6*x^6)/(Sqrt[-(b^2*x) + a^2*x^3]*(b^6 + a^6*x^6)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.83, size = 311, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}}{-b^2-2 a b x+a^2 x^2}\right )}{6 \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}}{-b^2-a b x+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {-\frac {b^{3/2}}{2 \sqrt {a}}+\sqrt {a} \sqrt {b} x+\frac {a^{3/2} x^2}{2 \sqrt {b}}}{\sqrt {-b^2 x+a^2 x^3}}\right )}{6 \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {-\frac {b^{3/2}}{\sqrt {2} \sqrt {a}}+\frac {\sqrt {a} \sqrt {b} x}{\sqrt {2}}+\frac {a^{3/2} x^2}{\sqrt {2} \sqrt {b}}}{\sqrt {-b^2 x+a^2 x^3}}\right )}{3 \sqrt {a} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-b^6 + a^6*x^6)/(Sqrt[-(b^2*x) + a^2*x^3]*(b^6 + a^6*x^6)),x]

[Out]

-1/6*ArcTan[(2*Sqrt[a]*Sqrt[b]*Sqrt[-(b^2*x) + a^2*x^3])/(-b^2 - 2*a*b*x + a^2*x^2)]/(Sqrt[a]*Sqrt[b]) - (Sqrt
[2]*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[-(b^2*x) + a^2*x^3])/(-b^2 - a*b*x + a^2*x^2)])/(3*Sqrt[a]*Sqrt[b]) -
ArcTanh[(-1/2*b^(3/2)/Sqrt[a] + Sqrt[a]*Sqrt[b]*x + (a^(3/2)*x^2)/(2*Sqrt[b]))/Sqrt[-(b^2*x) + a^2*x^3]]/(6*S
qrt[a]*Sqrt[b]) - (Sqrt[2]*ArcTanh[(-(b^(3/2)/(Sqrt[2]*Sqrt[a])) + (Sqrt[a]*Sqrt[b]*x)/Sqrt[2] + (a^(3/2)*x^2)
/(Sqrt[2]*Sqrt[b]))/Sqrt[-(b^2*x) + a^2*x^3]])/(3*Sqrt[a]*Sqrt[b])

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fricas [B]  time = 1.10, size = 2045, normalized size = 6.58

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^6*x^6-b^6)/(a^2*x^3-b^2*x)^(1/2)/(a^6*x^6+b^6),x, algorithm="fricas")

[Out]

-1/6*sqrt(2)*(1/4)^(1/4)*(1/(a^2*b^2))^(1/4)*arctan(1/2*((4*sqrt(2)*(1/4)^(3/4)*a^2*b^2*x*(1/(a^2*b^2))^(3/4)
- sqrt(2)*(1/4)^(1/4)*(a^2*x^2 - b^2)*(1/(a^2*b^2))^(1/4))*sqrt(a^2*x^3 - b^2*x) - (2*a^2*x^3 - 2*b^2*x - (4*s
qrt(2)*(1/4)^(3/4)*a^2*b^2*x*(1/(a^2*b^2))^(3/4) + sqrt(2)*(1/4)^(1/4)*(a^2*x^2 - b^2)*(1/(a^2*b^2))^(1/4))*sq
rt(a^2*x^3 - b^2*x))*sqrt((a^4*x^4 + 2*a^2*b^2*x^2 + b^4 + 8*(sqrt(2)*(1/4)^(1/4)*a^2*b^2*x*(1/(a^2*b^2))^(1/4
) + sqrt(2)*(1/4)^(3/4)*(a^4*b^2*x^2 - a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 - b^2*x) + 8*(a^4*b^2*x^3 -
a^2*b^4*x)*sqrt(1/(a^2*b^2)))/(a^4*x^4 + 2*a^2*b^2*x^2 + b^4)))/(a^2*x^3 - b^2*x)) - 1/6*sqrt(2)*(1/4)^(1/4)*(
1/(a^2*b^2))^(1/4)*arctan(1/2*((4*sqrt(2)*(1/4)^(3/4)*a^2*b^2*x*(1/(a^2*b^2))^(3/4) - sqrt(2)*(1/4)^(1/4)*(a^2
*x^2 - b^2)*(1/(a^2*b^2))^(1/4))*sqrt(a^2*x^3 - b^2*x) + (2*a^2*x^3 - 2*b^2*x + (4*sqrt(2)*(1/4)^(3/4)*a^2*b^2
*x*(1/(a^2*b^2))^(3/4) + sqrt(2)*(1/4)^(1/4)*(a^2*x^2 - b^2)*(1/(a^2*b^2))^(1/4))*sqrt(a^2*x^3 - b^2*x))*sqrt(
(a^4*x^4 + 2*a^2*b^2*x^2 + b^4 - 8*(sqrt(2)*(1/4)^(1/4)*a^2*b^2*x*(1/(a^2*b^2))^(1/4) + sqrt(2)*(1/4)^(3/4)*(a
^4*b^2*x^2 - a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 - b^2*x) + 8*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(1/(a^2*b^2
)))/(a^4*x^4 + 2*a^2*b^2*x^2 + b^4)))/(a^2*x^3 - b^2*x)) - 1/24*sqrt(2)*(1/4)^(1/4)*(1/(a^2*b^2))^(1/4)*log((a
^4*x^4 + 2*a^2*b^2*x^2 + b^4 + 8*(sqrt(2)*(1/4)^(1/4)*a^2*b^2*x*(1/(a^2*b^2))^(1/4) + sqrt(2)*(1/4)^(3/4)*(a^4
*b^2*x^2 - a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 - b^2*x) + 8*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(1/(a^2*b^2))
)/(a^4*x^4 + 2*a^2*b^2*x^2 + b^4)) + 1/24*sqrt(2)*(1/4)^(1/4)*(1/(a^2*b^2))^(1/4)*log((a^4*x^4 + 2*a^2*b^2*x^2
+ b^4 - 8*(sqrt(2)*(1/4)^(1/4)*a^2*b^2*x*(1/(a^2*b^2))^(1/4) + sqrt(2)*(1/4)^(3/4)*(a^4*b^2*x^2 - a^2*b^4)*(1
/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 - b^2*x) + 8*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(1/(a^2*b^2)))/(a^4*x^4 + 2*a^2*b^2
*x^2 + b^4)) - 1/3*sqrt(2)*(1/(a^2*b^2))^(1/4)*arctan(1/2*((sqrt(2)*a^2*b^2*x*(1/(a^2*b^2))^(3/4) - sqrt(2)*(a
^2*x^2 - b^2)*(1/(a^2*b^2))^(1/4))*sqrt(a^2*x^3 - b^2*x) - (2*a^2*x^3 - 2*b^2*x - (sqrt(2)*a^2*b^2*x*(1/(a^2*b
^2))^(3/4) + sqrt(2)*(a^2*x^2 - b^2)*(1/(a^2*b^2))^(1/4))*sqrt(a^2*x^3 - b^2*x))*sqrt((a^4*x^4 - a^2*b^2*x^2 +
b^4 + 2*(sqrt(2)*a^2*b^2*x*(1/(a^2*b^2))^(1/4) + sqrt(2)*(a^4*b^2*x^2 - a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^
2*x^3 - b^2*x) + 4*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(1/(a^2*b^2)))/(a^4*x^4 - a^2*b^2*x^2 + b^4)))/(a^2*x^3 - b^2
*x)) - 1/3*sqrt(2)*(1/(a^2*b^2))^(1/4)*arctan(1/2*((sqrt(2)*a^2*b^2*x*(1/(a^2*b^2))^(3/4) - sqrt(2)*(a^2*x^2 -
b^2)*(1/(a^2*b^2))^(1/4))*sqrt(a^2*x^3 - b^2*x) + (2*a^2*x^3 - 2*b^2*x + (sqrt(2)*a^2*b^2*x*(1/(a^2*b^2))^(3/
4) + sqrt(2)*(a^2*x^2 - b^2)*(1/(a^2*b^2))^(1/4))*sqrt(a^2*x^3 - b^2*x))*sqrt((a^4*x^4 - a^2*b^2*x^2 + b^4 - 2
*(sqrt(2)*a^2*b^2*x*(1/(a^2*b^2))^(1/4) + sqrt(2)*(a^4*b^2*x^2 - a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 -
b^2*x) + 4*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(1/(a^2*b^2)))/(a^4*x^4 - a^2*b^2*x^2 + b^4)))/(a^2*x^3 - b^2*x)) - 1
/12*sqrt(2)*(1/(a^2*b^2))^(1/4)*log((a^4*x^4 - a^2*b^2*x^2 + b^4 + 2*(sqrt(2)*a^2*b^2*x*(1/(a^2*b^2))^(1/4) +
sqrt(2)*(a^4*b^2*x^2 - a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 - b^2*x) + 4*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(
1/(a^2*b^2)))/(a^4*x^4 - a^2*b^2*x^2 + b^4)) + 1/12*sqrt(2)*(1/(a^2*b^2))^(1/4)*log((a^4*x^4 - a^2*b^2*x^2 + b
^4 - 2*(sqrt(2)*a^2*b^2*x*(1/(a^2*b^2))^(1/4) + sqrt(2)*(a^4*b^2*x^2 - a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^2*
x^3 - b^2*x) + 4*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(1/(a^2*b^2)))/(a^4*x^4 - a^2*b^2*x^2 + b^4))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{6} x^{6} - b^{6}}{{\left (a^{6} x^{6} + b^{6}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^6*x^6-b^6)/(a^2*x^3-b^2*x)^(1/2)/(a^6*x^6+b^6),x, algorithm="giac")

[Out]

integrate((a^6*x^6 - b^6)/((a^6*x^6 + b^6)*sqrt(a^2*x^3 - b^2*x)), x)

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maple [C]  time = 0.28, size = 475, normalized size = 1.53

method	result	size
default	\(\frac {b \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {2 \left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}-b^{2} x}}-\frac {2 b^{2} \left (-\frac {i \sqrt {1+\frac {a x}{b}}\, \sqrt {2-\frac {2 a x}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, -\frac {b}{a \left (-\frac {i b}{a}-\frac {b}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a^{2} x^{3}-b^{2} x}\, \left (-\frac {i b}{a}-\frac {b}{a}\right )}+\frac {i \sqrt {1+\frac {a x}{b}}\, \sqrt {2-\frac {2 a x}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, -\frac {b}{a \left (-\frac {b}{a}+\frac {i b}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a^{2} x^{3}-b^{2} x}\, \left (-\frac {b}{a}+\frac {i b}{a}\right )}\right )}{3}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}-\textit {\_Z}^{2} a^{2} b^{2}+b^{4}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+2 b^{2}\right ) \underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha a -b \right ) \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {\left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, -\frac {a^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha a -b \right )}{b^{3}}, \frac {\sqrt {2}}{2}\right )}{\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}\right ) \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}\right )}{3 b}\)	\(475\)
elliptic	\(\frac {b \sqrt {1+\frac {a x}{b}}\, \sqrt {2-\frac {2 a x}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}-b^{2} x}}+\frac {b \sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{2} \textit {\_Z}^{2}+b^{2}\right )}{\sum }\EllipticPi \left (\sqrt {\frac {a x +b}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a -b}{2 b}, \frac {\sqrt {2}}{2}\right )\right )}{6 a \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}-\frac {b^{2} \sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{2} \textit {\_Z}^{2}+b^{2}\right )}{\sum }\frac {\EllipticPi \left (\sqrt {\frac {a x +b}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a -b}{2 b}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha }\right )}{6 a^{2} \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}-\frac {\sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, a^{3} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}-\textit {\_Z}^{2} a^{2} b^{2}+b^{4}\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{4} \EllipticPi \left (\sqrt {\frac {a x +b}{b}}, -\frac {a^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha a -b \right )}{b^{3}}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}}\right )}{3 b \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}+\frac {\sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, a^{2} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}-\textit {\_Z}^{2} a^{2} b^{2}+b^{4}\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{3} \EllipticPi \left (\sqrt {\frac {a x +b}{b}}, -\frac {a^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha a -b \right )}{b^{3}}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}}\right )}{3 \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}+\frac {2 b \sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, a \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}-\textit {\_Z}^{2} a^{2} b^{2}+b^{4}\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \EllipticPi \left (\sqrt {\frac {a x +b}{b}}, -\frac {a^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha a -b \right )}{b^{3}}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}}\right )}{3 \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}-\frac {2 b^{2} \sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4} a^{4}-\textit {\_Z}^{2} a^{2} b^{2}+b^{4}\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \EllipticPi \left (\sqrt {\frac {a x +b}{b}}, -\frac {a^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha a -b \right )}{b^{3}}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-b^{2}}\right )}{3 \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}\)	\(847\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^6*x^6-b^6)/(a^2*x^3-b^2*x)^(1/2)/(a^6*x^6+b^6),x,method=_RETURNVERBOSE)

[Out]

b/a*((x+b/a)/b*a)^(1/2)*(-2*(x-b/a)/b*a)^(1/2)*(-a*x/b)^(1/2)/(a^2*x^3-b^2*x)^(1/2)*EllipticF(((x+b/a)/b*a)^(1
/2),1/2*2^(1/2))-2/3*b^2*(-1/2*I/a^2*(1+a*x/b)^(1/2)*(2-2*a*x/b)^(1/2)*(-a*x/b)^(1/2)/(a^2*x^3-b^2*x)^(1/2)/(-
I*b/a-b/a)*EllipticPi(((x+b/a)/b*a)^(1/2),-b/a/(-I*b/a-b/a),1/2*2^(1/2))+1/2*I/a^2*(1+a*x/b)^(1/2)*(2-2*a*x/b)
^(1/2)*(-a*x/b)^(1/2)/(a^2*x^3-b^2*x)^(1/2)/(-b/a+I*b/a)*EllipticPi(((x+b/a)/b*a)^(1/2),-b/a/(-b/a+I*b/a),1/2*
2^(1/2)))-1/3/b*2^(1/2)*sum((-_alpha^2*a^2+2*b^2)*_alpha/(2*_alpha^2*a^2-b^2)*(_alpha*a-b)*((x+b/a)/b*a)^(1/2)
*(-(x-b/a)/b*a)^(1/2)*(-a*x/b)^(1/2)/(x*(a^2*x^2-b^2))^(1/2)*EllipticPi(((x+b/a)/b*a)^(1/2),-a^2*_alpha^2*(_al
pha*a-b)/b^3,1/2*2^(1/2)),_alpha=RootOf(_Z^4*a^4-_Z^2*a^2*b^2+b^4))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{6} x^{6} - b^{6}}{{\left (a^{6} x^{6} + b^{6}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^6*x^6-b^6)/(a^2*x^3-b^2*x)^(1/2)/(a^6*x^6+b^6),x, algorithm="maxima")

[Out]

integrate((a^6*x^6 - b^6)/((a^6*x^6 + b^6)*sqrt(a^2*x^3 - b^2*x)), x)

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(b^6 - a^6*x^6)/((b^6 + a^6*x^6)*(a^2*x^3 - b^2*x)^(1/2)),x)

[Out]

\text{Hanged}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**6*x**6-b**6)/(a**2*x**3-b**2*x)**(1/2)/(a**6*x**6+b**6),x)

[Out]

Integral((a*x - b)*(a*x + b)*(a**2*x**2 - a*b*x + b**2)*(a**2*x**2 + a*b*x + b**2)/(sqrt(x*(a*x - b)*(a*x + b)
)*(a**2*x**2 + b**2)*(a**4*x**4 - a**2*b**2*x**2 + b**4)), x)

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