\(\int \frac {-b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} (b^6+a^6 x^6)} \, dx\) [2882]

Optimal result

Integrand size = 45, antiderivative size = 311 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=-\frac {\arctan \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}}{-b^2-2 a b x+a^2 x^2}\right )}{6 \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}}{-b^2-a b x+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {-\frac {b^{3/2}}{2 \sqrt {a}}+\sqrt {a} \sqrt {b} x+\frac {a^{3/2} x^2}{2 \sqrt {b}}}{\sqrt {-b^2 x+a^2 x^3}}\right )}{6 \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {-\frac {b^{3/2}}{\sqrt {2} \sqrt {a}}+\frac {\sqrt {a} \sqrt {b} x}{\sqrt {2}}+\frac {a^{3/2} x^2}{\sqrt {2} \sqrt {b}}}{\sqrt {-b^2 x+a^2 x^3}}\right )}{3 \sqrt {a} \sqrt {b}} \]

[Out]

-1/6*arctan(2*a^(1/2)*b^(1/2)*(a^2*x^3-b^2*x)^(1/2)/(a^2*x^2-2*a*b*x-b^2))/a^(1/2)/b^(1/2)-1/3*2^(1/2)*arctan(
2^(1/2)*a^(1/2)*b^(1/2)*(a^2*x^3-b^2*x)^(1/2)/(a^2*x^2-a*b*x-b^2))/a^(1/2)/b^(1/2)-1/6*arctanh((-1/2*b^(3/2)/a
^(1/2)+a^(1/2)*b^(1/2)*x+1/2*a^(3/2)*x^2/b^(1/2))/(a^2*x^3-b^2*x)^(1/2))/a^(1/2)/b^(1/2)-1/3*2^(1/2)*arctanh((
-1/2*b^(3/2)*2^(1/2)/a^(1/2)+1/2*a^(1/2)*b^(1/2)*x*2^(1/2)+1/2*a^(3/2)*x^2*2^(1/2)/b^(1/2))/(a^2*x^3-b^2*x)^(1
/2))/a^(1/2)/b^(1/2)

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 5.58 (sec) , antiderivative size = 1295, normalized size of antiderivative = 4.16, number of steps used = 209, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2081, 1600, 6847, 6857, 1743, 1223, 1215, 230, 227, 1214, 1213, 435, 1233, 1232, 1262, 749, 858, 223, 212, 739} \[ \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 {(-1)^{2/3} \left (a-(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} \sqrt [3]{-a^6} a^2-\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 a^{11/2} \sqrt {a^2 x^3-b^2 x}}-\frac {(-1)^{2/3} \left (a+(-1)^{2/3} \sqrt [6]{-a^6}\right ) \left (\sqrt [3]{-1} a^4-(-1)^{2/3} \sqrt [3]{-a^6} a^2-\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 a^{11/2} \sqrt {a^2 x^3-b^2 x}}-\frac {\sqrt [3]{-1} \left (a-\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left (\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+(-1)^{2/3} a^4+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 a^{11/2} \sqrt {a^2 x^3-b^2 x}}-\frac {\sqrt [3]{-1} \left (a+\sqrt [3]{-1} \sqrt [6]{-a^6}\right ) \left (\frac {\sqrt [3]{-1} a^8}{\left (-a^6\right )^{2/3}}+(-1)^{2/3} a^4+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 a^{11/2} \sqrt {a^2 x^3-b^2 x}}+\frac {\left (a-\sqrt [6]{-a^6}\right ) \left (a^4+\sqrt [3]{-a^6} a^2+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 a^{11/2} \sqrt {a^2 x^3-b^2 x}}+\frac {\left (a+\sqrt [6]{-a^6}\right ) \left (a^4+\sqrt [3]{-a^6} a^2+\left (-a^6\right )^{2/3}\right ) \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 a^{11/2} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {a^5}{\left (-a^6\right )^{5/6}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {\sqrt [3]{-1} a^5}{\left (-a^6\right )^{5/6}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {(-1)^{2/3} a^5}{\left (-a^6\right )^{5/6}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {\sqrt [6]{-a^6}}{a},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {\sqrt [3]{-1} \sqrt [6]{-a^6}}{a},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {(-1)^{2/3} \sqrt [6]{-a^6}}{a},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}} \]

[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 212

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

Rule 223

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 227

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 230

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 435

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

Rule 739

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 749

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 858

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 1213

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 1214

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 1215

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-c/a, 2]}, Dist[(d*q - e)/q, In
t[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 1223

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 1232

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

Rule 1233

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 1262

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 1600

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 1743

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 2081

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 6847

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 6857

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*} \text {integral}& = \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 ) \text {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 ) \text {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}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.99 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.76 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\frac {\left (\frac {1}{6}+\frac {i}{6}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2} \left (i \arctan \left (\frac {(1+i) \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {-b^2+a^2 x^2}}\right )+(2-2 i) (-1)^{3/4} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {-b^2+a^2 x^2}}\right )+(2-2 i) \sqrt [4]{-1} \arctan \left (\frac {(-1)^{3/4} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {-b^2+a^2 x^2}}\right )+\arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-b^2+a^2 x^2}}{\sqrt {a} \sqrt {b} \sqrt {x}}\right )\right )}{\sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}} \]

[In]

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

[Out]

((1/6 + I/6)*Sqrt[x]*Sqrt[-b^2 + a^2*x^2]*(I*ArcTan[((1 + I)*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[-b^2 + a^2*x^2]] +
(2 - 2*I)*(-1)^(3/4)*ArcTan[((-1)^(1/4)*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[-b^2 + a^2*x^2]] + (2 - 2*I)*(-1)^(1/4)*
ArcTan[((-1)^(3/4)*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[-b^2 + a^2*x^2]] + ArcTan[((1/2 + I/2)*Sqrt[-b^2 + a^2*x^2])/
(Sqrt[a]*Sqrt[b]*Sqrt[x])]))/(Sqrt[a]*Sqrt[b]*Sqrt[-(b^2*x) + a^2*x^3])

Maple [A] (verified)

Time = 2.50 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.38

method	result	size
default	\(-\frac {\left (4 \arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x -\sqrt {2}\, \sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )-2 \ln \left (\frac {a^{2} x^{2}-\left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}+\sqrt {a^{2} b^{2}}\, x -b^{2}}{a^{2} x^{2}+\left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}+\sqrt {a^{2} b^{2}}\, x -b^{2}}\right )-4 \arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x +\sqrt {2}\, \sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )\right ) \sqrt {2}-2 \arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x +\sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )-\ln \left (\frac {a^{2} x^{2}+2 \sqrt {a^{2} b^{2}}\, x -b^{2}-2 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}{a^{2} x^{2}+2 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}+2 \sqrt {a^{2} b^{2}}\, x -b^{2}}\right )+2 \arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x -\sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )}{12 \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\)	\(428\)
pseudoelliptic	\(-\frac {\left (4 \arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x -\sqrt {2}\, \sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )-2 \ln \left (\frac {a^{2} x^{2}-\left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}+\sqrt {a^{2} b^{2}}\, x -b^{2}}{a^{2} x^{2}+\left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}+\sqrt {a^{2} b^{2}}\, x -b^{2}}\right )-4 \arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x +\sqrt {2}\, \sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )\right ) \sqrt {2}-2 \arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x +\sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )-\ln \left (\frac {a^{2} x^{2}+2 \sqrt {a^{2} b^{2}}\, x -b^{2}-2 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}{a^{2} x^{2}+2 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}+2 \sqrt {a^{2} b^{2}}\, x -b^{2}}\right )+2 \arctan \left (\frac {\left (a^{2} b^{2}\right )^{\frac {1}{4}} x -\sqrt {a^{2} x^{3}-b^{2} x}}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )}{12 \left (a^{2} b^{2}\right )^{\frac {1}{4}}}\)	\(428\)
elliptic	\(\text {Expression too large to display}\)	\(847\)

[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]

-1/12*((4*arctan(((a^2*b^2)^(1/4)*x-2^(1/2)*(a^2*x^3-b^2*x)^(1/2))/(a^2*b^2)^(1/4)/x)-2*ln((a^2*x^2-(a^2*b^2)^
(1/4)*(a^2*x^3-b^2*x)^(1/2)*2^(1/2)+(a^2*b^2)^(1/2)*x-b^2)/(a^2*x^2+(a^2*b^2)^(1/4)*(a^2*x^3-b^2*x)^(1/2)*2^(1
/2)+(a^2*b^2)^(1/2)*x-b^2))-4*arctan(((a^2*b^2)^(1/4)*x+2^(1/2)*(a^2*x^3-b^2*x)^(1/2))/(a^2*b^2)^(1/4)/x))*2^(
1/2)-2*arctan(((a^2*b^2)^(1/4)*x+(a^2*x^3-b^2*x)^(1/2))/(a^2*b^2)^(1/4)/x)-ln((a^2*x^2+2*(a^2*b^2)^(1/2)*x-b^2
-2*(a^2*b^2)^(1/4)*(a^2*x^3-b^2*x)^(1/2))/(a^2*x^2+2*(a^2*b^2)^(1/4)*(a^2*x^3-b^2*x)^(1/2)+2*(a^2*b^2)^(1/2)*x
-b^2))+2*arctan(((a^2*b^2)^(1/4)*x-(a^2*x^3-b^2*x)^(1/2))/(a^2*b^2)^(1/4)/x))/(a^2*b^2)^(1/4)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.45 (sec) , antiderivative size = 1317, normalized size of antiderivative = 4.23 \[ \int \frac {-b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\text {Too large to display} \]

[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]

1/12*(1/4)^(1/4)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 6*a^2*b^2*x^2 + b^4 + 8*((1/4)^(1/4)*a^2*b^2*x*(-1/(a^2*b
^2))^(1/4) + (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) - 4*(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)) - 1/12*(1/4)^(1/4)*(-1/(a^2*b^2))^(1/4)*log(
(a^4*x^4 - 6*a^2*b^2*x^2 + b^4 - 8*((1/4)^(1/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (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) - 4*(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)) - 1/12*I*(1/4)^(1/4)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 6*a^2*b^2*x^2 + b^4 - 8*(I*(1
/4)^(1/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (1/4)^(3/4)*(-I*a^4*b^2*x^2 + I*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 + 2*a^2*b^2*x^2 + b^4)) + 1/12*I*
(1/4)^(1/4)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 6*a^2*b^2*x^2 + b^4 - 8*(-I*(1/4)^(1/4)*a^2*b^2*x*(-1/(a^2*b^2
))^(1/4) + (1/4)^(3/4)*(I*a^4*b^2*x^2 - I*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 + 2*a^2*b^2*x^2 + b^4)) + 1/6*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 -
3*a^2*b^2*x^2 + b^4 + 2*(a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (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) - 2*(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)) - 1/6*(-1/(a^
2*b^2))^(1/4)*log((a^4*x^4 - 3*a^2*b^2*x^2 + b^4 - 2*(a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (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) - 2*(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)) - 1/6*I*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 3*a^2*b^2*x^2 + b^4 - 2*(I*a^2*b^2*x*(-1/(a^2*b^2)
)^(1/4) + (-I*a^4*b^2*x^2 + I*a^2*b^4)*(-1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 - b^2*x) + 2*(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)) + 1/6*I*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 3*a^2*b^2*x^
2 + b^4 - 2*(-I*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (I*a^4*b^2*x^2 - I*a^2*b^4)*(-1/(a^2*b^2))^(3/4))*sqrt(a^2*x^
3 - b^2*x) + 2*(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))

Sympy [F]

\[ \int \frac {-b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\int \frac {\left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} - a b x + b^{2}\right ) \left (a^{2} x^{2} + a b x + b^{2}\right )}{\sqrt {x \left (a x - b\right ) \left (a x + b\right )} \left (a^{2} x^{2} + b^{2}\right ) \left (a^{4} x^{4} - a^{2} b^{2} x^{2} + b^{4}\right )}\, dx \]

[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)

Maxima [F]

\[ \int \frac {-b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} - b^{6}}{{\left (a^{6} x^{6} + b^{6}\right )} \sqrt {a^{2} x^{3} - b^{2} x}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {-b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} - b^{6}}{{\left (a^{6} x^{6} + b^{6}\right )} \sqrt {a^{2} x^{3} - b^{2} x}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {-b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (b^6+a^6 x^6\right )} \, dx=\text {Hanged} \]

[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}