\(\int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} (-b^6+a^6 x^6)} \, dx\) [2694]

   Optimal result
   Rubi [C] (verified)
   Mathematica [A] (verified)
   Maple [A] (verified)
   Fricas [C] (verification not implemented)
   Sympy [F]
   Maxima [F]
   Giac [F]
   Mupad [B] (verification not implemented)

Optimal result

Integrand size = 45, antiderivative size = 244 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=\frac {2 \sqrt {-b^2 x+a^2 x^3}}{3 \left (b^2-a^2 x^2\right )}-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} 3^{3/4} \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}}{-\sqrt {3} b^2-3 a b x+\sqrt {3} a^2 x^2}\right )}{3 \sqrt [4]{3} \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {-\frac {b^{3/2}}{\sqrt {2} \sqrt [4]{3} \sqrt {a}}+\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} x}{\sqrt {2}}+\frac {a^{3/2} x^2}{\sqrt {2} \sqrt [4]{3} \sqrt {b}}}{\sqrt {-b^2 x+a^2 x^3}}\right )}{3 \sqrt [4]{3} \sqrt {a} \sqrt {b}} \]

[Out]

2*(a^2*x^3-b^2*x)^(1/2)/(-3*a^2*x^2+3*b^2)-1/9*2^(1/2)*arctan(2^(1/2)*3^(3/4)*a^(1/2)*b^(1/2)*(a^2*x^3-b^2*x)^
(1/2)/(-3^(1/2)*b^2-3*a*b*x+3^(1/2)*a^2*x^2))*3^(3/4)/a^(1/2)/b^(1/2)-1/9*2^(1/2)*arctanh((-1/6*b^(3/2)*2^(1/2
)*3^(3/4)/a^(1/2)+1/2*3^(1/4)*a^(1/2)*b^(1/2)*x*2^(1/2)+1/6*a^(3/2)*x^2*2^(1/2)*3^(3/4)/b^(1/2))/(a^2*x^3-b^2*
x)^(1/2))*3^(3/4)/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 = 1.81 (sec) , antiderivative size = 519, normalized size of antiderivative = 2.13, number of steps used = 36, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.378, Rules used = {2081, 6847, 6857, 230, 227, 2098, 1166, 425, 21, 434, 438, 437, 435, 259, 6860, 1233, 1232} \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=\frac {4 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (-\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (-\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}}-\frac {x (b-a x)}{3 b \sqrt {a^2 x^3-b^2 x}}-\frac {x (a x+b)}{3 b \sqrt {a^2 x^3-b^2 x}} \]

[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*(x*(b - a*x))/(b*Sqrt[-(b^2*x) + a^2*x^3]) - (x*(b + a*x))/(3*b*Sqrt[-(b^2*x) + a^2*x^3]) + (4*Sqrt[b]*Sq
rt[x]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticF[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[(-2*a)/(a - Sqrt[3]*Sqrt[-a^2]), 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]*El
lipticPi[(2*a)/(a - Sqrt[3]*Sqrt[-a^2]), 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[(-2*a)/(a + Sqrt[3]*Sqrt[-a^2]), 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]*E
llipticPi[(2*a)/(a + Sqrt[3]*Sqrt[-a^2]), ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(3*Sqrt[a]*Sqrt[-(b^2*x) + a
^2*x^3])

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 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 259

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_)*((a2_.) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a1 + b1*x^n)^FracPar
t[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 425

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]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 434

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 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 437

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

Rule 438

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

Rule 1166

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/e)*x^2)^p, x], x] /; FreeQ[{
a, c, d, e, p, q}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p]

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 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 2098

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(PP /. x ->
x^2)^p*Q^q, x], x] /;  !SumQ[NonfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x^2] && PolyQ[Q, x] && ILtQ[p,
 0]

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]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \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 (2 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {b^6+a^6 x^{12}}{\sqrt {-b^2+a^2 x^4} \left (-b^6+a^6 x^{12}\right )} \, 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 {1}{\sqrt {-b^2+a^2 x^4}}+\frac {2 b^6}{\sqrt {-b^2+a^2 x^4} \left (-b^6+a^6 x^{12}\right )}\right ) \, 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 \frac {1}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}+\frac {\left (4 b^6 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4} \left (-b^6+a^6 x^{12}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}} \\ & = \frac {\left (4 b^6 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{6 b^5 \left (b-a x^2\right ) \sqrt {-b^2+a^2 x^4}}-\frac {1}{6 b^5 \left (b+a x^2\right ) \sqrt {-b^2+a^2 x^4}}+\frac {-2 b+a x^2}{6 b^5 \sqrt {-b^2+a^2 x^4} \left (b^2-a b x^2+a^2 x^4\right )}+\frac {-2 b-a x^2}{6 b^5 \sqrt {-b^2+a^2 x^4} \left (b^2+a b x^2+a^2 x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}} \\ & = \frac {2 \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 )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (2 b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (b-a x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (2 b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (b+a x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {-2 b+a x^2}{\sqrt {-b^2+a^2 x^4} \left (b^2-a b x^2+a^2 x^4\right )} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {-2 b-a x^2}{\sqrt {-b^2+a^2 x^4} \left (b^2+a b x^2+a^2 x^4\right )} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}} \\ & = \frac {2 \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 )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (2 b \sqrt {x} \sqrt {-b-a x} \sqrt {b-a x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b-a x^2} \left (b-a x^2\right )^{3/2}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (2 b \sqrt {x} \sqrt {-b+a x} \sqrt {b+a x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b+a x^2} \left (b+a x^2\right )^{3/2}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {a+\sqrt {3} \sqrt {-a^2}}{\left (-a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {-b^2+a^2 x^4}}+\frac {a-\sqrt {3} \sqrt {-a^2}}{\left (-a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {-b^2+a^2 x^4}}\right ) \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {-a+\sqrt {3} \sqrt {-a^2}}{\left (a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {-b^2+a^2 x^4}}+\frac {-a-\sqrt {3} \sqrt {-a^2}}{\left (a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {-b^2+a^2 x^4}}\right ) \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}} \\ & = -\frac {x (b-a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {x (b+a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}+\frac {2 \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 )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt {x} \sqrt {-b-a x} \sqrt {b-a x}\right ) \text {Subst}\left (\int \frac {-a b+a^2 x^2}{\sqrt {-b-a x^2} \sqrt {b-a x^2}} \, dx,x,\sqrt {x}\right )}{3 a b \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {-b+a x} \sqrt {b+a x}\right ) \text {Subst}\left (\int \frac {a b+a^2 x^2}{\sqrt {-b+a x^2} \sqrt {b+a x^2}} \, dx,x,\sqrt {x}\right )}{3 a b \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \left (-a-\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \left (-a+\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}} \\ & = -\frac {x (b-a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {x (b+a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}+\frac {2 \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 )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {-b-a x} \sqrt {b-a x}\right ) \text {Subst}\left (\int \frac {\sqrt {b-a x^2}}{\sqrt {-b-a x^2}} \, dx,x,\sqrt {x}\right )}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {-b+a x} \sqrt {b+a x}\right ) \text {Subst}\left (\int \frac {\sqrt {b+a x^2}}{\sqrt {-b+a x^2}} \, dx,x,\sqrt {x}\right )}{3 b \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \left (-a-\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \left (a-\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (-a b+\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \left (-a+\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \left (a+\sqrt {3} \sqrt {-a^2}\right ) b \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (-a b-\sqrt {3} \sqrt {-a^2} b+2 a^2 x^2\right ) \sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}} \\ & = -\frac {x (b-a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {x (b+a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}+\frac {2 \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 )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (-\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (-\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (2 \sqrt {x} \sqrt {-b-a x} \sqrt {b-a x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b-a x^2} \sqrt {b-a x^2}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {-b-a x} \sqrt {b-a x}\right ) \text {Subst}\left (\int \frac {\sqrt {-b-a x^2}}{\sqrt {b-a x^2}} \, dx,x,\sqrt {x}\right )}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {b+a x} \sqrt {1-\frac {a x}{b}}\right ) \text {Subst}\left (\int \frac {\sqrt {b+a x^2}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{3 b \sqrt {-b^2 x+a^2 x^3}} \\ & = -\frac {x (b-a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {x (b+a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}+\frac {2 \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 )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (-\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (-\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {-b-a x} \sqrt {1-\frac {a x}{b}}\right ) \text {Subst}\left (\int \frac {\sqrt {-b-a x^2}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} (b+a x) \sqrt {1-\frac {a x}{b}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {a x^2}{b}}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{3 b \sqrt {1+\frac {a x}{b}} \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 {1}{\sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}} \\ & = -\frac {x (b-a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {x (b+a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {\sqrt {x} (b+a x) \sqrt {1-\frac {a x}{b}} E\left (\left .\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {b} \sqrt {1+\frac {a x}{b}} \sqrt {-b^2 x+a^2 x^3}}+\frac {2 \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 )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (-\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (-\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} (-b-a x) \sqrt {1-\frac {a x}{b}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {a x^2}{b}}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{3 b \sqrt {1+\frac {a x}{b}} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (2 \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {a^2 x^4}{b^2}}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}} \\ & = -\frac {x (b-a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}-\frac {x (b+a x)}{3 b \sqrt {-b^2 x+a^2 x^3}}+\frac {4 \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 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (-\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (-\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {3} \sqrt {-a^2}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {-b^2 x+a^2 x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.34 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.02 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=-\frac {\sqrt {x} \left (6 \sqrt {a} \sqrt {b} \sqrt {x}+3^{3/4} \sqrt {-2 b^2+2 a^2 x^2} \arctan \left (\frac {\sqrt {3} b^2+3 a b x-\sqrt {3} a^2 x^2}{\sqrt {2} 3^{3/4} \sqrt {a} \sqrt {b} \sqrt {x} \sqrt {-b^2+a^2 x^2}}\right )-3^{3/4} \sqrt {-2 b^2+2 a^2 x^2} \text {arctanh}\left (\frac {\sqrt {3} b^2-3 a b x-\sqrt {3} a^2 x^2}{\sqrt {2} 3^{3/4} \sqrt {a} \sqrt {b} \sqrt {x} \sqrt {-b^2+a^2 x^2}}\right )\right )}{9 \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}} \]

[In]

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

[Out]

-1/9*(Sqrt[x]*(6*Sqrt[a]*Sqrt[b]*Sqrt[x] + 3^(3/4)*Sqrt[-2*b^2 + 2*a^2*x^2]*ArcTan[(Sqrt[3]*b^2 + 3*a*b*x - Sq
rt[3]*a^2*x^2)/(Sqrt[2]*3^(3/4)*Sqrt[a]*Sqrt[b]*Sqrt[x]*Sqrt[-b^2 + a^2*x^2])] - 3^(3/4)*Sqrt[-2*b^2 + 2*a^2*x
^2]*ArcTanh[(Sqrt[3]*b^2 - 3*a*b*x - Sqrt[3]*a^2*x^2)/(Sqrt[2]*3^(3/4)*Sqrt[a]*Sqrt[b]*Sqrt[x]*Sqrt[-b^2 + a^2
*x^2])]))/(Sqrt[a]*Sqrt[b]*Sqrt[-(b^2*x) + a^2*x^3])

Maple [A] (verified)

Time = 2.00 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.22

method result size
default \(\frac {\left (\frac {\sqrt {2}\, \left (2 \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {a^{2} x^{3}-b^{2} x}+3 \left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{3 \left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )-2 \arctan \left (\frac {-\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}{3}+\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )+\ln \left (\frac {a^{2} x^{2}+\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x -3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}-b^{2}}{a^{2} x^{2}+\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x +3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}-b^{2}}\right )\right ) \sqrt {a^{2} x^{3}-b^{2} x}}{2}-2 \,3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} x \right ) 3^{\frac {3}{4}}}{9 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}\) \(297\)
pseudoelliptic \(\frac {\left (\frac {\sqrt {2}\, \left (2 \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {a^{2} x^{3}-b^{2} x}+3 \left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{3 \left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )-2 \arctan \left (\frac {-\frac {\sqrt {2}\, 3^{\frac {3}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}{3}+\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}{\left (a^{2} b^{2}\right )^{\frac {1}{4}} x}\right )+\ln \left (\frac {a^{2} x^{2}+\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x -3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}-b^{2}}{a^{2} x^{2}+\sqrt {3}\, \sqrt {a^{2} b^{2}}\, x +3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}\, \sqrt {2}-b^{2}}\right )\right ) \sqrt {a^{2} x^{3}-b^{2} x}}{2}-2 \,3^{\frac {1}{4}} \left (a^{2} b^{2}\right )^{\frac {1}{4}} x \right ) 3^{\frac {3}{4}}}{9 \left (a^{2} b^{2}\right )^{\frac {1}{4}} \sqrt {a^{2} x^{3}-b^{2} x}}\) \(297\)
elliptic \(-\frac {2 x}{3 \sqrt {\left (x^{2}-\frac {b^{2}}{a^{2}}\right ) a^{2} x}}+\frac {2 b \sqrt {1+\frac {a x}{b}}\, \sqrt {-\frac {2 a x}{b}+2}\, \sqrt {-\frac {a x}{b}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{3 a \sqrt {a^{2} x^{3}-b^{2} x}}+\frac {\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 =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}+a b \textit {\_Z} +b^{2}\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \operatorname {EllipticPi}\left (\sqrt {\frac {a x +b}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a}{b}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a +b}\right )}{3 \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}+\frac {2 \sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}+a b \textit {\_Z} +b^{2}\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \operatorname {EllipticPi}\left (\sqrt {\frac {a x +b}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a}{b}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a +b}\right )}{3 \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}-\frac {a \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 =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}-a b \textit {\_Z} +b^{2}\right )}{\sum }\frac {\operatorname {EllipticPi}\left (\sqrt {\frac {a x +b}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a -2 b}{3 b}, \frac {\sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{2}}{2 \underline {\hspace {1.25 ex}}\alpha a -b}\right )}{9 \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}+\frac {4 \sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}-a b \textit {\_Z} +b^{2}\right )}{\sum }\frac {\operatorname {EllipticPi}\left (\sqrt {\frac {a x +b}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a -2 b}{3 b}, \frac {\sqrt {2}}{2}\right ) \underline {\hspace {1.25 ex}}\alpha }{2 \underline {\hspace {1.25 ex}}\alpha a -b}\right )}{9 \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}-\frac {4 \sqrt {2}\, \sqrt {\frac {a x +b}{b}}\, \sqrt {-\frac {a x -b}{b}}\, \sqrt {-\frac {a x}{b}}\, b^{2} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{2} \textit {\_Z}^{2}-a b \textit {\_Z} +b^{2}\right )}{\sum }\frac {\operatorname {EllipticPi}\left (\sqrt {\frac {a x +b}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a -2 b}{3 b}, \frac {\sqrt {2}}{2}\right )}{2 \underline {\hspace {1.25 ex}}\alpha a -b}\right )}{9 a \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}\) \(678\)

[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]

1/9*(1/2*2^(1/2)*(2*arctan(1/3*(2^(1/2)*3^(3/4)*(a^2*x^3-b^2*x)^(1/2)+3*(a^2*b^2)^(1/4)*x)/(a^2*b^2)^(1/4)/x)-
2*arctan((-1/3*2^(1/2)*3^(3/4)*(a^2*x^3-b^2*x)^(1/2)+(a^2*b^2)^(1/4)*x)/(a^2*b^2)^(1/4)/x)+ln((a^2*x^2+3^(1/2)
*(a^2*b^2)^(1/2)*x-3^(1/4)*(a^2*b^2)^(1/4)*(a^2*x^3-b^2*x)^(1/2)*2^(1/2)-b^2)/(a^2*x^2+3^(1/2)*(a^2*b^2)^(1/2)
*x+3^(1/4)*(a^2*b^2)^(1/4)*(a^2*x^3-b^2*x)^(1/2)*2^(1/2)-b^2)))*(a^2*x^3-b^2*x)^(1/2)-2*3^(1/4)*(a^2*b^2)^(1/4
)*x)/(a^2*b^2)^(1/4)/(a^2*x^3-b^2*x)^(1/2)*3^(3/4)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.37 (sec) , antiderivative size = 771, normalized size of antiderivative = 3.16 \[ \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}{3}\right )^{\frac {1}{4}} {\left (i \, a^{2} x^{2} - i \, b^{2}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 5 \, a^{2} b^{2} x^{2} + b^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}} - 6 \, {\left (i \, \left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (-i \, a^{4} b^{2} x^{2} + i \, a^{2} b^{4}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}{a^{4} x^{4} + a^{2} b^{2} x^{2} + b^{4}}\right ) + \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (-i \, a^{2} x^{2} + i \, b^{2}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 5 \, a^{2} b^{2} x^{2} + b^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}} - 6 \, {\left (-i \, \left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (i \, a^{4} b^{2} x^{2} - i \, a^{2} b^{4}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}{a^{4} x^{4} + a^{2} b^{2} x^{2} + b^{4}}\right ) - \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{2} x^{2} - b^{2}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 5 \, a^{2} b^{2} x^{2} + b^{4} - 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}} + 6 \, {\left (\left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (a^{4} b^{2} x^{2} - a^{2} b^{4}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}{a^{4} x^{4} + a^{2} b^{2} x^{2} + b^{4}}\right ) + \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (a^{2} x^{2} - b^{2}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x^{4} - 5 \, a^{2} b^{2} x^{2} + b^{4} - 6 \, \sqrt {\frac {1}{3}} {\left (a^{4} b^{2} x^{3} - a^{2} b^{4} x\right )} \sqrt {-\frac {1}{a^{2} b^{2}}} - 6 \, {\left (\left (\frac {1}{3}\right )^{\frac {1}{4}} a^{2} b^{2} x \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {1}{4}} + \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (a^{4} b^{2} x^{2} - a^{2} b^{4}\right )} \left (-\frac {1}{a^{2} b^{2}}\right )^{\frac {3}{4}}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}{a^{4} x^{4} + a^{2} b^{2} x^{2} + b^{4}}\right ) + 4 \, \sqrt {a^{2} x^{3} - b^{2} x}}{6 \, {\left (a^{2} x^{2} - b^{2}\right )}} \]

[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*((1/3)^(1/4)*(I*a^2*x^2 - I*b^2)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 5*a^2*b^2*x^2 + b^4 + 6*sqrt(1/3)*(a
^4*b^2*x^3 - a^2*b^4*x)*sqrt(-1/(a^2*b^2)) - 6*(I*(1/3)^(1/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (1/3)^(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))/(a^4*x^4 + a^2*b^2*x^2 + b^4)) + (1/3)^
(1/4)*(-I*a^2*x^2 + I*b^2)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 5*a^2*b^2*x^2 + b^4 + 6*sqrt(1/3)*(a^4*b^2*x^3
- a^2*b^4*x)*sqrt(-1/(a^2*b^2)) - 6*(-I*(1/3)^(1/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (1/3)^(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))/(a^4*x^4 + a^2*b^2*x^2 + b^4)) - (1/3)^(1/4)*(a^2*
x^2 - b^2)*(-1/(a^2*b^2))^(1/4)*log((a^4*x^4 - 5*a^2*b^2*x^2 + b^4 - 6*sqrt(1/3)*(a^4*b^2*x^3 - a^2*b^4*x)*sqr
t(-1/(a^2*b^2)) + 6*((1/3)^(1/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (1/3)^(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))/(a^4*x^4 + a^2*b^2*x^2 + b^4)) + (1/3)^(1/4)*(a^2*x^2 - b^2)*(-1/(a^2*b^2
))^(1/4)*log((a^4*x^4 - 5*a^2*b^2*x^2 + b^4 - 6*sqrt(1/3)*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(-1/(a^2*b^2)) - 6*((1
/3)^(1/4)*a^2*b^2*x*(-1/(a^2*b^2))^(1/4) + (1/3)^(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))/(a^4*x^4 + a^2*b^2*x^2 + b^4)) + 4*sqrt(a^2*x^3 - b^2*x))/(a^2*x^2 - b^2)

Sympy [F]

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

[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**2*x**2 + b**2)*(a**4*x**4 - a**2*b**2*x**2 + b**4)/(sqrt(x*(a*x - b)*(a*x + b))*(a*x - b)*(a*x +
b)*(a**2*x**2 - a*b*x + b**2)*(a**2*x**2 + a*b*x + b**2)), 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 [B] (verification not implemented)

Time = 13.24 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.97 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=\frac {2\,\sqrt {a^2\,x^3-b^2\,x}}{3\,\left (b^2-a^2\,x^2\right )}+\frac {3^{1/4}\,\sqrt {-\frac {1}{27}{}\mathrm {i}}\,\ln \left (\frac {{\left (-1\right )}^{1/4}\,3^{3/4}\,b^2-{\left (-1\right )}^{1/4}\,3^{3/4}\,a^2\,x^2-3\,{\left (-1\right )}^{3/4}\,3^{1/4}\,a\,b\,x+\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3-b^2\,x}\,6{}\mathrm {i}}{-a^2\,x^2+1{}\mathrm {i}\,\sqrt {3}\,a\,b\,x+b^2}\right )}{\sqrt {a}\,\sqrt {b}}+\frac {3^{1/4}\,\sqrt {\frac {1}{27}{}\mathrm {i}}\,\ln \left (\frac {{\left (-1\right )}^{3/4}\,3^{3/4}\,b^2-{\left (-1\right )}^{3/4}\,3^{3/4}\,a^2\,x^2-3\,{\left (-1\right )}^{1/4}\,3^{1/4}\,a\,b\,x+\sqrt {a}\,\sqrt {b}\,\sqrt {a^2\,x^3-b^2\,x}\,6{}\mathrm {i}}{a^2\,x^2+1{}\mathrm {i}\,\sqrt {3}\,a\,b\,x-b^2}\right )}{\sqrt {a}\,\sqrt {b}} \]

[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]

(2*(a^2*x^3 - b^2*x)^(1/2))/(3*(b^2 - a^2*x^2)) + (3^(1/4)*(-1i/27)^(1/2)*log((a^(1/2)*b^(1/2)*(a^2*x^3 - b^2*
x)^(1/2)*6i + (-1)^(1/4)*3^(3/4)*b^2 - (-1)^(1/4)*3^(3/4)*a^2*x^2 - 3*(-1)^(3/4)*3^(1/4)*a*b*x)/(b^2 - a^2*x^2
 + 3^(1/2)*a*b*x*1i)))/(a^(1/2)*b^(1/2)) + (3^(1/4)*(1i/27)^(1/2)*log((a^(1/2)*b^(1/2)*(a^2*x^3 - b^2*x)^(1/2)
*6i + (-1)^(3/4)*3^(3/4)*b^2 - (-1)^(3/4)*3^(3/4)*a^2*x^2 - 3*(-1)^(1/4)*3^(1/4)*a*b*x)/(a^2*x^2 - b^2 + 3^(1/
2)*a*b*x*1i)))/(a^(1/2)*b^(1/2))