3.8.87 \(\int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} (-b^6+c x^3+a^6 x^6)} \, dx\) [787]

Optimal. Leaf size=60 \[ \frac {1}{3} \text {RootSum}\left [c+3 a^2 b^2 \text {$\#$1}^2+\text {$\#$1}^6\& ,\frac {-\log (x)+\log \left (\sqrt {-b^2 x+a^2 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ] \]

[Out]

Unintegrable

________________________________________________________________________________________

Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(720\) vs. \(2(60)=120\).
time = 8.64, antiderivative size = 720, normalized size of antiderivative = 12.00, number of steps used = 92, number of rules used = 13, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.260, Rules used = {2081, 6847, 6860, 230, 227, 6857, 1739, 1233, 1232, 1262, 739, 210, 212} \begin {gather*} \frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\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}} \Pi \left (-\frac {\sqrt [3]{-2} a b}{\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}};\left .\text {ArcSin}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\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}} \Pi \left (\frac {\sqrt [3]{2} a b}{\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}};\left .\text {ArcSin}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\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}} \Pi \left (\frac {(-1)^{2/3} \sqrt [3]{2} a b}{\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}};\left .\text {ArcSin}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\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}} \Pi \left (-\frac {\sqrt [3]{-2} a b}{\sqrt [3]{\sqrt {4 a^6 b^6+c^2}-c}};\left .\text {ArcSin}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\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}} \Pi \left (\frac {\sqrt [3]{2} a b}{\sqrt [3]{\sqrt {4 a^6 b^6+c^2}-c}};\left .\text {ArcSin}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\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}} \Pi \left (\frac {(-1)^{2/3} \sqrt [3]{2} a b}{\sqrt [3]{\sqrt {4 a^6 b^6+c^2}-c}};\left .\text {ArcSin}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{3 \sqrt {a} \sqrt {a^2 x^3-b^2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b]*Sqrt[x]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticF[ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(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)^(1/3)*a*b)/(-c - Sqrt[4*a^6*
b^6 + c^2])^(1/3)), 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^(1/3)*a*b)/(-c - Sqrt[4*a^6*b^6 + c^2])^(1/3), ArcSin[(Sqrt[a]*S
qrt[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]*Ellip
ticPi[((-1)^(2/3)*2^(1/3)*a*b)/(-c - Sqrt[4*a^6*b^6 + c^2])^(1/3), 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)^(1/3)*a*b)/(
-c + Sqrt[4*a^6*b^6 + c^2])^(1/3)), 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^(1/3)*a*b)/(-c + Sqrt[4*a^6*b^6 + c^2])^(1/3), A
rcSin[(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)*2^(1/3)*a*b)/(-c + Sqrt[4*a^6*b^6 + c^2])^(1/3), ArcSin[(Sqrt[a]*Sqrt[x])/Sq
rt[b]], -1])/(3*Sqrt[a]*Sqrt[-(b^2*x) + a^2*x^3])

Rule 210

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

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 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 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 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 1739

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

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]

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*} \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+c x^3+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+c x^3+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+c x^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-c x^6}{\sqrt {-b^2+a^2 x^4} \left (-b^6+c x^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 (2 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {2 b^6-c x^6}{\sqrt {-b^2+a^2 x^4} \left (-b^6+c x^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 {-c+\sqrt {4 a^6 b^6+c^2}}{\sqrt {-b^2+a^2 x^4} \left (c-\sqrt {4 a^6 b^6+c^2}+2 a^6 x^6\right )}+\frac {-c-\sqrt {4 a^6 b^6+c^2}}{\sqrt {-b^2+a^2 x^4} \left (c+\sqrt {4 a^6 b^6+c^2}+2 a^6 x^6\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}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \left (-c-\sqrt {4 a^6 b^6+c^2}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4} \left (c+\sqrt {4 a^6 b^6+c^2}+2 a^6 x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \left (-c+\sqrt {4 a^6 b^6+c^2}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-b^2+a^2 x^4} \left (c-\sqrt {4 a^6 b^6+c^2}+2 a^6 x^6\right )} \, 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}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \left (-c-\sqrt {4 a^6 b^6+c^2}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {\sqrt {-c-\sqrt {4 a^6 b^6+c^2}}}{2 \left (c+\sqrt {4 a^6 b^6+c^2}\right ) \left (\sqrt {-c-\sqrt {4 a^6 b^6+c^2}}-\sqrt {2} a^3 x^3\right ) \sqrt {-b^2+a^2 x^4}}+\frac {\sqrt {-c-\sqrt {4 a^6 b^6+c^2}}}{2 \left (c+\sqrt {4 a^6 b^6+c^2}\right ) \left (\sqrt {-c-\sqrt {4 a^6 b^6+c^2}}+\sqrt {2} a^3 x^3\right ) \sqrt {-b^2+a^2 x^4}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}+\frac {\left (2 \left (-c+\sqrt {4 a^6 b^6+c^2}\right ) \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {\sqrt {-c+\sqrt {4 a^6 b^6+c^2}}}{2 \left (c-\sqrt {4 a^6 b^6+c^2}\right ) \left (\sqrt {-c+\sqrt {4 a^6 b^6+c^2}}-\sqrt {2} a^3 x^3\right ) \sqrt {-b^2+a^2 x^4}}+\frac {\sqrt {-c+\sqrt {4 a^6 b^6+c^2}}}{2 \left (c-\sqrt {4 a^6 b^6+c^2}\right ) \left (\sqrt {-c+\sqrt {4 a^6 b^6+c^2}}+\sqrt {2} a^3 x^3\right ) \sqrt {-b^2+a^2 x^4}}\right ) \, 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}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-c-\sqrt {4 a^6 b^6+c^2}}-\sqrt {2} a^3 x^3\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-c-\sqrt {4 a^6 b^6+c^2}}+\sqrt {2} a^3 x^3\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-c+\sqrt {4 a^6 b^6+c^2}}-\sqrt {2} a^3 x^3\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {-c+\sqrt {4 a^6 b^6+c^2}}+\sqrt {2} a^3 x^3\right ) \sqrt {-b^2+a^2 x^4}} \, 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}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{3 \sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \left (-\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}-\sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}-\frac {1}{3 \sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \left (-\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}-\frac {1}{3 \sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \left (-\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{3 \sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}-\sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}+\frac {1}{3 \sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}+\frac {1}{3 \sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{3 \sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \left (-\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}-\sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}-\frac {1}{3 \sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \left (-\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}-\frac {1}{3 \sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \left (-\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt {-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{3 \sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}-\sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}+\frac {1}{3 \sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}+\frac {1}{3 \sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}}\right ) \, 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}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}-\sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}-\sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [6]{-c-\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}-\sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}-\sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-1} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}+\frac {\left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-b^2 x+a^2 x^3}}-\frac {\left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [6]{-c+\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [6]{2} a x\right ) \sqrt {-b^2+a^2 x^4}} \, 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}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )\right |-1\right )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-2 \frac {\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-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}}-2 \frac {\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}-\sqrt [3]{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}}-2 \frac {\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [3]{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}}-2 \frac {\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-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}}-2 \frac {\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}}-\sqrt [3]{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}}-2 \frac {\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [3]{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 {2 \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 )}{\sqrt {a} \sqrt {-b^2 x+a^2 x^3}}-2 \frac {\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-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}}-2 \frac {\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}-\sqrt [3]{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}}-2 \frac {\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [3]{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}}-2 \frac {\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}}+\sqrt [3]{-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}}-2 \frac {\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}}-\sqrt [3]{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}}-2 \frac {\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}}-(-1)^{2/3} \sqrt [3]{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 {2 \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 )}{\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}} \Pi \left (-\frac {\sqrt [3]{-2} a b}{\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}};\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 {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [3]{2} a b}{\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}};\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 {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {(-1)^{2/3} \sqrt [3]{2} a b}{\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}};\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 {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (-\frac {\sqrt [3]{-2} a b}{\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}}};\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 {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {\sqrt [3]{2} a b}{\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}}};\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 {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {a^2 x^2}{b^2}} \Pi \left (\frac {(-1)^{2/3} \sqrt [3]{2} a b}{\sqrt [3]{-c+\sqrt {4 a^6 b^6+c^2}}};\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}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 11.16, size = 544, normalized size = 9.07 \begin {gather*} -\frac {2 i \sqrt {1-\frac {b^2}{a^2 x^2}} x^{3/2} \left (3 F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {\sqrt [3]{2} a}{b \sqrt [3]{\frac {c-\sqrt {4 a^6 b^6+c^2}}{b^6}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {\left (1-i \sqrt {3}\right ) a b^5 \left (\frac {c-\sqrt {4 a^6 b^6+c^2}}{b^6}\right )^{2/3}}{2^{2/3} \left (-c+\sqrt {4 a^6 b^6+c^2}\right )};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {\left (1+i \sqrt {3}\right ) a b^5 \left (\frac {c-\sqrt {4 a^6 b^6+c^2}}{b^6}\right )^{2/3}}{2^{2/3} \left (-c+\sqrt {4 a^6 b^6+c^2}\right )};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {\sqrt [3]{2} a}{b \sqrt [3]{\frac {c+\sqrt {4 a^6 b^6+c^2}}{b^6}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-\frac {i \left (-i+\sqrt {3}\right ) a}{2^{2/3} b \sqrt [3]{\frac {c+\sqrt {4 a^6 b^6+c^2}}{b^6}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {i \left (i+\sqrt {3}\right ) a}{2^{2/3} b \sqrt [3]{\frac {c+\sqrt {4 a^6 b^6+c^2}}{b^6}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )\right )}{3 \sqrt {-\frac {b}{a}} \sqrt {-b^2 x+a^2 x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b^6 + a^6*x^6)/(Sqrt[-(b^2*x) + a^2*x^3]*(-b^6 + c*x^3 + 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[(2^
(1/3)*a)/(b*((c - Sqrt[4*a^6*b^6 + c^2])/b^6)^(1/3)), I*ArcSinh[Sqrt[-(b/a)]/Sqrt[x]], -1] - EllipticPi[((1 -
I*Sqrt[3])*a*b^5*((c - Sqrt[4*a^6*b^6 + c^2])/b^6)^(2/3))/(2^(2/3)*(-c + Sqrt[4*a^6*b^6 + c^2])), I*ArcSinh[Sq
rt[-(b/a)]/Sqrt[x]], -1] - EllipticPi[((1 + I*Sqrt[3])*a*b^5*((c - Sqrt[4*a^6*b^6 + c^2])/b^6)^(2/3))/(2^(2/3)
*(-c + Sqrt[4*a^6*b^6 + c^2])), I*ArcSinh[Sqrt[-(b/a)]/Sqrt[x]], -1] - EllipticPi[(2^(1/3)*a)/(b*((c + Sqrt[4*
a^6*b^6 + c^2])/b^6)^(1/3)), I*ArcSinh[Sqrt[-(b/a)]/Sqrt[x]], -1] - EllipticPi[((-I)*(-I + Sqrt[3])*a)/(2^(2/3
)*b*((c + Sqrt[4*a^6*b^6 + c^2])/b^6)^(1/3)), I*ArcSinh[Sqrt[-(b/a)]/Sqrt[x]], -1] - EllipticPi[(I*(I + Sqrt[3
])*a)/(2^(2/3)*b*((c + Sqrt[4*a^6*b^6 + c^2])/b^6)^(1/3)), I*ArcSinh[Sqrt[-(b/a)]/Sqrt[x]], -1]))/(Sqrt[-(b/a)
]*Sqrt[-(b^2*x) + a^2*x^3])

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 0.30, size = 371, normalized size = 6.18

method	result	size
default	\(\frac {b \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {2 \left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}-b^{2} x}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{6} \textit {\_Z}^{6}-b^{6}+c \,\textit {\_Z}^{3}\right )}{\sum }\frac {\left (-2 b^{6}+\underline {\hspace {1.25 ex}}\alpha ^{3} c \right ) \left (a^{8} \underline {\hspace {1.25 ex}}\alpha ^{5}-a^{7} \underline {\hspace {1.25 ex}}\alpha ^{4} b +a^{6} \underline {\hspace {1.25 ex}}\alpha ^{3} b^{2}-a^{5} \underline {\hspace {1.25 ex}}\alpha ^{2} b^{3}+a^{4} b^{4} \underline {\hspace {1.25 ex}}\alpha -a^{3} b^{5}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} c -\underline {\hspace {1.25 ex}}\alpha a b c +b^{2} c \right ) \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {\left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {a^{8} \underline {\hspace {1.25 ex}}\alpha ^{5}-a^{7} \underline {\hspace {1.25 ex}}\alpha ^{4} b +a^{6} \underline {\hspace {1.25 ex}}\alpha ^{3} b^{2}-a^{5} \underline {\hspace {1.25 ex}}\alpha ^{2} b^{3}+a^{4} b^{4} \underline {\hspace {1.25 ex}}\alpha -a^{3} b^{5}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} c -\underline {\hspace {1.25 ex}}\alpha a b c +b^{2} c}{b^{2} c}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{2} \left (2 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{6}+c \right ) \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}\right )}{3 b^{2} c}\)	\(371\)
elliptic	\(\frac {b \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {2 \left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}-b^{2} x}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{6} \textit {\_Z}^{6}-b^{6}+c \,\textit {\_Z}^{3}\right )}{\sum }\frac {\left (-2 b^{6}+\underline {\hspace {1.25 ex}}\alpha ^{3} c \right ) \left (a^{8} \underline {\hspace {1.25 ex}}\alpha ^{5}-a^{7} \underline {\hspace {1.25 ex}}\alpha ^{4} b +a^{6} \underline {\hspace {1.25 ex}}\alpha ^{3} b^{2}-a^{5} \underline {\hspace {1.25 ex}}\alpha ^{2} b^{3}+a^{4} b^{4} \underline {\hspace {1.25 ex}}\alpha -a^{3} b^{5}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} c -\underline {\hspace {1.25 ex}}\alpha a b c +b^{2} c \right ) \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {\left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {a^{8} \underline {\hspace {1.25 ex}}\alpha ^{5}-a^{7} \underline {\hspace {1.25 ex}}\alpha ^{4} b +a^{6} \underline {\hspace {1.25 ex}}\alpha ^{3} b^{2}-a^{5} \underline {\hspace {1.25 ex}}\alpha ^{2} b^{3}+a^{4} b^{4} \underline {\hspace {1.25 ex}}\alpha -a^{3} b^{5}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} c -\underline {\hspace {1.25 ex}}\alpha a b c +b^{2} c}{b^{2} c}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{2} \left (2 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{6}+c \right ) \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}\right )}{3 b^{2} c}\)	\(371\)

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+c*x^3),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))+1/3/b^2/c*2^(1/2)*sum((-2*b^6+_alpha^3*c)/_alpha^2/(2*_alpha^3*a^6+c)*(_alpha^5*a^8-_alpha^4*
a^7*b+_alpha^3*a^6*b^2-_alpha^2*a^5*b^3+_alpha*a^4*b^4-a^3*b^5+_alpha^2*a^2*c-_alpha*a*b*c+b^2*c)*((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),(_alpha^5*a
^8-_alpha^4*a^7*b+_alpha^3*a^6*b^2-_alpha^2*a^5*b^3+_alpha*a^4*b^4-a^3*b^5+_alpha^2*a^2*c-_alpha*a*b*c+b^2*c)/
b^2/c,1/2*2^(1/2)),_alpha=RootOf(_Z^6*a^6-b^6+_Z^3*c))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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+c*x^3),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 1.98, size = 11793, normalized size = 196.55 \begin {gather*} \text {Too large to display} \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+c*x^3),x, algorithm="fricas")

[Out]

-1/12*sqrt(2/3)*sqrt(1/6)*sqrt(-(54*a^2*b^2 - (a^4*b^4*(-I*sqrt(3) + 1)/((-1/729*a^6*b^6/c^3 - 1/1458/c + 1/14
58*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*c^2) + 18*a^2*b^2/c + 81*(-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^
6*b^6 + c^2)/c^2)^(1/3)*(I*sqrt(3) + 1))*c + 3*sqrt(1/3)*c*sqrt((972*a^4*b^4 + 36*(a^4*b^4*(-I*sqrt(3) + 1)/((
-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*c^2) + 18*a^2*b^2/c + 81*(-1/729*a^6*b
^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*(I*sqrt(3) + 1))*a^2*b^2*c - (a^4*b^4*(-I*sqrt(3)
+ 1)/((-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*c^2) + 18*a^2*b^2/c + 81*(-1/72
9*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*(I*sqrt(3) + 1))^2*c^2)/c^2))/c)*log(1/324*
(648*a^6*x^6 - 1944*a^4*b^2*x^4 + 1944*a^2*b^4*x^2 - 648*b^6 - 648*c*x^3 + 2*(a^4*c*x^5 - 2*a^2*b^2*c*x^3 + b^
4*c*x)*(a^4*b^4*(-I*sqrt(3) + 1)/((-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*c^2
) + 18*a^2*b^2/c + 81*(-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*(I*sqrt(3) + 1)
)^2 + sqrt(2/3)*sqrt(1/6)*((a^4*c*x^4 - 2*a^2*b^2*c*x^2 + b^4*c)*(a^4*b^4*(-I*sqrt(3) + 1)/((-1/729*a^6*b^6/c^
3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*c^2) + 18*a^2*b^2/c + 81*(-1/729*a^6*b^6/c^3 - 1/1458/c
+ 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*(I*sqrt(3) + 1))^2*sqrt(a^2*x^3 - b^2*x) - 18*(3*a^6*b^2*x^4 - 6*a^
4*b^4*x^2 + 3*a^2*b^6 + a^2*c*x^3 - b^2*c*x)*(a^4*b^4*(-I*sqrt(3) + 1)/((-1/729*a^6*b^6/c^3 - 1/1458/c + 1/145
8*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*c^2) + 18*a^2*b^2/c + 81*(-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6
*b^6 + c^2)/c^2)^(1/3)*(I*sqrt(3) + 1))*sqrt(a^2*x^3 - b^2*x) + 3*sqrt(1/3)*((a^4*c*x^4 - 2*a^2*b^2*c*x^2 + b^
4*c)*(a^4*b^4*(-I*sqrt(3) + 1)/((-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*c^2)
+ 18*a^2*b^2/c + 81*(-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*(I*sqrt(3) + 1))*
sqrt(a^2*x^3 - b^2*x) + 18*(a^2*c*x^3 - b^2*c*x)*sqrt(a^2*x^3 - b^2*x))*sqrt((972*a^4*b^4 + 36*(a^4*b^4*(-I*sq
rt(3) + 1)/((-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*c^2) + 18*a^2*b^2/c + 81*
(-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*(I*sqrt(3) + 1))*a^2*b^2*c - (a^4*b^4
*(-I*sqrt(3) + 1)/((-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*c^2) + 18*a^2*b^2/
c + 81*(-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*(I*sqrt(3) + 1))^2*c^2)/c^2) -
324*(3*a^4*b^2*x^3 - 3*a^2*b^4*x + 2*c*x^2)*sqrt(a^2*x^3 - b^2*x))*sqrt(-(54*a^2*b^2 - (a^4*b^4*(-I*sqrt(3) +
1)/((-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*c^2) + 18*a^2*b^2/c + 81*(-1/729
*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*(I*sqrt(3) + 1))*c + 3*sqrt(1/3)*c*sqrt((972
*a^4*b^4 + 36*(a^4*b^4*(-I*sqrt(3) + 1)/((-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1
/3)*c^2) + 18*a^2*b^2/c + 81*(-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*(I*sqrt(
3) + 1))*a^2*b^2*c - (a^4*b^4*(-I*sqrt(3) + 1)/((-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/
c^2)^(1/3)*c^2) + 18*a^2*b^2/c + 81*(-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*(
I*sqrt(3) + 1))^2*c^2)/c^2))/c) - 36*(3*a^6*b^2*x^5 - 6*a^4*b^4*x^3 + 3*a^2*b^6*x + a^2*c*x^4 - b^2*c*x^2)*(a^
4*b^4*(-I*sqrt(3) + 1)/((-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*c^2) + 18*a^2
*b^2/c + 81*(-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*(I*sqrt(3) + 1)) + 6*sqrt
(1/3)*(18*a^2*c*x^4 - 18*b^2*c*x^2 + (a^4*c*x^5 - 2*a^2*b^2*c*x^3 + b^4*c*x)*(a^4*b^4*(-I*sqrt(3) + 1)/((-1/72
9*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*c^2) + 18*a^2*b^2/c + 81*(-1/729*a^6*b^6/c^
3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*(I*sqrt(3) + 1)))*sqrt((972*a^4*b^4 + 36*(a^4*b^4*(-I*s
qrt(3) + 1)/((-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*c^2) + 18*a^2*b^2/c + 81
*(-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*(I*sqrt(3) + 1))*a^2*b^2*c - (a^4*b^
4*(-I*sqrt(3) + 1)/((-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*c^2) + 18*a^2*b^2
/c + 81*(-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*(I*sqrt(3) + 1))^2*c^2)/c^2))
/(a^6*x^6 - b^6 + c*x^3)) + 1/12*sqrt(2/3)*sqrt(1/6)*sqrt(-(54*a^2*b^2 - (a^4*b^4*(-I*sqrt(3) + 1)/((-1/729*a^
6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*c^2) + 18*a^2*b^2/c + 81*(-1/729*a^6*b^6/c^3 -
1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*(I*sqrt(3) + 1))*c + 3*sqrt(1/3)*c*sqrt((972*a^4*b^4 + 36*(
a^4*b^4*(-I*sqrt(3) + 1)/((-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*c^2) + 18*a
^2*b^2/c + 81*(-1/729*a^6*b^6/c^3 - 1/1458/c + 1/1458*sqrt(4*a^6*b^6 + c^2)/c^2)^(1/3)*(I*sqrt(3) + 1))*a^2*b^
2*c - (a^4*b^4*(-I*sqrt(3) + 1)/((-1/729*a^6*b^...

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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+c*x^3),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {a^6\,x^6+b^6}{\sqrt {a^2\,x^3-b^2\,x}\,\left (a^6\,x^6-b^6+c\,x^3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________