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

Optimal. Leaf size=244 \[ \frac {2 \sqrt {a^2 x^3-b^2 x}}{3 \left (b^2-a^2 x^2\right )}-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} 3^{3/4} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3-b^2 x}}{\sqrt {3} a^2 x^2-3 a b x-\sqrt {3} b^2}\right )}{3 \sqrt [4]{3} \sqrt {a} \sqrt {b}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\frac {a^{3/2} x^2}{\sqrt {2} \sqrt [4]{3} \sqrt {b}}-\frac {b^{3/2}}{\sqrt {2} \sqrt [4]{3} \sqrt {a}}+\frac {\sqrt [4]{3} \sqrt {a} \sqrt {b} x}{\sqrt {2}}}{\sqrt {a^2 x^3-b^2 x}}\right )}{3 \sqrt [4]{3} \sqrt {a} \sqrt {b}} \]

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Rubi [C]  time = 3.10, antiderivative size = 519, normalized size of antiderivative = 2.13, number of steps used = 36, number of rules used = 17, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.378, Rules used = {2056, 6715, 6725, 224, 221, 2073, 1152, 414, 21, 423, 427, 426, 424, 253, 6728, 1219, 1218} \begin {gather*} -\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}}+\frac {4 \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 )}{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 {2 a}{a-\sqrt {3} \sqrt {-a^2}};\left .\sin ^{-1}\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 {2 a}{a-\sqrt {3} \sqrt {-a^2}};\left .\sin ^{-1}\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 {2 a}{a+\sqrt {3} \sqrt {-a^2}};\left .\sin ^{-1}\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 {2 a}{a+\sqrt {3} \sqrt {-a^2}};\left .\sin ^{-1}\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 + 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 221

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

Rule 224

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

Rule 253

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

Rule 414

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

Rule 423

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

Rule 424

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

Rule 426

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

Rule 427

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

Rule 1152

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

Rule 1218

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

Rule 1219

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

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 2073

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 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6725

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

Rule 6728

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+a^6 x^6\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \int \frac {b^6+a^6 x^6}{\sqrt {x} \sqrt {-b^2+a^2 x^2} \left (-b^6+a^6 x^6\right )} \, dx}{\sqrt {-b^2 x+a^2 x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 b \sqrt {x} \sqrt {-b^2+a^2 x^2}\right ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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}} 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 b \sqrt {x} \sqrt {-b-a x} \sqrt {b-a x}\right ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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}} 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 {x} \sqrt {-b-a x} \sqrt {b-a x}\right ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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}} 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 {x} \sqrt {-b-a x} \sqrt {b-a x}\right ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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}} 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 {2 a}{a-\sqrt {3} \sqrt {-a^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 {2 a}{a-\sqrt {3} \sqrt {-a^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 {2 a}{a+\sqrt {3} \sqrt {-a^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 {2 a}{a+\sqrt {3} \sqrt {-a^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 {\left (2 \sqrt {x} \sqrt {-b-a x} \sqrt {b-a x}\right ) \operatorname {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 ) \operatorname {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 ) \operatorname {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}} 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 {2 a}{a-\sqrt {3} \sqrt {-a^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 {2 a}{a-\sqrt {3} \sqrt {-a^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 {2 a}{a+\sqrt {3} \sqrt {-a^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 {2 a}{a+\sqrt {3} \sqrt {-a^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 {\left (\sqrt {x} \sqrt {-b-a x} \sqrt {1-\frac {a x}{b}}\right ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 .\sin ^{-1}\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}} 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 {2 a}{a-\sqrt {3} \sqrt {-a^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 {2 a}{a-\sqrt {3} \sqrt {-a^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 {2 a}{a+\sqrt {3} \sqrt {-a^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 {2 a}{a+\sqrt {3} \sqrt {-a^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 {\left (\sqrt {x} (-b-a x) \sqrt {1-\frac {a x}{b}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {a x^2}{b}}}{\sqrt {1-\frac {a x^2}{b}}} \, dx,x,\sqrt {x}\right )}{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 ) \operatorname {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}} F\left (\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 {2 a}{a-\sqrt {3} \sqrt {-a^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 {2 a}{a-\sqrt {3} \sqrt {-a^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 {2 a}{a+\sqrt {3} \sqrt {-a^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 {2 a}{a+\sqrt {3} \sqrt {-a^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*}

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Mathematica [C]  time = 2.11, size = 255, normalized size = 1.05 \begin {gather*} \frac {2 \left (-x^{3/2}-\frac {i x^2 \sqrt {1-\frac {b^2}{a^2 x^2}} \left (2 F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-\frac {2 i}{-i+\sqrt {3}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {2 i}{-i+\sqrt {3}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-\frac {2 i}{i+\sqrt {3}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {2 i}{i+\sqrt {3}};\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {b}{a}}}{\sqrt {x}}\right )\right |-1\right )\right )}{\sqrt {-\frac {b}{a}}}\right )}{3 \sqrt {x} \sqrt {a^2 x^3-b^2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.96, size = 244, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {-b^2 x+a^2 x^3}}{3 \left (b^2-a^2 x^2\right )}-\frac {\sqrt {2} \tan ^{-1}\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} \tanh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[-(b^2*x) + a^2*x^3])/(3*(b^2 - a^2*x^2)) - (Sqrt[2]*ArcTan[(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)])/(3*3^(1/4)*Sqrt[a]*Sqrt[b]) - (Sqrt[2]*ArcTanh[
(-(b^(3/2)/(Sqrt[2]*3^(1/4)*Sqrt[a])) + (3^(1/4)*Sqrt[a]*Sqrt[b]*x)/Sqrt[2] + (a^(3/2)*x^2)/(Sqrt[2]*3^(1/4)*S
qrt[b]))/Sqrt[-(b^2*x) + a^2*x^3]])/(3*3^(1/4)*Sqrt[a]*Sqrt[b])

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fricas [B]  time = 0.72, size = 1145, normalized size = 4.69

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.32, size = 678, normalized size = 2.78

method	result	size
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 {2-\frac {2 a x}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{3 a \sqrt {a^{2} x^{3}-b^{2} x}}-\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 =\RootOf \left (\textit {\_Z}^{2} a^{2}-\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\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 =\RootOf \left (\textit {\_Z}^{2} a^{2}-\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\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 =\RootOf \left (\textit {\_Z}^{2} a^{2}-\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\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 )}}+\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 =\RootOf \left (\textit {\_Z}^{2} a^{2}+\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \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 =\RootOf \left (\textit {\_Z}^{2} a^{2}+\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \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 )}}\)	\(678\)
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 (\textit {\_Z}^{2} a^{2}-\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha a +2 b \right ) \left (\underline {\hspace {1.25 ex}}\alpha a -2 b \right ) \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {\left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, -\frac {\underline {\hspace {1.25 ex}}\alpha a -2 b}{3 b}, \frac {\sqrt {2}}{2}\right )}{\left (2 \underline {\hspace {1.25 ex}}\alpha a -b \right ) \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}\right )}{9 a}-\frac {b \left (-\frac {a^{2} x^{2}-a b x}{b^{2} a \sqrt {\left (x +\frac {b}{a}\right ) \left (a^{2} x^{2}-a b x \right )}}+\frac {\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 )}{2 a \sqrt {a^{2} x^{3}-b^{2} x}}+\frac {\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}}\, \left (-\frac {2 b \EllipticE \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {b \EllipticF \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{2 b \sqrt {a^{2} x^{3}-b^{2} x}}\right )}{3}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2} a^{2}+\textit {\_Z} a b +b^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha a -2 b \right ) \underline {\hspace {1.25 ex}}\alpha \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 {\underline {\hspace {1.25 ex}}\alpha a}{b}, \frac {\sqrt {2}}{2}\right )}{\left (2 \underline {\hspace {1.25 ex}}\alpha a +b \right ) \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}\right )}{3}+\frac {b \left (-\frac {a^{2} x^{2}+a b x}{b^{2} a \sqrt {\left (x -\frac {b}{a}\right ) \left (a^{2} x^{2}+a b x \right )}}-\frac {\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 )}{2 a \sqrt {a^{2} x^{3}-b^{2} x}}+\frac {\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}}\, \left (-\frac {2 b \EllipticE \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {b \EllipticF \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{2 b \sqrt {a^{2} x^{3}-b^{2} x}}\right )}{3}\)	\(828\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*x/((x^2-b^2/a^2)*a^2*x)^(1/2)+2/3*b/a*(1+a*x/b)^(1/2)*(2-2*a*x/b)^(1/2)*(-a*x/b)^(1/2)/(a^2*x^3-b^2*x)^(1
/2)*EllipticF(((x+b/a)/b*a)^(1/2),1/2*2^(1/2))-1/9*a*2^(1/2)*((a*x+b)/b)^(1/2)*(-(a*x-b)/b)^(1/2)*(-a*x/b)^(1/
2)/(x*(a^2*x^2-b^2))^(1/2)*sum(1/(2*_alpha*a-b)*EllipticPi(((a*x+b)/b)^(1/2),-1/3*(_alpha*a-2*b)/b,1/2*2^(1/2)
)*_alpha^2,_alpha=RootOf(_Z^2*a^2-_Z*a*b+b^2))+4/9*2^(1/2)*((a*x+b)/b)^(1/2)*(-(a*x-b)/b)^(1/2)*(-a*x/b)^(1/2)
/(x*(a^2*x^2-b^2))^(1/2)*b*sum(1/(2*_alpha*a-b)*EllipticPi(((a*x+b)/b)^(1/2),-1/3*(_alpha*a-2*b)/b,1/2*2^(1/2)
)*_alpha,_alpha=RootOf(_Z^2*a^2-_Z*a*b+b^2))-4/9/a*2^(1/2)*((a*x+b)/b)^(1/2)*(-(a*x-b)/b)^(1/2)*(-a*x/b)^(1/2)
/(x*(a^2*x^2-b^2))^(1/2)*b^2*sum(1/(2*_alpha*a-b)*EllipticPi(((a*x+b)/b)^(1/2),-1/3*(_alpha*a-2*b)/b,1/2*2^(1/
2)),_alpha=RootOf(_Z^2*a^2-_Z*a*b+b^2))+1/3*2^(1/2)*((a*x+b)/b)^(1/2)*(-(a*x-b)/b)^(1/2)*(-a*x/b)^(1/2)/(x*(a^
2*x^2-b^2))^(1/2)*a*sum(1/(2*_alpha*a+b)*_alpha^2*EllipticPi(((a*x+b)/b)^(1/2),-_alpha/b*a,1/2*2^(1/2)),_alpha
=RootOf(_Z^2*a^2+_Z*a*b+b^2))+2/3*2^(1/2)*((a*x+b)/b)^(1/2)*(-(a*x-b)/b)^(1/2)*(-a*x/b)^(1/2)/(x*(a^2*x^2-b^2)
)^(1/2)*b*sum(1/(2*_alpha*a+b)*_alpha*EllipticPi(((a*x+b)/b)^(1/2),-_alpha/b*a,1/2*2^(1/2)),_alpha=RootOf(_Z^2
*a^2+_Z*a*b+b^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 8.57, size = 236, normalized size = 0.97 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a**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)

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