\(\int \frac {b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} (-b^6+a^6 x^6)} \, dx\) [2696]

Optimal result

Integrand size = 44, antiderivative size = 245 \[ \int \frac {b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {2} \sqrt {a} \sqrt {b}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {a} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{b^2+a^2 x^2}\right )}{3 \sqrt {2} \sqrt {a} \sqrt {b}} \]

[Out]

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

Rubi [C] (verified)

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

Time = 5.11 (sec) , antiderivative size = 956, normalized size of antiderivative = 3.90, number of steps used = 185, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.477, Rules used = {2081, 1600, 6847, 6857, 1743, 1223, 1212, 226, 1210, 1225, 1713, 214, 1262, 749, 858, 223, 212, 739, 211, 1231, 1721} \[ \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 {x} \sqrt {b^2+a^2 x^2} \arctan \left (\frac {\sqrt {-a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {-a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {2 \sqrt {x} \sqrt {b^2+a^2 x^2} \arctan \left (\frac {\sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {\sqrt {x} \sqrt {b^2+a^2 x^2} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {\sqrt {x} \sqrt {b^2+a^2 x^2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )}{3 \sqrt {2} \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}+\frac {\left (3+i \sqrt {3}\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{6 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}+\frac {\left (3-i \sqrt {3}\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{6 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \left (3 i-\sqrt {3}\right ) \sqrt {a} \sqrt {b} \sqrt {a^2 x^3+b^2 x}}-\frac {\sqrt {x} (b+a x) \sqrt {\frac {b^2+a^2 x^2}{(b+a x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),\frac {1}{2}\right )}{3 \left (1+\sqrt [3]{-1}\right ) \sqrt {a} \sqrt {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]

(-2*Sqrt[x]*Sqrt[b^2 + a^2*x^2]*ArcTan[(Sqrt[-a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]])/(3*Sqrt[-a]*Sqrt[b]*Sq
rt[b^2*x + a^2*x^3]) - (2*Sqrt[x]*Sqrt[b^2 + a^2*x^2]*ArcTan[(Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]])/(
3*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3]) - (Sqrt[x]*Sqrt[b^2 + a^2*x^2]*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[x
])/Sqrt[b^2 + a^2*x^2]])/(3*Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3]) - (Sqrt[x]*Sqrt[b^2 + a^2*x^2]*ArcT
anh[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]])/(3*Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3])
- (Sqrt[x]*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(3
*(1 + (-1)^(1/3))*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3]) - ((I - Sqrt[3])*Sqrt[x]*(b + a*x)*Sqrt[(b^2 + a^2*x^
2)/(b + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(3*(3*I - Sqrt[3])*Sqrt[a]*Sqrt[b]*Sqrt[b
^2*x + a^2*x^3]) - ((1 - I*Sqrt[3])*Sqrt[x]*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticF[2*ArcTan[(Sq
rt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(3*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3]) + ((3 - I*Sqrt[3])*Sqrt[x]*(b + a*x)*
Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(6*Sqrt[a]*Sqrt[b]*Sqrt
[b^2*x + a^2*x^3]) - ((1 + I*Sqrt[3])*Sqrt[x]*(b + a*x)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticF[2*ArcTan[(
Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(3*Sqrt[a]*Sqrt[b]*Sqrt[b^2*x + a^2*x^3]) + ((3 + I*Sqrt[3])*Sqrt[x]*(b + a*x
)*Sqrt[(b^2 + a^2*x^2)/(b + a*x)^2]*EllipticF[2*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]], 1/2])/(6*Sqrt[a]*Sqrt[b]*Sq
rt[b^2*x + a^2*x^3])

Rule 211

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

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 214

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

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 739

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

Rule 749

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

Rule 858

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

Rule 1210

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

Rule 1212

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

Rule 1223

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

Rule 1225

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

Rule 1231

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

Rule 1262

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

Rule 1600

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

Rule 1713

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

Rule 1721

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]
}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2]))
, x] + Simp[(B*d + A*e)*(A + B*x^2)*(Sqrt[A^2*((a + c*x^4)/(a*(A + B*x^2)^2))]/(4*d*e*A*q*Sqrt[a + c*x^4]))*El
lipticPi[Cancel[-(B*d - A*e)^2/(4*d*e*A*B)], 2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c
*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0]

Rule 1743

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

Rule 2081

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

Rule 6847

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

Rule 6857

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \int \frac {b^6+a^6 x^6}{\sqrt {x} \sqrt {b^2+a^2 x^2} \left (-b^6+a^6 x^6\right )} \, dx}{\sqrt {b^2 x+a^2 x^3}} \\ & = \frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \int \frac {\sqrt {b^2+a^2 x^2} \left (b^4-a^2 b^2 x^2+a^4 x^4\right )}{\sqrt {x} \left (-b^6+a^6 x^6\right )} \, dx}{\sqrt {b^2 x+a^2 x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x^4} \left (b^4-a^2 b^2 x^4+a^4 x^8\right )}{-b^6+a^6 x^{12}} \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {\sqrt {b^2+a^2 x^4}}{12 b^{3/2} \left (\sqrt {b}-\sqrt {a} x\right )}-\frac {\sqrt {b^2+a^2 x^4}}{12 b^{3/2} \left (\sqrt {b}-i \sqrt {a} x\right )}-\frac {\sqrt {b^2+a^2 x^4}}{12 b^{3/2} \left (\sqrt {b}+i \sqrt {a} x\right )}-\frac {\sqrt {b^2+a^2 x^4}}{12 b^{3/2} \left (\sqrt {b}+\sqrt {a} x\right )}-\frac {\left (b^{9/2}+\sqrt [3]{-1} b^{9/2}+(-1)^{2/3} b^{9/2}\right ) \sqrt {b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-\sqrt [6]{-1} \sqrt {a} x\right )}-\frac {\left (b^{9/2}+\sqrt [3]{-1} b^{9/2}+(-1)^{2/3} b^{9/2}\right ) \sqrt {b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+\sqrt [6]{-1} \sqrt {a} x\right )}-\frac {\left (b^{9/2}-\sqrt [3]{-1} b^{9/2}-(-1)^{2/3} b^{9/2}\right ) \sqrt {b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-\sqrt [3]{-1} \sqrt {a} x\right )}-\frac {\left (b^{9/2}-\sqrt [3]{-1} b^{9/2}-(-1)^{2/3} b^{9/2}\right ) \sqrt {b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+\sqrt [3]{-1} \sqrt {a} x\right )}-\frac {\left (b^{9/2}+\sqrt [3]{-1} b^{9/2}+(-1)^{2/3} b^{9/2}\right ) \sqrt {b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-(-1)^{2/3} \sqrt {a} x\right )}-\frac {\left (b^{9/2}+\sqrt [3]{-1} b^{9/2}+(-1)^{2/3} b^{9/2}\right ) \sqrt {b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+(-1)^{2/3} \sqrt {a} x\right )}-\frac {\left (b^{9/2}-\sqrt [3]{-1} b^{9/2}-(-1)^{2/3} b^{9/2}\right ) \sqrt {b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}-(-1)^{5/6} \sqrt {a} x\right )}-\frac {\left (b^{9/2}-\sqrt [3]{-1} b^{9/2}-(-1)^{2/3} b^{9/2}\right ) \sqrt {b^2+a^2 x^4}}{12 b^6 \left (\sqrt {b}+(-1)^{5/6} \sqrt {a} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {b^2 x+a^2 x^3}} \\ & = -\frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x^4}}{\sqrt {b}-\sqrt {a} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x^4}}{\sqrt {b}-i \sqrt {a} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x^4}}{\sqrt {b}+i \sqrt {a} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x^4}}{\sqrt {b}+\sqrt {a} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x^4}}{\sqrt {b}-\sqrt [3]{-1} \sqrt {a} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x^4}}{\sqrt {b}+\sqrt [3]{-1} \sqrt {a} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x^4}}{\sqrt {b}-(-1)^{5/6} \sqrt {a} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x^4}}{\sqrt {b}+(-1)^{5/6} \sqrt {a} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x^4}}{\sqrt {b}-\sqrt [6]{-1} \sqrt {a} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x^4}}{\sqrt {b}+\sqrt [6]{-1} \sqrt {a} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x^4}}{\sqrt {b}-(-1)^{2/3} \sqrt {a} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {b^2 x+a^2 x^3}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x^4}}{\sqrt {b}+(-1)^{2/3} \sqrt {a} x} \, dx,x,\sqrt {x}\right )}{6 b^{3/2} \sqrt {b^2 x+a^2 x^3}} \\ & = -2 \frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x^4}}{b-a x^2} \, dx,x,\sqrt {x}\right )}{6 b \sqrt {b^2 x+a^2 x^3}}-2 \frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x^4}}{b+a x^2} \, dx,x,\sqrt {x}\right )}{6 b \sqrt {b^2 x+a^2 x^3}}-2 \frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x^4}}{b-(-1)^{2/3} a x^2} \, dx,x,\sqrt {x}\right )}{6 b \sqrt {b^2 x+a^2 x^3}}-2 \frac {\left (\left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x^4}}{b+(-1)^{2/3} a x^2} \, dx,x,\sqrt {x}\right )}{6 b \sqrt {b^2 x+a^2 x^3}}-2 \frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x^4}}{b-\sqrt [3]{-1} a x^2} \, dx,x,\sqrt {x}\right )}{6 b \sqrt {b^2 x+a^2 x^3}}-2 \frac {\left (\left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x^4}}{b+\sqrt [3]{-1} a x^2} \, dx,x,\sqrt {x}\right )}{6 b \sqrt {b^2 x+a^2 x^3}} \\ & = -2 \left (-\frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {a^2 b+a^3 x^2}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^2 b \sqrt {b^2 x+a^2 x^3}}+\frac {\left (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}}\right )-2 \left (-\frac {\left (\sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {a^2 b-a^3 x^2}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^2 b \sqrt {b^2 x+a^2 x^3}}+\frac {\left (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}}\right )-2 \left (-\frac {\left ((-1)^{2/3} \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {a^2 b+(-1)^{2/3} a^3 x^2}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^2 b \sqrt {b^2 x+a^2 x^3}}+\frac {\left (\left (1+(-1)^{2/3}\right ) \left (1-i \sqrt {3}\right ) b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (b-(-1)^{2/3} a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {b^2 x+a^2 x^3}}\right )-2 \left (-\frac {\left ((-1)^{2/3} \left (1-i \sqrt {3}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {a^2 b-(-1)^{2/3} a^3 x^2}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^2 b \sqrt {b^2 x+a^2 x^3}}+\frac {\left (\left (1+(-1)^{2/3}\right ) \left (1-i \sqrt {3}\right ) b \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (b+(-1)^{2/3} a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 \sqrt {b^2 x+a^2 x^3}}\right )-2 \left (\frac {\left (\sqrt [3]{-1} \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {a^2 b+\sqrt [3]{-1} a^3 x^2}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^2 b \sqrt {b^2 x+a^2 x^3}}-\frac {\left (\sqrt [3]{-1} \left (1+i \sqrt {3}\right ) \left (a^2 b^2+(-1)^{2/3} a^2 b^2\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (b-\sqrt [3]{-1} a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^2 b \sqrt {b^2 x+a^2 x^3}}\right )-2 \left (\frac {\left (\sqrt [3]{-1} \left (1+i \sqrt {3}\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {a^2 b-\sqrt [3]{-1} a^3 x^2}{\sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^2 b \sqrt {b^2 x+a^2 x^3}}-\frac {\left (\sqrt [3]{-1} \left (1+i \sqrt {3}\right ) \left (a^2 b^2+(-1)^{2/3} a^2 b^2\right ) \sqrt {x} \sqrt {b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (b+\sqrt [3]{-1} a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx,x,\sqrt {x}\right )}{6 a^2 b \sqrt {b^2 x+a^2 x^3}}\right ) \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.82 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.84 \[ \int \frac {b^6+a^6 x^6}{\sqrt {b^2 x+a^2 x^3} \left (-b^6+a^6 x^6\right )} \, dx=-\frac {\sqrt {x} \sqrt {b^2+a^2 x^2} \left (4 \arctan \left (\frac {\sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )+\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )+4 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {x}}{\sqrt {b^2+a^2 x^2}}\right )\right )}{6 \sqrt {a} \sqrt {b} \sqrt {x \left (b^2+a^2 x^2\right )}} \]

[In]

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

[Out]

-1/6*(Sqrt[x]*Sqrt[b^2 + a^2*x^2]*(4*ArcTan[(Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]] + Sqrt[2]*ArcTan[(S
qrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]] + 4*ArcTanh[(Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]
] + Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[b]*Sqrt[x])/Sqrt[b^2 + a^2*x^2]]))/(Sqrt[a]*Sqrt[b]*Sqrt[x*(b^2 + a^
2*x^2)])

Maple [A] (verified)

Time = 1.03 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.53

method	result	size
default	\(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \sqrt {2}}{2 x \sqrt {a b}}\right )-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \sqrt {2}}{2 x \sqrt {a b}}\right )+4 \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {a b}}\right )-4 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {a b}}\right )}{6 \sqrt {a b}}\)	\(130\)
pseudoelliptic	\(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \sqrt {2}}{2 x \sqrt {a b}}\right )-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}\, \sqrt {2}}{2 x \sqrt {a b}}\right )+4 \arctan \left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {a b}}\right )-4 \,\operatorname {arctanh}\left (\frac {\sqrt {x \left (a^{2} x^{2}+b^{2}\right )}}{x \sqrt {a b}}\right )}{6 \sqrt {a b}}\)	\(130\)
elliptic	\(\text {Expression too large to display}\)	\(1660\)

[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]

1/6*(2^(1/2)*arctan(1/2*(x*(a^2*x^2+b^2))^(1/2)/x*2^(1/2)/(a*b)^(1/2))-2^(1/2)*arctanh(1/2*(x*(a^2*x^2+b^2))^(
1/2)/x*2^(1/2)/(a*b)^(1/2))+4*arctan((x*(a^2*x^2+b^2))^(1/2)/x/(a*b)^(1/2))-4*arctanh((x*(a^2*x^2+b^2))^(1/2)/
x/(a*b)^(1/2)))/(a*b)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (189) = 378\).

Time = 0.37 (sec) , antiderivative size = 828, normalized size of antiderivative = 3.38 \[ \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=\left [-\frac {2 \, \sqrt {2} a b \sqrt {\frac {1}{a b}} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {a^{2} x^{3} + b^{2} x} a b \sqrt {\frac {1}{a b}}}{a^{2} x^{2} - 2 \, a b x + b^{2}}\right ) - \sqrt {2} a b \sqrt {\frac {1}{a b}} \log \left (\frac {a^{4} x^{4} + 12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 12 \, a b^{3} x + b^{4} - 4 \, \sqrt {2} {\left (a^{3} b x^{2} + 2 \, a^{2} b^{2} x + a b^{3}\right )} \sqrt {a^{2} x^{3} + b^{2} x} \sqrt {\frac {1}{a b}}}{a^{4} x^{4} - 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 4 \, a b^{3} x + b^{4}}\right ) - 8 \, \sqrt {a b} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} - a b x + b^{2}\right )} \sqrt {a b}}{2 \, {\left (a^{3} b x^{3} + a b^{3} x\right )}}\right ) - 4 \, \sqrt {a b} \log \left (\frac {a^{4} x^{4} + 6 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} + 6 \, a b^{3} x + b^{4} - 4 \, \sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} + a b x + b^{2}\right )} \sqrt {a b}}{a^{4} x^{4} - 2 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} - 2 \, a b^{3} x + b^{4}}\right )}{24 \, a b}, \frac {2 \, \sqrt {2} a b \sqrt {-\frac {1}{a b}} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {a^{2} x^{3} + b^{2} x} a b \sqrt {-\frac {1}{a b}}}{a^{2} x^{2} + 2 \, a b x + b^{2}}\right ) + \sqrt {2} a b \sqrt {-\frac {1}{a b}} \log \left (\frac {a^{4} x^{4} - 12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 12 \, a b^{3} x + b^{4} + 4 \, \sqrt {2} {\left (a^{3} b x^{2} - 2 \, a^{2} b^{2} x + a b^{3}\right )} \sqrt {a^{2} x^{3} + b^{2} x} \sqrt {-\frac {1}{a b}}}{a^{4} x^{4} + 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x + b^{4}}\right ) + 8 \, \sqrt {-a b} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} + a b x + b^{2}\right )} \sqrt {-a b}}{2 \, {\left (a^{3} b x^{3} + a b^{3} x\right )}}\right ) - 4 \, \sqrt {-a b} \log \left (\frac {a^{4} x^{4} - 6 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} - 6 \, a b^{3} x + b^{4} - 4 \, \sqrt {a^{2} x^{3} + b^{2} x} {\left (a^{2} x^{2} - a b x + b^{2}\right )} \sqrt {-a b}}{a^{4} x^{4} + 2 \, a^{3} b x^{3} + 3 \, a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}}\right )}{24 \, a b}\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]

[-1/24*(2*sqrt(2)*a*b*sqrt(1/(a*b))*arctan(2*sqrt(2)*sqrt(a^2*x^3 + b^2*x)*a*b*sqrt(1/(a*b))/(a^2*x^2 - 2*a*b*
x + b^2)) - sqrt(2)*a*b*sqrt(1/(a*b))*log((a^4*x^4 + 12*a^3*b*x^3 + 6*a^2*b^2*x^2 + 12*a*b^3*x + b^4 - 4*sqrt(
2)*(a^3*b*x^2 + 2*a^2*b^2*x + a*b^3)*sqrt(a^2*x^3 + b^2*x)*sqrt(1/(a*b)))/(a^4*x^4 - 4*a^3*b*x^3 + 6*a^2*b^2*x
^2 - 4*a*b^3*x + b^4)) - 8*sqrt(a*b)*arctan(1/2*sqrt(a^2*x^3 + b^2*x)*(a^2*x^2 - a*b*x + b^2)*sqrt(a*b)/(a^3*b
*x^3 + a*b^3*x)) - 4*sqrt(a*b)*log((a^4*x^4 + 6*a^3*b*x^3 + 3*a^2*b^2*x^2 + 6*a*b^3*x + b^4 - 4*sqrt(a^2*x^3 +
b^2*x)*(a^2*x^2 + a*b*x + b^2)*sqrt(a*b))/(a^4*x^4 - 2*a^3*b*x^3 + 3*a^2*b^2*x^2 - 2*a*b^3*x + b^4)))/(a*b),
1/24*(2*sqrt(2)*a*b*sqrt(-1/(a*b))*arctan(2*sqrt(2)*sqrt(a^2*x^3 + b^2*x)*a*b*sqrt(-1/(a*b))/(a^2*x^2 + 2*a*b*
x + b^2)) + sqrt(2)*a*b*sqrt(-1/(a*b))*log((a^4*x^4 - 12*a^3*b*x^3 + 6*a^2*b^2*x^2 - 12*a*b^3*x + b^4 + 4*sqrt
(2)*(a^3*b*x^2 - 2*a^2*b^2*x + a*b^3)*sqrt(a^2*x^3 + b^2*x)*sqrt(-1/(a*b)))/(a^4*x^4 + 4*a^3*b*x^3 + 6*a^2*b^2
*x^2 + 4*a*b^3*x + b^4)) + 8*sqrt(-a*b)*arctan(1/2*sqrt(a^2*x^3 + b^2*x)*(a^2*x^2 + a*b*x + b^2)*sqrt(-a*b)/(a
^3*b*x^3 + a*b^3*x)) - 4*sqrt(-a*b)*log((a^4*x^4 - 6*a^3*b*x^3 + 3*a^2*b^2*x^2 - 6*a*b^3*x + b^4 - 4*sqrt(a^2*
x^3 + b^2*x)*(a^2*x^2 - a*b*x + b^2)*sqrt(-a*b))/(a^4*x^4 + 2*a^3*b*x^3 + 3*a^2*b^2*x^2 + 2*a*b^3*x + b^4)))/(
a*b)]

Sympy [F]

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

[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**2*x**2 + b**2))*(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 [F(-1)]

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

[In]

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

[Out]

\text{Hanged}