3.3.99 \(\int \frac {2+x}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx\)

Optimal. Leaf size=27 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {x^3+3 x-1}}\right )}{\sqrt {3}} \]

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Rubi [C]  time = 3.94, antiderivative size = 1340, normalized size of antiderivative = 49.63, number of steps used = 11, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6742, 2066, 718, 419, 2080, 934, 169, 538, 537}

result too large to display

Warning: Unable to verify antiderivative.

[In]

Int[(2 + x)/((-1 + x)*Sqrt[-1 + 3*x + x^3]),x]

[Out]

((2*I)*2^(5/6)*Sqrt[((2/(1 + Sqrt[5]))^(1/3) - ((1 + Sqrt[5])/2)^(1/3) + x)/(6/(1 + Sqrt[5])^(1/3) - 3*(2*(1 +
Sqrt[5]))^(1/3) - I*2^(1/6)*Sqrt[3*(4 + 2*(2/(1 + Sqrt[5]))^(2/3) + 2^(1/3)*(1 + Sqrt[5])^(2/3))])]*Sqrt[1 +
(2/(1 + Sqrt[5]))^(2/3) + ((1 + Sqrt[5])/2)^(2/3) - ((2/(1 + Sqrt[5]))^(1/3) - ((1 + Sqrt[5])/2)^(1/3))*x + x^
2]*EllipticF[ArcSin[Sqrt[(I*((2/(1 + Sqrt[5]))^(1/3) - ((1 + Sqrt[5])/2)^(1/3) - I*Sqrt[6 + 3*(2/(1 + Sqrt[5])
)^(2/3) + 3*((1 + Sqrt[5])/2)^(2/3)] - 2*x))/Sqrt[6*(4 + 2*(2/(1 + Sqrt[5]))^(2/3) + 2^(1/3)*(1 + Sqrt[5])^(2/
3))]]], (2*2^(1/6)*Sqrt[3*(4 + 2*(2/(1 + Sqrt[5]))^(2/3) + 2^(1/3)*(1 + Sqrt[5])^(2/3))])/((6*I)/(1 + Sqrt[5])
^(1/3) - (3*I)*(2*(1 + Sqrt[5]))^(1/3) + 2^(1/6)*Sqrt[3*(4 + 2*(2/(1 + Sqrt[5]))^(2/3) + 2^(1/3)*(1 + Sqrt[5])
^(2/3))])])/Sqrt[-1 + 3*x + x^3] - (3*(2/(1 + Sqrt[5]))^(1/6)*Sqrt[6 - 3*2^(1/3)*(1 + Sqrt[5])^(2/3) + I*2^(1/
6)*(1 + Sqrt[5])^(1/3)*Sqrt[3*(4 + 2*(2/(1 + Sqrt[5]))^(2/3) + 2^(1/3)*(1 + Sqrt[5])^(2/3))]]*Sqrt[(2/(1 + Sqr
t[5]))^(1/3) - ((1 + Sqrt[5])/2)^(1/3) + x]*Sqrt[1 - (2*((2/(1 + Sqrt[5]))^(1/3) - ((1 + Sqrt[5])/2)^(1/3) + x
))/(3*(2/(1 + Sqrt[5]))^(1/3) - 3*((1 + Sqrt[5])/2)^(1/3) - I*Sqrt[6 + 3*(2/(1 + Sqrt[5]))^(2/3) + 3*((1 + Sqr
t[5])/2)^(2/3)])]*Sqrt[1 - (2*((2/(1 + Sqrt[5]))^(1/3) - ((1 + Sqrt[5])/2)^(1/3) + x))/(3*(2/(1 + Sqrt[5]))^(1
/3) - 3*((1 + Sqrt[5])/2)^(1/3) + I*Sqrt[6 + 3*(2/(1 + Sqrt[5]))^(2/3) + 3*((1 + Sqrt[5])/2)^(2/3)])]*Elliptic
Pi[(3*(2/(1 + Sqrt[5]))^(1/3) - 3*((1 + Sqrt[5])/2)^(1/3) + I*Sqrt[6 + 3*(2/(1 + Sqrt[5]))^(2/3) + 3*((1 + Sqr
t[5])/2)^(2/3)])/(2*(1 + (2/(1 + Sqrt[5]))^(1/3) - ((1 + Sqrt[5])/2)^(1/3))), ArcSin[(2^(5/6)*(1 + Sqrt[5])^(1
/6)*Sqrt[(2/(1 + Sqrt[5]))^(1/3) - ((1 + Sqrt[5])/2)^(1/3) + x])/Sqrt[6 - 3*2^(1/3)*(1 + Sqrt[5])^(2/3) + I*2^
(1/6)*(1 + Sqrt[5])^(1/3)*Sqrt[3*(4 + 2*(2/(1 + Sqrt[5]))^(2/3) + 2^(1/3)*(1 + Sqrt[5])^(2/3))]]], (3*(2/(1 +
Sqrt[5]))^(1/3) - 3*((1 + Sqrt[5])/2)^(1/3) + I*Sqrt[6 + 3*(2/(1 + Sqrt[5]))^(2/3) + 3*((1 + Sqrt[5])/2)^(2/3)
])/(3*(2/(1 + Sqrt[5]))^(1/3) - 3*((1 + Sqrt[5])/2)^(1/3) - I*Sqrt[6 + 3*(2/(1 + Sqrt[5]))^(2/3) + 3*((1 + Sqr
t[5])/2)^(2/3)])])/((1 + (2/(1 + Sqrt[5]))^(1/3) - ((1 + Sqrt[5])/2)^(1/3))*Sqrt[-1 + 3*x + x^3])

Rule 169

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] &&  !SimplerQ[e
+ f*x, c + d*x] &&  !SimplerQ[g + h*x, c + d*x]

Rule 419

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

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 538

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

Rule 718

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

Rule 934

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x])/Sqrt[a + b*x + c*x^2], Int[1/((d +
e*x)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 2066

Int[((a_.) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27
*a^2*d^2], 3]}, Dist[(a + b*x + d*x^3)^p/(Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/3) + d*x, x]^p*Simp[(b*d)/3 + (1
2^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r/18^(1/3))*x + d^2*x^2, x]^p), I
nt[Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/3) + d*x, x]^p*Simp[(b*d)/3 + (12^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1
/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, p}, x] && NeQ[
4*b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]

Rule 2080

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + S
qrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Dist[(a + b*x + d*x^3)^p/(Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/3) + d
*x, x]^p*Simp[(b*d)/3 + (12^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r/18^(1
/3))*x + d^2*x^2, x]^p), Int[(e + f*x)^m*Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/3) + d*x, x]^p*Simp[(b*d)/3 + (12
^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r/18^(1/3))*x + d^2*x^2, x]^p, x],
x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && NeQ[4*b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx &=\int \left (\frac {1}{\sqrt {-1+3 x+x^3}}+\frac {3}{(-1+x) \sqrt {-1+3 x+x^3}}\right ) \, dx\\ &=3 \int \frac {1}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx+\int \frac {1}{\sqrt {-1+3 x+x^3}} \, dx\\ &=\frac {\left (\sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2}\right ) \int \frac {1}{\sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2}} \, dx}{\sqrt {-1+3 x+x^3}}+\frac {\left (3 \sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2}\right ) \int \frac {1}{(-1+x) \sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}+\left (-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2}} \, dx}{\sqrt {-1+3 x+x^3}}\\ &=\frac {\left (3 \sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2}\right ) \int \frac {1}{(-1+x) \sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x}} \, dx}{\sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}+\left (-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} \sqrt {-1+3 x+x^3}}+\frac {\left (2 i \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )} \sqrt {\frac {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x}{\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+2 \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right )-i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} \sqrt {-\frac {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}+\left (-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2}{\left (-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right )^2-4 \left (1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}\right )}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {i \sqrt {6 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )} x^2}{\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+2 \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right )-i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}}} \, dx,x,\sqrt {-\frac {i \left (-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x\right )}{\sqrt {6 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}}\right )}{\sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}+\left (-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} \sqrt {-1+3 x+x^3}}\\ &=\frac {2 i 2^{5/6} \sqrt {\frac {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x}{\frac {6}{\sqrt [3]{1+\sqrt {5}}}-3 \sqrt [3]{2 \left (1+\sqrt {5}\right )}-i \sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} F\left (\sin ^{-1}\left (\sqrt {\frac {i \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}-2 x\right )}{\sqrt {6 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}}\right )|\frac {2 \sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}{\frac {6 i}{\sqrt [3]{1+\sqrt {5}}}-3 i \sqrt [3]{2 \left (1+\sqrt {5}\right )}+\sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}\right )}{\sqrt {-1+3 x+x^3}}-\frac {\left (6 \sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-x^2\right ) \sqrt {-3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}+3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}+2 x^2} \sqrt {-3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}+3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}+2 x^2}} \, dx,x,\sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x}\right )}{\sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}+\left (-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} \sqrt {-1+3 x+x^3}}\\ &=\frac {2 i 2^{5/6} \sqrt {\frac {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x}{\frac {6}{\sqrt [3]{1+\sqrt {5}}}-3 \sqrt [3]{2 \left (1+\sqrt {5}\right )}-i \sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} F\left (\sin ^{-1}\left (\sqrt {\frac {i \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}-2 x\right )}{\sqrt {6 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}}\right )|\frac {2 \sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}{\frac {6 i}{\sqrt [3]{1+\sqrt {5}}}-3 i \sqrt [3]{2 \left (1+\sqrt {5}\right )}+\sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}\right )}{\sqrt {-1+3 x+x^3}}-\frac {\left (6 \sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} \sqrt {1+\frac {2 \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x\right )}{-3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}+3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-x^2\right ) \sqrt {-3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}+3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}+2 x^2} \sqrt {1+\frac {2 x^2}{-3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}+3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}}} \, dx,x,\sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x}\right )}{\sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}+2 x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}+\left (-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} \sqrt {-1+3 x+x^3}}\\ &=\frac {2 i 2^{5/6} \sqrt {\frac {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x}{\frac {6}{\sqrt [3]{1+\sqrt {5}}}-3 \sqrt [3]{2 \left (1+\sqrt {5}\right )}-i \sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} F\left (\sin ^{-1}\left (\sqrt {\frac {i \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}-2 x\right )}{\sqrt {6 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}}\right )|\frac {2 \sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}{\frac {6 i}{\sqrt [3]{1+\sqrt {5}}}-3 i \sqrt [3]{2 \left (1+\sqrt {5}\right )}+\sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}\right )}{\sqrt {-1+3 x+x^3}}-\frac {\left (6 \sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {\frac {3}{2} \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}+2 x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} \sqrt {1+\frac {2 \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x\right )}{-3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}+3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}} \sqrt {1+\frac {2 \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x\right )}{-3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}+3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-x^2\right ) \sqrt {1+\frac {2 x^2}{-3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}+3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}} \sqrt {1+\frac {2 x^2}{-3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}+3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}}} \, dx,x,\sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x}\right )}{\sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}+2 x} \sqrt {-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}+2 x} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}+\left (-\sqrt [3]{\frac {2}{1+\sqrt {5}}}+\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} \sqrt {-1+3 x+x^3}}\\ &=\frac {2 i 2^{5/6} \sqrt {\frac {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x}{\frac {6}{\sqrt [3]{1+\sqrt {5}}}-3 \sqrt [3]{2 \left (1+\sqrt {5}\right )}-i \sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}} \sqrt {1+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+x^2} F\left (\sin ^{-1}\left (\sqrt {\frac {i \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}-2 x\right )}{\sqrt {6 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}}\right )|\frac {2 \sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}{\frac {6 i}{\sqrt [3]{1+\sqrt {5}}}-3 i \sqrt [3]{2 \left (1+\sqrt {5}\right )}+\sqrt [6]{2} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}\right )}{\sqrt {-1+3 x+x^3}}-\frac {3 \sqrt [6]{\frac {2}{1+\sqrt {5}}} \sqrt {6-3 \sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}+i \sqrt [6]{2} \sqrt [3]{1+\sqrt {5}} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}} \sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x} \sqrt {1-\frac {2 \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x\right )}{3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}-3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}} \sqrt {1-\frac {2 \left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x\right )}{3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}-3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}} \Pi \left (\frac {3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}-3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}{2 \left (1+\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right )};\sin ^{-1}\left (\frac {2^{5/6} \sqrt [6]{1+\sqrt {5}} \sqrt {\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+x}}{\sqrt {6-3 \sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}+i \sqrt [6]{2} \sqrt [3]{1+\sqrt {5}} \sqrt {3 \left (4+2 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+\sqrt [3]{2} \left (1+\sqrt {5}\right )^{2/3}\right )}}}\right )|\frac {3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}-3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}{3 \sqrt [3]{\frac {2}{1+\sqrt {5}}}-3 \sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}-i \sqrt {6+3 \left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+3 \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}}}\right )}{\left (1+\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) \sqrt {-1+3 x+x^3}}\\ \end {align*}

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Mathematica [C]  time = 1.06, size = 812, normalized size = 30.07 \begin {gather*} \frac {2 \sqrt {\frac {-x+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,1\right ]+1}{\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}} \left (\frac {3 \sqrt {-\frac {\left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]-1\right ) \left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]-1\right )}{\left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]\right )^2}} \left (\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]\right ) \Pi \left (1-\frac {\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]}{\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]};\sin ^{-1}\left (\sqrt {\frac {-x+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]+1}{-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}}\right )|\frac {\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}{\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}\right )}{\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}+\frac {F\left (\sin ^{-1}\left (\sqrt {\frac {-x+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]+1}{-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}}\right )|\frac {\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}{\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}\right ) \left (x-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]-1\right ) \sqrt {\frac {-x+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]+1}{\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}}}{\sqrt {\frac {-x+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]+1}{-\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,2\right ]+\text {Root}\left [\text {$\#$1}^3+3 \text {$\#$1}^2+6 \text {$\#$1}+3\&,3\right ]}}}\right )}{\sqrt {x^3+3 x-1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(2 + x)/((-1 + x)*Sqrt[-1 + 3*x + x^3]),x]

[Out]

(2*Sqrt[(1 - x + Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 1, 0])/(Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 1, 0] - Root[3 +
6*#1 + 3*#1^2 + #1^3 & , 3, 0])]*((3*EllipticPi[1 - Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 2, 0]/Root[3 + 6*#1 + 3*
#1^2 + #1^3 & , 3, 0], ArcSin[Sqrt[(1 - x + Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 3, 0])/(-Root[3 + 6*#1 + 3*#1^2
+ #1^3 & , 2, 0] + Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 3, 0])]], (Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 2, 0] - Root
[3 + 6*#1 + 3*#1^2 + #1^3 & , 3, 0])/(Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 1, 0] - Root[3 + 6*#1 + 3*#1^2 + #1^3
& , 3, 0])]*Sqrt[-(((-1 + x - Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 2, 0])*(-1 + x - Root[3 + 6*#1 + 3*#1^2 + #1^3
& , 3, 0]))/(Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 2, 0] - Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 3, 0])^2)]*(Root[3 +
6*#1 + 3*#1^2 + #1^3 & , 2, 0] - Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 3, 0]))/Root[3 + 6*#1 + 3*#1^2 + #1^3 & ,
3, 0] + (EllipticF[ArcSin[Sqrt[(1 - x + Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 3, 0])/(-Root[3 + 6*#1 + 3*#1^2 + #1
^3 & , 2, 0] + Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 3, 0])]], (Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 2, 0] - Root[3 +
6*#1 + 3*#1^2 + #1^3 & , 3, 0])/(Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 1, 0] - Root[3 + 6*#1 + 3*#1^2 + #1^3 & ,
3, 0])]*(-1 + x - Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 3, 0])*Sqrt[(1 - x + Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 2,
0])/(Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 2, 0] - Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 3, 0])])/Sqrt[(1 - x + Root[3
+ 6*#1 + 3*#1^2 + #1^3 & , 3, 0])/(-Root[3 + 6*#1 + 3*#1^2 + #1^3 & , 2, 0] + Root[3 + 6*#1 + 3*#1^2 + #1^3 &
, 3, 0])]))/Sqrt[-1 + 3*x + x^3]

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IntegrateAlgebraic [A]  time = 0.21, size = 27, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {-1+3 x+x^3}}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2 + x)/((-1 + x)*Sqrt[-1 + 3*x + x^3]),x]

[Out]

(-2*ArcTanh[(Sqrt[3]*x)/Sqrt[-1 + 3*x + x^3]])/Sqrt[3]

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fricas [B]  time = 0.49, size = 97, normalized size = 3.59 \begin {gather*} \frac {1}{6} \, \sqrt {3} \log \left (\frac {x^{6} + 18 \, x^{5} + 15 \, x^{4} + 52 \, x^{3} - 4 \, \sqrt {3} {\left (x^{4} + 3 \, x^{3} + 3 \, x^{2} - x\right )} \sqrt {x^{3} + 3 \, x - 1} - 9 \, x^{2} - 6 \, x + 1}{x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3)*log((x^6 + 18*x^5 + 15*x^4 + 52*x^3 - 4*sqrt(3)*(x^4 + 3*x^3 + 3*x^2 - x)*sqrt(x^3 + 3*x - 1) - 9*
x^2 - 6*x + 1)/(x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 2}{\sqrt {x^{3} + 3 \, x - 1} {\left (x - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x + 2)/(sqrt(x^3 + 3*x - 1)*(x - 1)), x)

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maple [C]  time = 0.54, size = 69, normalized size = 2.56

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) x^{3}+3 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{2}-3\right ) x -6 \sqrt {x^{3}+3 x -1}\, x -\RootOf \left (\textit {\_Z}^{2}-3\right )}{\left (-1+x \right )^{3}}\right )}{3}\)	\(69\)
default	\(\frac {2 i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right ) \sqrt {-\frac {i \left (x +\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}-\frac {1}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}}}\, \sqrt {\frac {x -\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}}{-\frac {3 \left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}+\frac {3}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}}}\, \sqrt {\frac {i \left (x +\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}-\frac {1}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}-\frac {1}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{-\frac {3 \left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}+\frac {3}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}}}\right )}{3 \sqrt {x^{3}+3 x -1}}+\frac {2 i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right ) \sqrt {-\frac {i \left (x +\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}-\frac {1}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}}}\, \sqrt {\frac {x -\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}}{-\frac {3 \left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}+\frac {3}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}}}\, \sqrt {\frac {i \left (x +\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}-\frac {1}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}-\frac {1}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}}}}{3}, -\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{-\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}-1}, \sqrt {-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{-\frac {3 \left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}+\frac {3}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}}}\right )}{\sqrt {x^{3}+3 x -1}\, \left (-\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}-1\right )}\)	\(1075\)
elliptic	\(\frac {2 i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right ) \sqrt {-\frac {i \left (x +\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}-\frac {1}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}}}\, \sqrt {\frac {x -\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}}{-\frac {3 \left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}+\frac {3}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}}}\, \sqrt {\frac {i \left (x +\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}-\frac {1}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}}}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}-\frac {1}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{-\frac {3 \left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}+\frac {3}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}}}\right )}{3 \sqrt {x^{3}+3 x -1}}+\frac {2 i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right ) \sqrt {-\frac {i \left (x +\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}-\frac {1}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}}}\, \sqrt {\frac {x -\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}}{-\frac {3 \left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}+\frac {3}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}}}\, \sqrt {\frac {i \left (x +\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}-\frac {1}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}}}\, \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}-\frac {1}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}}}}{3}, -\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{-\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}-1}, \sqrt {-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{-\frac {3 \left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}+\frac {3}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}}}\right )}{\sqrt {x^{3}+3 x -1}\, \left (-\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{4}+\frac {1}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (4+4 \sqrt {5}\right )^{\frac {1}{3}}}\right )}{2}-1\right )}\)	\(1075\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*RootOf(_Z^2-3)*ln(-(RootOf(_Z^2-3)*x^3+3*RootOf(_Z^2-3)*x^2+3*RootOf(_Z^2-3)*x-6*(x^3+3*x-1)^(1/2)*x-RootO
f(_Z^2-3))/(-1+x)^3)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 2}{\sqrt {x^{3} + 3 \, x - 1} {\left (x - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x + 2)/(sqrt(x^3 + 3*x - 1)*(x - 1)), x)

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mupad [B]  time = 1.37, size = 1872, normalized size = 69.33

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(-(x + 1/(5^(1/2)/2 + 1/2)^(1/3) - (5^(1/2)/2 + 1/2)^(1/3))/((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)
/2 + 1/2)^(1/3))*1i)/2 - 3/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (3*(5^(1/2)/2 + 1/2)^(1/3))/2))^(1/2)*ellipticF(asin(
((x + (3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 1/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (
5^(1/2)/2 + 1/2)^(1/3)/2)/((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 3/(2*(5^(1/2
)/2 + 1/2)^(1/3)) + (3*(5^(1/2)/2 + 1/2)^(1/3))/2))^(1/2)), -(3^(1/2)*((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (
5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 3/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (3*(5^(1/2)/2 + 1/2)^(1/3))/2)*1i)/(3*(1/(5^(1
/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))))*((x + (3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^
(1/3))*1i)/2 - 1/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (5^(1/2)/2 + 1/2)^(1/3)/2)/((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3)
+ (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 3/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (3*(5^(1/2)/2 + 1/2)^(1/3))/2))^(1/2)*((3^
(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 3/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (3*(5^(1/2
)/2 + 1/2)^(1/3))/2)*((3^(1/2)*(x - (3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 1/(
2*(5^(1/2)/2 + 1/2)^(1/3)) + (5^(1/2)/2 + 1/2)^(1/3)/2)*1i)/(3*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^
(1/3))))^(1/2))/(x^3 - x*((1/(5^(1/2)/2 + 1/2)^(1/3) - (5^(1/2)/2 + 1/2)^(1/3))*((3^(1/2)*(1/(5^(1/2)/2 + 1/2)
^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 + 1/(2*(5^(1/2)/2 + 1/2)^(1/3)) - (5^(1/2)/2 + 1/2)^(1/3)/2) - (1/(5^(
1/2)/2 + 1/2)^(1/3) - (5^(1/2)/2 + 1/2)^(1/3))*((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))
*1i)/2 - 1/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (5^(1/2)/2 + 1/2)^(1/3)/2) + ((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (
5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 1/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (5^(1/2)/2 + 1/2)^(1/3)/2)*((3^(1/2)*(1/(5^(1/
2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 + 1/(2*(5^(1/2)/2 + 1/2)^(1/3)) - (5^(1/2)/2 + 1/2)^(1/3)/2
)) - (1/(5^(1/2)/2 + 1/2)^(1/3) - (5^(1/2)/2 + 1/2)^(1/3))*((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 +
1/2)^(1/3))*1i)/2 - 1/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (5^(1/2)/2 + 1/2)^(1/3)/2)*((3^(1/2)*(1/(5^(1/2)/2 + 1/2)
^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 + 1/(2*(5^(1/2)/2 + 1/2)^(1/3)) - (5^(1/2)/2 + 1/2)^(1/3)/2))^(1/2) -
(6*(-(x + 1/(5^(1/2)/2 + 1/2)^(1/3) - (5^(1/2)/2 + 1/2)^(1/3))/((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)
/2 + 1/2)^(1/3))*1i)/2 - 3/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (3*(5^(1/2)/2 + 1/2)^(1/3))/2))^(1/2)*((x + (3^(1/2)*
(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 1/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (5^(1/2)/2 + 1/2
)^(1/3)/2)/((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 3/(2*(5^(1/2)/2 + 1/2)^(1/3
)) + (3*(5^(1/2)/2 + 1/2)^(1/3))/2))^(1/2)*((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)
/2 - 3/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (3*(5^(1/2)/2 + 1/2)^(1/3))/2)*ellipticPi(((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^
(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 3/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (3*(5^(1/2)/2 + 1/2)^(1/3))/2)/((3^(1
/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 1/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (5^(1/2)/2 +
1/2)^(1/3)/2 + 1), asin(((x + (3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 1/(2*(5^
(1/2)/2 + 1/2)^(1/3)) + (5^(1/2)/2 + 1/2)^(1/3)/2)/((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1
/3))*1i)/2 - 3/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (3*(5^(1/2)/2 + 1/2)^(1/3))/2))^(1/2)), -(3^(1/2)*((3^(1/2)*(1/(5
^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 3/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (3*(5^(1/2)/2 + 1/2)^
(1/3))/2)*1i)/(3*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))))*((3^(1/2)*(x - (3^(1/2)*(1/(5^(1/2)/2
+ 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 1/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (5^(1/2)/2 + 1/2)^(1/3)/2)*1i
)/(3*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))))^(1/2))/((x^3 - x*((1/(5^(1/2)/2 + 1/2)^(1/3) - (5
^(1/2)/2 + 1/2)^(1/3))*((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 + 1/(2*(5^(1/2)/2
+ 1/2)^(1/3)) - (5^(1/2)/2 + 1/2)^(1/3)/2) - (1/(5^(1/2)/2 + 1/2)^(1/3) - (5^(1/2)/2 + 1/2)^(1/3))*((3^(1/2)*
(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 1/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (5^(1/2)/2 + 1/2
)^(1/3)/2) + ((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 1/(2*(5^(1/2)/2 + 1/2)^(1
/3)) + (5^(1/2)/2 + 1/2)^(1/3)/2)*((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 + 1/(2
*(5^(1/2)/2 + 1/2)^(1/3)) - (5^(1/2)/2 + 1/2)^(1/3)/2)) - (1/(5^(1/2)/2 + 1/2)^(1/3) - (5^(1/2)/2 + 1/2)^(1/3)
)*((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 - 1/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (5^(
1/2)/2 + 1/2)^(1/3)/2)*((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1/3))*1i)/2 + 1/(2*(5^(1/2)/2
+ 1/2)^(1/3)) - (5^(1/2)/2 + 1/2)^(1/3)/2))^(1/2)*((3^(1/2)*(1/(5^(1/2)/2 + 1/2)^(1/3) + (5^(1/2)/2 + 1/2)^(1
/3))*1i)/2 - 1/(2*(5^(1/2)/2 + 1/2)^(1/3)) + (5^(1/2)/2 + 1/2)^(1/3)/2 + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 2}{\left (x - 1\right ) \sqrt {x^{3} + 3 x - 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x + 2)/((x - 1)*sqrt(x**3 + 3*x - 1)), x)

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