\(\int \frac {2+x}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx\) [299]

Optimal result

Integrand size = 21, antiderivative size = 27 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {3} x}{\sqrt {-1+3 x+x^3}}\right )}{\sqrt {3}} \]

[Out]

-2/3*arctanh(3^(1/2)*x/(x^3+3*x-1)^(1/2))*3^(1/2)

Rubi [C] (warning: unable to verify)

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

Time = 2.83 (sec) , antiderivative size = 1340, normalized size of antiderivative = 49.63, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6874, 2091, 732, 430, 2105, 948, 175, 552, 551} \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx=\frac {2 i 2^{5/6} \sqrt {\frac {x-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [3]{\frac {2}{1+\sqrt {5}}}}{\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 {x^2-\left (\sqrt [3]{\frac {2}{1+\sqrt {5}}}-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}\right ) x+\left (\frac {1}{2} \left (1+\sqrt {5}\right )\right )^{2/3}+\left (\frac {2}{1+\sqrt {5}}\right )^{2/3}+1} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {i \left (-2 x-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 [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [3]{\frac {2}{1+\sqrt {5}}}\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 {x^3+3 x-1}}-\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 {x-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [3]{\frac {2}{1+\sqrt {5}}}} \sqrt {1-\frac {2 \left (x-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [3]{\frac {2}{1+\sqrt {5}}}\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 (x-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [3]{\frac {2}{1+\sqrt {5}}}\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}}}} \operatorname {EllipticPi}\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 )},\arcsin \left (\frac {2^{5/6} \sqrt [6]{1+\sqrt {5}} \sqrt {x-\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )}+\sqrt [3]{\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 )}}}\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 {x^3+3 x-1}} \]

[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 175

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 430

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

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], 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 552

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

Rule 732

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 948

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 2091

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) + 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), 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 2105

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 6874

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

Rubi steps \begin{align*} \text {integral}& = \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}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {3} x}{\sqrt {-1+3 x+x^3}}\right )}{\sqrt {3}} \]

[In]

Integrate[(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]

Maple [A] (verified)

Time = 2.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

method	result	size
default	\(-\frac {2 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{3}+3 x -1}\, \sqrt {3}}{3 x}\right )}{3}\)	\(25\)
pseudoelliptic	\(-\frac {2 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{3}+3 x -1}\, \sqrt {3}}{3 x}\right )}{3}\)	\(25\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x -6 \sqrt {x^{3}+3 x -1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{\left (x -1\right )^{3}}\right )}{3}\)	\(69\)
elliptic	\(\text {Expression too large to display}\)	\(1075\)

[In]

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

[Out]

-2/3*3^(1/2)*arctanh(1/3*(x^3+3*x-1)^(1/2)/x*3^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (21) = 42\).

Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.59 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx=\frac {1}{6} \, \sqrt {3} \log \left (\frac {x^{6} + 18 \, x^{5} + 15 \, x^{4} + 52 \, x^{3} - 4 \, \sqrt {3} {\left (x^{4} + 3 \, x^{3} + 3 \, x^{2} - x\right )} \sqrt {x^{3} + 3 \, x - 1} - 9 \, x^{2} - 6 \, x + 1}{x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1}\right ) \]

[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))

Sympy [F]

\[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx=\int \frac {x + 2}{\left (x - 1\right ) \sqrt {x^{3} + 3 x - 1}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx=\int { \frac {x + 2}{\sqrt {x^{3} + 3 \, x - 1} {\left (x - 1\right )}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx=\int { \frac {x + 2}{\sqrt {x^{3} + 3 \, x - 1} {\left (x - 1\right )}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 6.23 (sec) , antiderivative size = 1872, normalized size of antiderivative = 69.33 \[ \int \frac {2+x}{(-1+x) \sqrt {-1+3 x+x^3}} \, dx=\text {Too large to display} \]

[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))