\(\int \frac {2 x+x^2}{(-2-2 x+x^2) \sqrt {-1+x^3}} \, dx\) [1372]

Optimal result

Integrand size = 27, antiderivative size = 99 \[ \int \frac {2 x+x^2}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=\frac {1}{3} \sqrt {-3+2 \sqrt {3}} \arctan \left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {-1+x^3}}{1+x+x^2}\right )-\frac {1}{3} \sqrt {3+2 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {-1+x^3}}{1+x+x^2}\right ) \]

[Out]

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

Rubi [C] (verified)

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

Time = 0.48 (sec) , antiderivative size = 450, normalized size of antiderivative = 4.55, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1607, 6860, 225, 2160, 2165, 212, 209} \[ \int \frac {2 x+x^2}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=\frac {\sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {\left (2-\sqrt {3}\right )^{3/2} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {1}{3} \sqrt {2 \sqrt {3}-3} \arctan \left (\frac {\sqrt {2 \sqrt {3}-3} (1-x)}{\sqrt {x^3-1}}\right )+\frac {1}{3} \sqrt {3+2 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {x^3-1}}\right ) \]

[In]

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

[Out]

-1/3*(Sqrt[-3 + 2*Sqrt[3]]*ArcTan[(Sqrt[-3 + 2*Sqrt[3]]*(1 - x))/Sqrt[-1 + x^3]]) + (Sqrt[3 + 2*Sqrt[3]]*ArcTa
nh[(Sqrt[3 + 2*Sqrt[3]]*(1 - x))/Sqrt[-1 + x^3]])/3 - (2*Sqrt[2 - Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqr
t[3] - x)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[-((1 - x)/(
1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3]) - ((2 - Sqrt[3])^(3/2)*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*El
lipticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt[3]])/(3^(3/4)*Sqrt[-((1 - x)/(1 - Sqrt[3] - x
)^2)]*Sqrt[-1 + x^3]) + (Sqrt[2 + Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*EllipticF[ArcSin[(1
+ Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt[3]])/(3^(3/4)*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x
^3])

Rule 209

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

Rule 212

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

Rule 225

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt
[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sq
rt[(-s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r
*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2160

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[-6*a*(d^3/(c*(b*c^3 - 28*a*d^3))), In
t[1/Sqrt[a + b*x^3], x], x] + Dist[1/(c*(b*c^3 - 28*a*d^3)), Int[Simp[c*(b*c^3 - 22*a*d^3) + 6*a*d^4*x, x]/((c
+ d*x)*Sqrt[a + b*x^3]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0]

Rule 2165

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{k = Simplify[(d*e
+ 2*c*f)/(c*f)]}, Dist[(1 + k)*(e/d), Subst[Int[1/(1 + (3 + 2*k)*a*x^2), x], x, (1 + (1 + k)*d*(x/c))/Sqrt[a +
b*x^3]], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6
, 0] && EqQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3), 0]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x (2+x)}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx \\ & = \int \left (\frac {1}{\sqrt {-1+x^3}}+\frac {2 (1+2 x)}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}}\right ) \, dx \\ & = 2 \int \frac {1+2 x}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx+\int \frac {1}{\sqrt {-1+x^3}} \, dx \\ & = -\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+2 \int \left (\frac {2+\sqrt {3}}{\left (-2-2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}}+\frac {2-\sqrt {3}}{\left (-2+2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}}\right ) \, dx \\ & = -\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\left (2 \left (2-\sqrt {3}\right )\right ) \int \frac {1}{\left (-2+2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx+\left (2 \left (2+\sqrt {3}\right )\right ) \int \frac {1}{\left (-2-2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx \\ & = -\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {1}{288} \left (-3+2 \sqrt {3}\right ) \int \frac {96 \left (1+\sqrt {3}\right )-96 x}{\left (-2+2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx+\frac {1}{6} \left (-3+2 \sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^3}} \, dx-\frac {1}{288} \left (3+2 \sqrt {3}\right ) \int \frac {96 \left (1-\sqrt {3}\right )-96 x}{\left (-2-2 \sqrt {3}+2 x\right ) \sqrt {-1+x^3}} \, dx-\frac {1}{6} \left (3+2 \sqrt {3}\right ) \int \frac {1}{\sqrt {-1+x^3}} \, dx \\ & = -\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {\left (2-\sqrt {3}\right )^{3/2} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {\sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {1}{3} \left (3-2 \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{1-\left (3-2 \sqrt {3}\right ) x^2} \, dx,x,\frac {1+\frac {2 \left (1-\sqrt {3}\right ) x}{-2+2 \sqrt {3}}}{\sqrt {-1+x^3}}\right )+\frac {1}{3} \left (3+2 \sqrt {3}\right ) \text {Subst}\left (\int \frac {1}{1-\left (3+2 \sqrt {3}\right ) x^2} \, dx,x,\frac {1+\frac {2 \left (1+\sqrt {3}\right ) x}{-2-2 \sqrt {3}}}{\sqrt {-1+x^3}}\right ) \\ & = -\frac {1}{3} \sqrt {-3+2 \sqrt {3}} \arctan \left (\frac {\sqrt {-3+2 \sqrt {3}} (1-x)}{\sqrt {-1+x^3}}\right )+\frac {1}{3} \sqrt {3+2 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {3+2 \sqrt {3}} (1-x)}{\sqrt {-1+x^3}}\right )-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {\left (2-\sqrt {3}\right )^{3/2} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {\sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1-\sqrt {3}-x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-x}{1-\sqrt {3}-x}\right ),-7+4 \sqrt {3}\right )}{3^{3/4} \sqrt {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.38 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int \frac {2 x+x^2}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=\frac {1}{3} \sqrt {-3+2 \sqrt {3}} \arctan \left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {-1+x^3}}{1+x+x^2}\right )-\frac {1}{3} \sqrt {3+2 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {-1+x^3}}{1+x+x^2}\right ) \]

[In]

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

[Out]

(Sqrt[-3 + 2*Sqrt[3]]*ArcTan[(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[-1 + x^3])/(1 + x + x^2)])/3 - (Sqrt[3 + 2*Sqrt[3]]*Ar
cTanh[(Sqrt[3 + 2*Sqrt[3]]*Sqrt[-1 + x^3])/(1 + x + x^2)])/3

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 8.80 (sec) , antiderivative size = 584, normalized size of antiderivative = 5.90

method	result	size
trager	\(\frac {\operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right ) \ln \left (\frac {-243 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{5} x^{2}-486 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{5} x +306 x^{2} \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{3}+396 x \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{3}+48 \sqrt {x^{3}-1}\, \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}+216 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{3}-95 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right ) x^{2}-38 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right ) x -32 \sqrt {x^{3}-1}-152 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )}{{\left (9 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2} x -5 x -4\right )}^{2}}\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+9 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}-6\right ) \ln \left (\frac {243 \operatorname {RootOf}\left (\textit {\_Z}^{2}+9 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}-6\right ) \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{4} x^{2}+486 \operatorname {RootOf}\left (\textit {\_Z}^{2}+9 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}-6\right ) \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{4} x -18 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+9 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}-6\right ) x^{2}-252 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+9 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}-6\right ) x +216 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+9 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}-6\right )-\operatorname {RootOf}\left (\textit {\_Z}^{2}+9 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}-6\right ) x^{2}-144 \sqrt {x^{3}-1}\, \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}-10 \operatorname {RootOf}\left (\textit {\_Z}^{2}+9 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}-6\right ) x +8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+9 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2}-6\right )}{{\left (9 \operatorname {RootOf}\left (27 \textit {\_Z}^{4}-18 \textit {\_Z}^{2}-1\right )^{2} x -x +4\right )}^{2}}\right )}{6}\)	\(584\)
default	\(\text {Expression too large to display}\)	\(1517\)
elliptic	\(\text {Expression too large to display}\)	\(1726\)

[In]

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

[Out]

1/2*RootOf(27*_Z^4-18*_Z^2-1)*ln((-243*RootOf(27*_Z^4-18*_Z^2-1)^5*x^2-486*RootOf(27*_Z^4-18*_Z^2-1)^5*x+306*x
^2*RootOf(27*_Z^4-18*_Z^2-1)^3+396*x*RootOf(27*_Z^4-18*_Z^2-1)^3+48*(x^3-1)^(1/2)*RootOf(27*_Z^4-18*_Z^2-1)^2+
216*RootOf(27*_Z^4-18*_Z^2-1)^3-95*RootOf(27*_Z^4-18*_Z^2-1)*x^2-38*RootOf(27*_Z^4-18*_Z^2-1)*x-32*(x^3-1)^(1/
2)-152*RootOf(27*_Z^4-18*_Z^2-1))/(9*RootOf(27*_Z^4-18*_Z^2-1)^2*x-5*x-4)^2)-1/6*RootOf(_Z^2+9*RootOf(27*_Z^4-
18*_Z^2-1)^2-6)*ln((243*RootOf(_Z^2+9*RootOf(27*_Z^4-18*_Z^2-1)^2-6)*RootOf(27*_Z^4-18*_Z^2-1)^4*x^2+486*RootO
f(_Z^2+9*RootOf(27*_Z^4-18*_Z^2-1)^2-6)*RootOf(27*_Z^4-18*_Z^2-1)^4*x-18*RootOf(27*_Z^4-18*_Z^2-1)^2*RootOf(_Z
^2+9*RootOf(27*_Z^4-18*_Z^2-1)^2-6)*x^2-252*RootOf(27*_Z^4-18*_Z^2-1)^2*RootOf(_Z^2+9*RootOf(27*_Z^4-18*_Z^2-1
)^2-6)*x+216*RootOf(27*_Z^4-18*_Z^2-1)^2*RootOf(_Z^2+9*RootOf(27*_Z^4-18*_Z^2-1)^2-6)-RootOf(_Z^2+9*RootOf(27*
_Z^4-18*_Z^2-1)^2-6)*x^2-144*(x^3-1)^(1/2)*RootOf(27*_Z^4-18*_Z^2-1)^2-10*RootOf(_Z^2+9*RootOf(27*_Z^4-18*_Z^2
-1)^2-6)*x+8*RootOf(_Z^2+9*RootOf(27*_Z^4-18*_Z^2-1)^2-6))/(9*RootOf(27*_Z^4-18*_Z^2-1)^2*x-x+4)^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (75) = 150\).

Time = 0.29 (sec) , antiderivative size = 360, normalized size of antiderivative = 3.64 \[ \int \frac {2 x+x^2}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=-\frac {1}{12} \, \sqrt {2 \, \sqrt {3} + 3} \log \left (\frac {x^{4} + 2 \, x^{3} + 6 \, x^{2} + 2 \, \sqrt {x^{3} - 1} {\left (x^{2} + 2 \, \sqrt {3} {\left (x - 1\right )} - 2 \, x + 4\right )} \sqrt {2 \, \sqrt {3} + 3} + 4 \, \sqrt {3} {\left (x^{3} - 1\right )} - 4 \, x + 4}{x^{4} - 4 \, x^{3} + 8 \, x + 4}\right ) + \frac {1}{12} \, \sqrt {2 \, \sqrt {3} + 3} \log \left (\frac {x^{4} + 2 \, x^{3} + 6 \, x^{2} - 2 \, \sqrt {x^{3} - 1} {\left (x^{2} + 2 \, \sqrt {3} {\left (x - 1\right )} - 2 \, x + 4\right )} \sqrt {2 \, \sqrt {3} + 3} + 4 \, \sqrt {3} {\left (x^{3} - 1\right )} - 4 \, x + 4}{x^{4} - 4 \, x^{3} + 8 \, x + 4}\right ) - \frac {1}{24} \, \sqrt {-8 \, \sqrt {3} + 12} \log \left (\frac {x^{4} + 2 \, x^{3} + 6 \, x^{2} + \sqrt {x^{3} - 1} {\left (x^{2} - 2 \, \sqrt {3} {\left (x - 1\right )} - 2 \, x + 4\right )} \sqrt {-8 \, \sqrt {3} + 12} - 4 \, \sqrt {3} {\left (x^{3} - 1\right )} - 4 \, x + 4}{x^{4} - 4 \, x^{3} + 8 \, x + 4}\right ) + \frac {1}{24} \, \sqrt {-8 \, \sqrt {3} + 12} \log \left (\frac {x^{4} + 2 \, x^{3} + 6 \, x^{2} - \sqrt {x^{3} - 1} {\left (x^{2} - 2 \, \sqrt {3} {\left (x - 1\right )} - 2 \, x + 4\right )} \sqrt {-8 \, \sqrt {3} + 12} - 4 \, \sqrt {3} {\left (x^{3} - 1\right )} - 4 \, x + 4}{x^{4} - 4 \, x^{3} + 8 \, x + 4}\right ) \]

[In]

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

[Out]

-1/12*sqrt(2*sqrt(3) + 3)*log((x^4 + 2*x^3 + 6*x^2 + 2*sqrt(x^3 - 1)*(x^2 + 2*sqrt(3)*(x - 1) - 2*x + 4)*sqrt(
2*sqrt(3) + 3) + 4*sqrt(3)*(x^3 - 1) - 4*x + 4)/(x^4 - 4*x^3 + 8*x + 4)) + 1/12*sqrt(2*sqrt(3) + 3)*log((x^4 +
2*x^3 + 6*x^2 - 2*sqrt(x^3 - 1)*(x^2 + 2*sqrt(3)*(x - 1) - 2*x + 4)*sqrt(2*sqrt(3) + 3) + 4*sqrt(3)*(x^3 - 1)
- 4*x + 4)/(x^4 - 4*x^3 + 8*x + 4)) - 1/24*sqrt(-8*sqrt(3) + 12)*log((x^4 + 2*x^3 + 6*x^2 + sqrt(x^3 - 1)*(x^
2 - 2*sqrt(3)*(x - 1) - 2*x + 4)*sqrt(-8*sqrt(3) + 12) - 4*sqrt(3)*(x^3 - 1) - 4*x + 4)/(x^4 - 4*x^3 + 8*x + 4
)) + 1/24*sqrt(-8*sqrt(3) + 12)*log((x^4 + 2*x^3 + 6*x^2 - sqrt(x^3 - 1)*(x^2 - 2*sqrt(3)*(x - 1) - 2*x + 4)*s
qrt(-8*sqrt(3) + 12) - 4*sqrt(3)*(x^3 - 1) - 4*x + 4)/(x^4 - 4*x^3 + 8*x + 4))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 509, normalized size of antiderivative = 5.14 \[ \int \frac {2 x+x^2}{\left (-2-2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=-\frac {2\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}-\frac {\left (4\,\sqrt {3}-6\right )\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{3\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}+\frac {\left (4\,\sqrt {3}+6\right )\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (-\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{3\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]

[In]

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

[Out]

((4*3^(1/2) + 6)*((3^(1/2)*1i)/2 + 3/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + (3^(1
/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi(-(3^(1/2)*((
3^(1/2)*1i)/2 + 3/2))/3, asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/
2 - 3/2)))/(3*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2
) + 1) + x^3)^(1/2)) - ((4*3^(1/2) - 6)*((3^(1/2)*1i)/2 + 3/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 -
3/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*
ellipticPi((3^(1/2)*((3^(1/2)*1i)/2 + 3/2))/3, asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2
+ 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(3*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i)/2 - 1/2)
*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2)) - (2*((3^(1/2)*1i)/2 + 3/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*
1i)/2 - 3/2))^(1/2)*((x + (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2)
)^(1/2)*ellipticF(asin((-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2
)))/(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) + x
^3)^(1/2)