\(\int \frac {-2-2 x+x^2}{(2 x+x^2) \sqrt {-1+x^3}} \, dx\) [193]

Optimal result

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

[Out]

-2*arctan((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 = 0.41 (sec) , antiderivative size = 402, normalized size of antiderivative = 19.14, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1607, 6874, 225, 272, 65, 209, 2161, 2164, 2163} \[ \int \frac {-2-2 x+x^2}{\left (2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=-\frac {2 \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {\sqrt [4]{3} \sqrt {2 \left (7-4 \sqrt {3}\right )} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {\sqrt {2 \left (7-4 \sqrt {3}\right )} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {2}{3} \arctan \left (\frac {(1-x)^2}{3 \sqrt {x^3-1}}\right )-\frac {2}{3} \arctan \left (\sqrt {x^3-1}\right ) \]

[In]

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

[Out]

(2*ArcTan[(1 - x)^2/(3*Sqrt[-1 + x^3])])/3 - (2*ArcTan[Sqrt[-1 + x^3]])/3 + (Sqrt[2*(7 - 4*Sqrt[3])]*(1 - x)*S
qrt[(1 + x + x^2)/(1 - Sqrt[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]) + (3^(1/4)*Sqrt[2*(7 - 4*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]])/(S
qrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^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])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 2161

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{q = Rt[b/a, 3]}, Dist[-q/((1 + Sqrt[
3])*d - c*q), Int[1/Sqrt[a + b*x^3], x], x] + Dist[d/((1 + Sqrt[3])*d - c*q), Int[(1 + Sqrt[3] + q*x)/((c + d*
x)*Sqrt[a + b*x^3]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0]

Rule 2163

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[-2*(e/d), Subst[Int
[1/(9 - a*x^2), x], x, (1 + f*(x/e))^2/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f,
0] && EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]

Rule 2164

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {-2-2 x+x^2}{x (2+x) \sqrt {-1+x^3}} \, dx \\ & = \int \left (\frac {1}{\sqrt {-1+x^3}}-\frac {1}{x \sqrt {-1+x^3}}-\frac {3}{(2+x) \sqrt {-1+x^3}}\right ) \, dx \\ & = -\left (3 \int \frac {1}{(2+x) \sqrt {-1+x^3}} \, dx\right )+\int \frac {1}{\sqrt {-1+x^3}} \, dx-\int \frac {1}{x \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}{3} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^3\right )-\frac {3 \int \frac {1}{\sqrt {-1+x^3}} \, dx}{3+\sqrt {3}}-\frac {3 \int \frac {1+\sqrt {3}-x}{(2+x) \sqrt {-1+x^3}} \, dx}{3+\sqrt {3}} \\ & = \frac {2\ 3^{3/4} \sqrt {\frac {7}{6}-\frac {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 {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\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}{2} \int \frac {2-2 x}{(2+x) \sqrt {-1+x^3}} \, dx-\frac {2}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^3}\right )-\frac {\sqrt {3} \int \frac {1}{\sqrt {-1+x^3}} \, dx}{3+\sqrt {3}} \\ & = -\frac {2}{3} \arctan \left (\sqrt {-1+x^3}\right )+\frac {2 \sqrt [4]{3} \sqrt {\frac {7}{6}-\frac {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 {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {2\ 3^{3/4} \sqrt {\frac {7}{6}-\frac {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 {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\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 \text {Subst}\left (\int \frac {1}{9+x^2} \, dx,x,\frac {(1-x)^2}{\sqrt {-1+x^3}}\right ) \\ & = \frac {2}{3} \arctan \left (\frac {(1-x)^2}{3 \sqrt {-1+x^3}}\right )-\frac {2}{3} \arctan \left (\sqrt {-1+x^3}\right )+\frac {2 \sqrt [4]{3} \sqrt {\frac {7}{6}-\frac {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 {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}+\frac {2\ 3^{3/4} \sqrt {\frac {7}{6}-\frac {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 {-\frac {1-x}{\left (1-\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.10 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-2-2 x+x^2}{\left (2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=-2 \arctan \left (\frac {\sqrt {-1+x^3}}{1+x+x^2}\right ) \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 2.57 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.24

method	result	size
trager	\(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-2 \sqrt {x^{3}-1}}{x \left (x +2\right )}\right )\)	\(47\)
default	\(\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}-\frac {2 \arctan \left (\sqrt {x^{3}-1}\right )}{3}-\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\)	\(250\)
elliptic	\(-\frac {3 \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \sqrt {3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}-\frac {i \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \sqrt {3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {3}}{\sqrt {x^{3}-1}}-\frac {2 \arctan \left (\sqrt {x^{3}-1}\right )}{3}+\frac {3 \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}+\frac {i \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {1}{3-i \sqrt {3}}-\frac {i \sqrt {3}}{2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{3+i \sqrt {3}}+\frac {i \sqrt {3}}{3+i \sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {3}}{\sqrt {x^{3}-1}}\)	\(676\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {-2-2 x+x^2}{\left (2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=\arctan \left (\frac {x^{2} + 2}{2 \, \sqrt {x^{3} - 1}}\right ) \]

[In]

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

[Out]

arctan(1/2*(x^2 + 2)/sqrt(x^3 - 1))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.34 (sec) , antiderivative size = 252, normalized size of antiderivative = 12.00 \[ \int \frac {-2-2 x+x^2}{\left (2 x+x^2\right ) \sqrt {-1+x^3}} \, dx=\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]

[In]

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

[Out]

((3^(1/2)*1i + 3)*(-(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)*1i)/2 + 3/2, 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)) - 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)) + ellipticPi((3^(1/2)*1i)/6 +
1/2, 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)