\(\int \frac {3+x+x^2}{(-2+2 x+x^2) \sqrt {1+x^3}} \, dx\) [1459]

Optimal result

Integrand size = 26, antiderivative size = 103 \[ \int \frac {3+x+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {1}{6} \sqrt {-15+14 \sqrt {3}} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right )-\frac {1}{6} \sqrt {15+14 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right ) \]

[Out]

-1/6*(-15+14*3^(1/2))^(1/2)*arctan((3+2*3^(1/2))^(1/2)*(x^3+1)^(1/2)/(x^2-x+1))-1/6*(15+14*3^(1/2))^(1/2)*arct
anh((-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 = 0.42 (sec) , antiderivative size = 406, normalized size of antiderivative = 3.94, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6860, 224, 2160, 2165, 212, 209} \[ \int \frac {3+x+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {\sqrt {38+21 \sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{2\ 3^{3/4} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {\sqrt {14+5 \sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{2\ 3^{3/4} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {1}{6} \sqrt {14 \sqrt {3}-15} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (x+1)}{\sqrt {x^3+1}}\right )-\frac {1}{6} \sqrt {15+14 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {2 \sqrt {3}-3} (x+1)}{\sqrt {x^3+1}}\right ) \]

[In]

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

[Out]

-1/6*(Sqrt[-15 + 14*Sqrt[3]]*ArcTan[(Sqrt[3 + 2*Sqrt[3]]*(1 + x))/Sqrt[1 + x^3]]) - (Sqrt[15 + 14*Sqrt[3]]*Arc
Tanh[(Sqrt[-3 + 2*Sqrt[3]]*(1 + x))/Sqrt[1 + x^3]])/6 + (2*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + S
qrt[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]) - (Sqrt[14 + 5*Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*Ell
ipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(2*3^(3/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)
^2]*Sqrt[1 + x^3]) - (Sqrt[38 + 21*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]])/(2*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 224

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] && PosQ[a]

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 \left (\frac {1}{\sqrt {1+x^3}}+\frac {5-x}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}}\right ) \, dx \\ & = \int \frac {1}{\sqrt {1+x^3}} \, dx+\int \frac {5-x}{\left (-2+2 x+x^2\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}}+\int \left (\frac {-1+2 \sqrt {3}}{\left (2-2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}}+\frac {-1-2 \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 (-1-2 \sqrt {3}\right ) \int \frac {1}{\left (2+2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx+\left (-1+2 \sqrt {3}\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}{576} \left (6-\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}{12} \left (-6+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx+\frac {1}{576} \left (6+\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}{12} \left (6+\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 {\sqrt {14+5 \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 )}{2\ 3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {\sqrt {38+21 \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 )}{2\ 3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {1}{6} \left (-6+\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}{6} \left (6+\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}{6} \sqrt {-15+14 \sqrt {3}} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} (1+x)}{\sqrt {1+x^3}}\right )-\frac {1}{6} \sqrt {15+14 \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 {\sqrt {14+5 \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 )}{2\ 3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {\sqrt {38+21 \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 )}{2\ 3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.59 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00 \[ \int \frac {3+x+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {1}{6} \sqrt {-15+14 \sqrt {3}} \arctan \left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right )-\frac {1}{6} \sqrt {15+14 \sqrt {3}} \text {arctanh}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right ) \]

[In]

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

[Out]

-1/6*(Sqrt[-15 + 14*Sqrt[3]]*ArcTan[(Sqrt[3 + 2*Sqrt[3]]*Sqrt[1 + x^3])/(1 - x + x^2)]) - (Sqrt[15 + 14*Sqrt[3
]]*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*Sqrt[1 + x^3])/(1 - x + x^2)])/6

Maple [C] (warning: unable to verify)

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

Time = 3.06 (sec) , antiderivative size = 594, normalized size of antiderivative = 5.77

method	result	size
trager	\(\frac {\operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right ) \ln \left (-\frac {-3888 x^{2} \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{5}+7776 x \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{5}-1800 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{3} x^{2}-2448 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{3} x +7392 \sqrt {x^{3}+1}\, \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{2}-6048 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{3}+53 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right ) x^{2}-3074 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right ) x +1232 \sqrt {x^{3}+1}-2968 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )}{{\left (36 x \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{2}-x -28\right )}^{2}}\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{2}-30\right ) \ln \left (\frac {-3888 \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{2}-30\right ) \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{4} x^{2}+7776 x \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{4} \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{2}-30\right )+8280 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{2}-30\right ) x^{2}-10512 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{2}-30\right ) x +44352 \sqrt {x^{3}+1}\, \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{2}+6048 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{2}-30\right )-4147 \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{2}-30\right ) x^{2}+286 \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{2}-30\right ) x -44352 \sqrt {x^{3}+1}-8008 \operatorname {RootOf}\left (\textit {\_Z}^{2}+36 \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{2}-30\right )}{{\left (36 x \operatorname {RootOf}\left (432 \textit {\_Z}^{4}-360 \textit {\_Z}^{2}-121\right )^{2}-29 x +28\right )}^{2}}\right )}{12}\)	\(594\)
default	\(\text {Expression too large to display}\)	\(1501\)
elliptic	\(\text {Expression too large to display}\)	\(1706\)

[In]

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

[Out]

1/2*RootOf(432*_Z^4-360*_Z^2-121)*ln(-(-3888*x^2*RootOf(432*_Z^4-360*_Z^2-121)^5+7776*x*RootOf(432*_Z^4-360*_Z
^2-121)^5-1800*RootOf(432*_Z^4-360*_Z^2-121)^3*x^2-2448*RootOf(432*_Z^4-360*_Z^2-121)^3*x+7392*(x^3+1)^(1/2)*R
ootOf(432*_Z^4-360*_Z^2-121)^2-6048*RootOf(432*_Z^4-360*_Z^2-121)^3+53*RootOf(432*_Z^4-360*_Z^2-121)*x^2-3074*
RootOf(432*_Z^4-360*_Z^2-121)*x+1232*(x^3+1)^(1/2)-2968*RootOf(432*_Z^4-360*_Z^2-121))/(36*x*RootOf(432*_Z^4-3
60*_Z^2-121)^2-x-28)^2)-1/12*RootOf(_Z^2+36*RootOf(432*_Z^4-360*_Z^2-121)^2-30)*ln((-3888*RootOf(_Z^2+36*RootO
f(432*_Z^4-360*_Z^2-121)^2-30)*RootOf(432*_Z^4-360*_Z^2-121)^4*x^2+7776*x*RootOf(432*_Z^4-360*_Z^2-121)^4*Root
Of(_Z^2+36*RootOf(432*_Z^4-360*_Z^2-121)^2-30)+8280*RootOf(432*_Z^4-360*_Z^2-121)^2*RootOf(_Z^2+36*RootOf(432*
_Z^4-360*_Z^2-121)^2-30)*x^2-10512*RootOf(432*_Z^4-360*_Z^2-121)^2*RootOf(_Z^2+36*RootOf(432*_Z^4-360*_Z^2-121
)^2-30)*x+44352*(x^3+1)^(1/2)*RootOf(432*_Z^4-360*_Z^2-121)^2+6048*RootOf(432*_Z^4-360*_Z^2-121)^2*RootOf(_Z^2
+36*RootOf(432*_Z^4-360*_Z^2-121)^2-30)-4147*RootOf(_Z^2+36*RootOf(432*_Z^4-360*_Z^2-121)^2-30)*x^2+286*RootOf
(_Z^2+36*RootOf(432*_Z^4-360*_Z^2-121)^2-30)*x-44352*(x^3+1)^(1/2)-8008*RootOf(_Z^2+36*RootOf(432*_Z^4-360*_Z^
2-121)^2-30))/(36*x*RootOf(432*_Z^4-360*_Z^2-121)^2-29*x+28)^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (79) = 158\).

Time = 0.29 (sec) , antiderivative size = 403, normalized size of antiderivative = 3.91 \[ \int \frac {3+x+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx=\frac {1}{24} \, \sqrt {14 \, \sqrt {3} + 15} \log \left (\frac {11 \, x^{4} - 22 \, x^{3} + 66 \, x^{2} + 2 \, \sqrt {x^{3} + 1} {\left (4 \, x^{2} - \sqrt {3} {\left (3 \, x^{2} - 2 \, x + 4\right )} - 10 \, x - 2\right )} \sqrt {14 \, \sqrt {3} + 15} + 44 \, \sqrt {3} {\left (x^{3} + 1\right )} + 44 \, x + 44}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) - \frac {1}{24} \, \sqrt {14 \, \sqrt {3} + 15} \log \left (\frac {11 \, x^{4} - 22 \, x^{3} + 66 \, x^{2} - 2 \, \sqrt {x^{3} + 1} {\left (4 \, x^{2} - \sqrt {3} {\left (3 \, x^{2} - 2 \, x + 4\right )} - 10 \, x - 2\right )} \sqrt {14 \, \sqrt {3} + 15} + 44 \, \sqrt {3} {\left (x^{3} + 1\right )} + 44 \, x + 44}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) + \frac {1}{24} \, \sqrt {-14 \, \sqrt {3} + 15} \log \left (\frac {11 \, x^{4} - 22 \, x^{3} + 66 \, x^{2} + 2 \, \sqrt {x^{3} + 1} {\left (4 \, x^{2} + \sqrt {3} {\left (3 \, x^{2} - 2 \, x + 4\right )} - 10 \, x - 2\right )} \sqrt {-14 \, \sqrt {3} + 15} - 44 \, \sqrt {3} {\left (x^{3} + 1\right )} + 44 \, x + 44}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) - \frac {1}{24} \, \sqrt {-14 \, \sqrt {3} + 15} \log \left (\frac {11 \, x^{4} - 22 \, x^{3} + 66 \, x^{2} - 2 \, \sqrt {x^{3} + 1} {\left (4 \, x^{2} + \sqrt {3} {\left (3 \, x^{2} - 2 \, x + 4\right )} - 10 \, x - 2\right )} \sqrt {-14 \, \sqrt {3} + 15} - 44 \, \sqrt {3} {\left (x^{3} + 1\right )} + 44 \, x + 44}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) \]

[In]

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

[Out]

1/24*sqrt(14*sqrt(3) + 15)*log((11*x^4 - 22*x^3 + 66*x^2 + 2*sqrt(x^3 + 1)*(4*x^2 - sqrt(3)*(3*x^2 - 2*x + 4)
- 10*x - 2)*sqrt(14*sqrt(3) + 15) + 44*sqrt(3)*(x^3 + 1) + 44*x + 44)/(x^4 + 4*x^3 - 8*x + 4)) - 1/24*sqrt(14*
sqrt(3) + 15)*log((11*x^4 - 22*x^3 + 66*x^2 - 2*sqrt(x^3 + 1)*(4*x^2 - sqrt(3)*(3*x^2 - 2*x + 4) - 10*x - 2)*s
qrt(14*sqrt(3) + 15) + 44*sqrt(3)*(x^3 + 1) + 44*x + 44)/(x^4 + 4*x^3 - 8*x + 4)) + 1/24*sqrt(-14*sqrt(3) + 15
)*log((11*x^4 - 22*x^3 + 66*x^2 + 2*sqrt(x^3 + 1)*(4*x^2 + sqrt(3)*(3*x^2 - 2*x + 4) - 10*x - 2)*sqrt(-14*sqrt
(3) + 15) - 44*sqrt(3)*(x^3 + 1) + 44*x + 44)/(x^4 + 4*x^3 - 8*x + 4)) - 1/24*sqrt(-14*sqrt(3) + 15)*log((11*x
^4 - 22*x^3 + 66*x^2 - 2*sqrt(x^3 + 1)*(4*x^2 + sqrt(3)*(3*x^2 - 2*x + 4) - 10*x - 2)*sqrt(-14*sqrt(3) + 15) -
44*sqrt(3)*(x^3 + 1) + 44*x + 44)/(x^4 + 4*x^3 - 8*x + 4))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.04 (sec) , antiderivative size = 505, normalized size of antiderivative = 4.90 \[ \int \frac {3+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+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\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 (\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+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (\sqrt {3}-6\right )\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\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 (\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+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (\sqrt {3}+6\right )\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\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((x + x^2 + 3)/((x^3 + 1)^(1/2)*(2*x + x^2 - 2)),x)

[Out]

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