\(\int \frac {2+x+x^2}{(3+2 x+x^2) \sqrt [3]{x^2+x^3}} \, dx\) [2047]

Optimal result

Integrand size = 28, antiderivative size = 147 \[ \int \frac {2+x+x^2}{\left (3+2 x+x^2\right ) \sqrt [3]{x^2+x^3}} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x^2+x^3}+\left (x^2+x^3\right )^{2/3}\right )+\frac {1}{2} \text {RootSum}\left [2-4 \text {$\#$1}^3+3 \text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-2+3 \text {$\#$1}^3}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.58 (sec) , antiderivative size = 599, normalized size of antiderivative = 4.07, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2081, 6860, 61, 925, 129, 399, 245, 384} \[ \int \frac {2+x+x^2}{\left (3+2 x+x^2\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {i \sqrt {3} x^{2/3} \sqrt [3]{x+1} \arctan \left (\frac {1+\frac {2 \sqrt [6]{2} \sqrt [3]{x}}{\sqrt [3]{\sqrt {2}-i} \sqrt [3]{x+1}}}{\sqrt {3}}\right )}{2 \sqrt [6]{2} \left (\sqrt {2}-i\right )^{2/3} \sqrt [3]{x^3+x^2}}-\frac {i \sqrt {3} x^{2/3} \sqrt [3]{x+1} \arctan \left (\frac {1+\frac {2 \sqrt [6]{2} \sqrt [3]{x}}{\sqrt [3]{\sqrt {2}+i} \sqrt [3]{x+1}}}{\sqrt {3}}\right )}{2 \sqrt [6]{2} \left (\sqrt {2}+i\right )^{2/3} \sqrt [3]{x^3+x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{x+1} \arctan \left (\frac {2 \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{x^3+x^2}}-\frac {x^{2/3} \sqrt [3]{x+1} \log (x)}{2 \sqrt [3]{x^3+x^2}}-\frac {i x^{2/3} \sqrt [3]{x+1} \log \left (x-i \sqrt {2}+1\right )}{4 \sqrt [6]{2} \left (\sqrt {2}+i\right )^{2/3} \sqrt [3]{x^3+x^2}}+\frac {i x^{2/3} \sqrt [3]{x+1} \log \left (x+i \sqrt {2}+1\right )}{4 \sqrt [6]{2} \left (\sqrt {2}-i\right )^{2/3} \sqrt [3]{x^3+x^2}}-\frac {3 i x^{2/3} \sqrt [3]{x+1} \log \left (\sqrt [6]{2} \sqrt [3]{x}-\sqrt [3]{\sqrt {2}-i} \sqrt [3]{x+1}\right )}{4 \sqrt [6]{2} \left (\sqrt {2}-i\right )^{2/3} \sqrt [3]{x^3+x^2}}+\frac {3 i x^{2/3} \sqrt [3]{x+1} \log \left (\sqrt [6]{2} \sqrt [3]{x}-\sqrt [3]{\sqrt {2}+i} \sqrt [3]{x+1}\right )}{4 \sqrt [6]{2} \left (\sqrt {2}+i\right )^{2/3} \sqrt [3]{x^3+x^2}}-\frac {3 x^{2/3} \sqrt [3]{x+1} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{x}}-1\right )}{2 \sqrt [3]{x^3+x^2}} \]

[In]

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

[Out]

((I/2)*Sqrt[3]*x^(2/3)*(1 + x)^(1/3)*ArcTan[(1 + (2*2^(1/6)*x^(1/3))/((-I + Sqrt[2])^(1/3)*(1 + x)^(1/3)))/Sqr
t[3]])/(2^(1/6)*(-I + Sqrt[2])^(2/3)*(x^2 + x^3)^(1/3)) - ((I/2)*Sqrt[3]*x^(2/3)*(1 + x)^(1/3)*ArcTan[(1 + (2*
2^(1/6)*x^(1/3))/((I + Sqrt[2])^(1/3)*(1 + x)^(1/3)))/Sqrt[3]])/(2^(1/6)*(I + Sqrt[2])^(2/3)*(x^2 + x^3)^(1/3)
) - (Sqrt[3]*x^(2/3)*(1 + x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(1 + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(x^2 + x^3)^(1/3)
- (x^(2/3)*(1 + x)^(1/3)*Log[x])/(2*(x^2 + x^3)^(1/3)) - ((I/4)*x^(2/3)*(1 + x)^(1/3)*Log[1 - I*Sqrt[2] + x])/
(2^(1/6)*(I + Sqrt[2])^(2/3)*(x^2 + x^3)^(1/3)) + ((I/4)*x^(2/3)*(1 + x)^(1/3)*Log[1 + I*Sqrt[2] + x])/(2^(1/6
)*(-I + Sqrt[2])^(2/3)*(x^2 + x^3)^(1/3)) - (((3*I)/4)*x^(2/3)*(1 + x)^(1/3)*Log[2^(1/6)*x^(1/3) - (-I + Sqrt[
2])^(1/3)*(1 + x)^(1/3)])/(2^(1/6)*(-I + Sqrt[2])^(2/3)*(x^2 + x^3)^(1/3)) + (((3*I)/4)*x^(2/3)*(1 + x)^(1/3)*
Log[2^(1/6)*x^(1/3) - (I + Sqrt[2])^(1/3)*(1 + x)^(1/3)])/(2^(1/6)*(I + Sqrt[2])^(2/3)*(x^2 + x^3)^(1/3)) - (3
*x^(2/3)*(1 + x)^(1/3)*Log[-1 + (1 + x)^(1/3)/x^(1/3)])/(2*(x^2 + x^3)^(1/3))

Rule 61

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, Simp[(-Sqrt
[3])*(q/d)*ArcTan[2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3))) + 1/Sqrt[3]], x] + (-Simp[3*(q/(2*d))*Log[q*
((a + b*x)^(1/3)/(c + d*x)^(1/3)) - 1], x] - Simp[(q/(2*d))*Log[c + d*x], x])] /; FreeQ[{a, b, c, d}, x] && Ne
Q[b*c - a*d, 0] && PosQ[d/b]

Rule 129

Int[((e_.)*(x_))^(p_)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> With[{k = Denominator[p]
}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + b*(x^k/e))^m*(c + d*(x^k/e))^n, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]

Rule 245

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 384

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

Rule 399

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[(a + b*x^n)^(p - 1), x]
, x] - Dist[(b*c - a*d)/d, Int[(a + b*x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c
- a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]

Rule 925

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n, 1/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x
] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.18 \[ \int \frac {2+x+x^2}{\left (3+2 x+x^2\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {x^{2/3} \sqrt [3]{1+x} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{1+x}}\right )-2 \log \left (-\sqrt [3]{x}+\sqrt [3]{1+x}\right )+\log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{1+x}+(1+x)^{2/3}\right )+\text {RootSum}\left [2-4 \text {$\#$1}^3+3 \text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right ) \text {$\#$1}^2+\log \left (\sqrt [3]{1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^2}{-2+3 \text {$\#$1}^3}\&\right ]\right )}{2 \sqrt [3]{x^2 (1+x)}} \]

[In]

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

[Out]

(x^(2/3)*(1 + x)^(1/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2*(1 + x)^(1/3))] - 2*Log[-x^(1/3) + (1
+ x)^(1/3)] + Log[x^(2/3) + x^(1/3)*(1 + x)^(1/3) + (1 + x)^(2/3)] + RootSum[2 - 4*#1^3 + 3*#1^6 & , (-(Log[x^
(1/3)]*#1^2) + Log[(1 + x)^(1/3) - x^(1/3)*#1]*#1^2)/(-2 + 3*#1^3) & ]))/(2*(x^2*(1 + x))^(1/3))

Maple [N/A] (verified)

Time = 19.64 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(\frac {\ln \left (\frac {\left (x^{2} \left (1+x \right )\right )^{\frac {2}{3}}+\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{6}-4 \textit {\_Z}^{3}+2\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}}{x}\right )}{3 \textit {\_R}^{3}-2}\right )}{2}-\ln \left (\frac {\left (x^{2} \left (1+x \right )\right )^{\frac {1}{3}}-x}{x}\right )\)	\(130\)
trager	\(\text {Expression too large to display}\)	\(8784\)

[In]

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

[Out]

1/2*ln(((x^2*(1+x))^(2/3)+(x^2*(1+x))^(1/3)*x+x^2)/x^2)-3^(1/2)*arctan(1/3*(2*(x^2*(1+x))^(1/3)+x)*3^(1/2)/x)+
1/2*sum(_R^2*ln((-_R*x+(x^2*(1+x))^(1/3))/x)/(3*_R^3-2),_R=RootOf(3*_Z^6-4*_Z^3+2))-ln(((x^2*(1+x))^(1/3)-x)/x
)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.28 (sec) , antiderivative size = 484, normalized size of antiderivative = 3.29 \[ \int \frac {2+x+x^2}{\left (3+2 x+x^2\right ) \sqrt [3]{x^2+x^3}} \, dx=\frac {1}{72} \cdot 18^{\frac {2}{3}} {\left (i \, \sqrt {2} - 4\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (\frac {18^{\frac {1}{3}} {\left (\sqrt {2} {\left (i \, \sqrt {-3} x + i \, x\right )} - 2 \, \sqrt {-3} x - 2 \, x\right )} {\left (i \, \sqrt {2} - 4\right )}^{\frac {2}{3}} + 36 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{72} \cdot 18^{\frac {2}{3}} {\left (i \, \sqrt {2} - 4\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (\frac {18^{\frac {1}{3}} {\left (\sqrt {2} {\left (-i \, \sqrt {-3} x + i \, x\right )} + 2 \, \sqrt {-3} x - 2 \, x\right )} {\left (i \, \sqrt {2} - 4\right )}^{\frac {2}{3}} + 36 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{72} \cdot 18^{\frac {2}{3}} {\left (-i \, \sqrt {2} - 4\right )}^{\frac {1}{3}} {\left (\sqrt {-3} + 1\right )} \log \left (\frac {18^{\frac {1}{3}} {\left (\sqrt {2} {\left (i \, \sqrt {-3} x - i \, x\right )} + 2 \, \sqrt {-3} x - 2 \, x\right )} {\left (-i \, \sqrt {2} - 4\right )}^{\frac {2}{3}} + 36 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{72} \cdot 18^{\frac {2}{3}} {\left (-i \, \sqrt {2} - 4\right )}^{\frac {1}{3}} {\left (\sqrt {-3} - 1\right )} \log \left (\frac {18^{\frac {1}{3}} {\left (\sqrt {2} {\left (-i \, \sqrt {-3} x - i \, x\right )} - 2 \, \sqrt {-3} x - 2 \, x\right )} {\left (-i \, \sqrt {2} - 4\right )}^{\frac {2}{3}} + 36 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{36} \cdot 18^{\frac {2}{3}} {\left (i \, \sqrt {2} - 4\right )}^{\frac {1}{3}} \log \left (\frac {18^{\frac {1}{3}} {\left (-i \, \sqrt {2} x + 2 \, x\right )} {\left (i \, \sqrt {2} - 4\right )}^{\frac {2}{3}} + 18 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{36} \cdot 18^{\frac {2}{3}} {\left (-i \, \sqrt {2} - 4\right )}^{\frac {1}{3}} \log \left (\frac {18^{\frac {1}{3}} {\left (i \, \sqrt {2} x + 2 \, x\right )} {\left (-i \, \sqrt {2} - 4\right )}^{\frac {2}{3}} + 18 \, {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \log \left (-\frac {x - {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{2} \, \log \left (\frac {x^{2} + {\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} + x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]

[In]

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

[Out]

1/72*18^(2/3)*(I*sqrt(2) - 4)^(1/3)*(sqrt(-3) - 1)*log((18^(1/3)*(sqrt(2)*(I*sqrt(-3)*x + I*x) - 2*sqrt(-3)*x
- 2*x)*(I*sqrt(2) - 4)^(2/3) + 36*(x^3 + x^2)^(1/3))/x) - 1/72*18^(2/3)*(I*sqrt(2) - 4)^(1/3)*(sqrt(-3) + 1)*l
og((18^(1/3)*(sqrt(2)*(-I*sqrt(-3)*x + I*x) + 2*sqrt(-3)*x - 2*x)*(I*sqrt(2) - 4)^(2/3) + 36*(x^3 + x^2)^(1/3)
)/x) - 1/72*18^(2/3)*(-I*sqrt(2) - 4)^(1/3)*(sqrt(-3) + 1)*log((18^(1/3)*(sqrt(2)*(I*sqrt(-3)*x - I*x) + 2*sqr
t(-3)*x - 2*x)*(-I*sqrt(2) - 4)^(2/3) + 36*(x^3 + x^2)^(1/3))/x) + 1/72*18^(2/3)*(-I*sqrt(2) - 4)^(1/3)*(sqrt(
-3) - 1)*log((18^(1/3)*(sqrt(2)*(-I*sqrt(-3)*x - I*x) - 2*sqrt(-3)*x - 2*x)*(-I*sqrt(2) - 4)^(2/3) + 36*(x^3 +
x^2)^(1/3))/x) + 1/36*18^(2/3)*(I*sqrt(2) - 4)^(1/3)*log((18^(1/3)*(-I*sqrt(2)*x + 2*x)*(I*sqrt(2) - 4)^(2/3)
+ 18*(x^3 + x^2)^(1/3))/x) + 1/36*18^(2/3)*(-I*sqrt(2) - 4)^(1/3)*log((18^(1/3)*(I*sqrt(2)*x + 2*x)*(-I*sqrt(
2) - 4)^(2/3) + 18*(x^3 + x^2)^(1/3))/x) - sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 + x^2)^(1/3))/x) - l
og(-(x - (x^3 + x^2)^(1/3))/x) + 1/2*log((x^2 + (x^3 + x^2)^(1/3)*x + (x^3 + x^2)^(2/3))/x^2)

Sympy [N/A]

Not integrable

Time = 2.65 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.18 \[ \int \frac {2+x+x^2}{\left (3+2 x+x^2\right ) \sqrt [3]{x^2+x^3}} \, dx=\int \frac {x^{2} + x + 2}{\sqrt [3]{x^{2} \left (x + 1\right )} \left (x^{2} + 2 x + 3\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.19 \[ \int \frac {2+x+x^2}{\left (3+2 x+x^2\right ) \sqrt [3]{x^2+x^3}} \, dx=\int { \frac {x^{2} + x + 2}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 2 \, x + 3\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {2+x+x^2}{\left (3+2 x+x^2\right ) \sqrt [3]{x^2+x^3}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:Invalid _EXT in r
eplace_ext Error: Bad Argument Value-ln(abs((1/sageVARx+1)^(1/3)-1))+1/2*ln(((1/sageVARx+1)^(1/3))^2+(1/sageVA
Rx+1)^(1/3)+1)

Mupad [N/A]

Not integrable

Time = 5.62 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.19 \[ \int \frac {2+x+x^2}{\left (3+2 x+x^2\right ) \sqrt [3]{x^2+x^3}} \, dx=\int \frac {x^2+x+2}{{\left (x^3+x^2\right )}^{1/3}\,\left (x^2+2\,x+3\right )} \,d x \]

[In]

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

[Out]

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