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

Optimal. Leaf size=147 \[ \frac {1}{2} \text {RootSum}\left [3 \text {$\#$1}^6-4 \text {$\#$1}^3+2\& ,\frac {\text {$\#$1}^2 \log \left (\sqrt [3]{x^3+x^2}-\text {$\#$1} x\right )-\text {$\#$1}^2 \log (x)}{3 \text {$\#$1}^3-2}\& \right ]-\log \left (\sqrt [3]{x^3+x^2}-x\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^3+x^2} x+\left (x^3+x^2\right )^{2/3}\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x^2}+x}\right ) \]

________________________________________________________________________________________

Rubi [C]  time = 0.70, antiderivative size = 627, normalized size of antiderivative = 4.27, number of steps used = 12, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2056, 6728, 59, 911, 105, 91} \begin {gather*} -\frac {i x^{2/3} \sqrt [3]{x+1} \log \left (-2 x-2 \left (1-i \sqrt {2}\right )\right )}{2\ 2^{5/6} \left (2+i \sqrt {2}\right )^{2/3} \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 (2 x+2 \left (1+i \sqrt {2}\right )\right )}{2\ 2^{5/6} \left (2-i \sqrt {2}\right )^{2/3} \sqrt [3]{x^3+x^2}}-\frac {3 i x^{2/3} \sqrt [3]{x+1} \log \left (-\sqrt [3]{x}+\sqrt [3]{\frac {1}{2} \left (2-i \sqrt {2}\right )} \sqrt [3]{x+1}\right )}{2\ 2^{5/6} \left (2-i \sqrt {2}\right )^{2/3} \sqrt [3]{x^3+x^2}}+\frac {3 i x^{2/3} \sqrt [3]{x+1} \log \left (-\sqrt [3]{x}+\sqrt [3]{\frac {1}{2} \left (2+i \sqrt {2}\right )} \sqrt [3]{x+1}\right )}{2\ 2^{5/6} \left (2+i \sqrt {2}\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}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{x+1} \tan ^{-1}\left (\frac {2 \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{x^3+x^2}}-\frac {i \sqrt {3} x^{2/3} \sqrt [3]{x+1} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \sqrt [3]{2-i \sqrt {2}} \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{x}}\right )}{2^{5/6} \left (2-i \sqrt {2}\right )^{2/3} \sqrt [3]{x^3+x^2}}+\frac {i \sqrt {3} x^{2/3} \sqrt [3]{x+1} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \sqrt [3]{2+i \sqrt {2}} \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{x}}\right )}{2^{5/6} \left (2+i \sqrt {2}\right )^{2/3} \sqrt [3]{x^3+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((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)) -
(I*Sqrt[3]*x^(2/3)*(1 + x)^(1/3)*ArcTan[1/Sqrt[3] + (2^(2/3)*(2 - I*Sqrt[2])^(1/3)*(1 + x)^(1/3))/(Sqrt[3]*x^
(1/3))])/(2^(5/6)*(2 - I*Sqrt[2])^(2/3)*(x^2 + x^3)^(1/3)) + (I*Sqrt[3]*x^(2/3)*(1 + x)^(1/3)*ArcTan[1/Sqrt[3]
+ (2^(2/3)*(2 + I*Sqrt[2])^(1/3)*(1 + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(2^(5/6)*(2 + I*Sqrt[2])^(2/3)*(x^2 + x^3
)^(1/3)) - ((I/2)*x^(2/3)*(1 + x)^(1/3)*Log[-2*(1 - I*Sqrt[2]) - 2*x])/(2^(5/6)*(2 + I*Sqrt[2])^(2/3)*(x^2 + x
^3)^(1/3)) - (x^(2/3)*(1 + x)^(1/3)*Log[x])/(2*(x^2 + x^3)^(1/3)) + ((I/2)*x^(2/3)*(1 + x)^(1/3)*Log[2*(1 + I*
Sqrt[2]) + 2*x])/(2^(5/6)*(2 - I*Sqrt[2])^(2/3)*(x^2 + x^3)^(1/3)) - (((3*I)/2)*x^(2/3)*(1 + x)^(1/3)*Log[-x^(
1/3) + ((2 - I*Sqrt[2])/2)^(1/3)*(1 + x)^(1/3)])/(2^(5/6)*(2 - I*Sqrt[2])^(2/3)*(x^2 + x^3)^(1/3)) + (((3*I)/2
)*x^(2/3)*(1 + x)^(1/3)*Log[-x^(1/3) + ((2 + I*Sqrt[2])/2)^(1/3)*(1 + x)^(1/3)])/(2^(5/6)*(2 + 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 59

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

Rule 91

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

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

Rule 911

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 2056

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 6728

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*} \int \frac {2+x+x^2}{\left (3+2 x+x^2\right ) \sqrt [3]{x^2+x^3}} \, dx &=\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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 \frac {1}{\left (-2+2 i \sqrt {2}-2 x\right ) 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} \sqrt [3]{1+x} \left (2+2 i \sqrt {2}+2 x\right )} \, dx}{\sqrt [3]{x^2+x^3}}\\ &=-\frac {\sqrt {3} x^{2/3} \sqrt [3]{1+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{x^2+x^3}}+\frac {\sqrt {3} \sqrt [3]{\frac {1}{2} \left (2-i \sqrt {2}\right )} x^{2/3} \sqrt [3]{1+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \sqrt [3]{2-i \sqrt {2}} \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{2 \left (1+i \sqrt {2}\right ) \sqrt [3]{x^2+x^3}}+\frac {\sqrt {3} \sqrt [3]{\frac {1}{2} \left (2+i \sqrt {2}\right )} x^{2/3} \sqrt [3]{1+x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} \sqrt [3]{2+i \sqrt {2}} \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{2 \left (1-i \sqrt {2}\right ) \sqrt [3]{x^2+x^3}}-\frac {\sqrt [3]{\frac {1}{2} \left (2+i \sqrt {2}\right )} x^{2/3} \sqrt [3]{1+x} \log \left (-2 \left (1-i \sqrt {2}\right )-2 x\right )}{4 \left (1-i \sqrt {2}\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 {\sqrt [3]{\frac {1}{2} \left (2-i \sqrt {2}\right )} x^{2/3} \sqrt [3]{1+x} \log \left (2 \left (1+i \sqrt {2}\right )+2 x\right )}{4 \left (1+i \sqrt {2}\right ) \sqrt [3]{x^2+x^3}}+\frac {3 \sqrt [3]{\frac {1}{2} \left (2-i \sqrt {2}\right )} x^{2/3} \sqrt [3]{1+x} \log \left (-\sqrt [3]{x}+\sqrt [3]{\frac {1}{2} \left (2-i \sqrt {2}\right )} \sqrt [3]{1+x}\right )}{4 \left (1+i \sqrt {2}\right ) \sqrt [3]{x^2+x^3}}+\frac {3 \sqrt [3]{\frac {1}{2} \left (2+i \sqrt {2}\right )} x^{2/3} \sqrt [3]{1+x} \log \left (-\sqrt [3]{x}+\sqrt [3]{\frac {1}{2} \left (2+i \sqrt {2}\right )} \sqrt [3]{1+x}\right )}{4 \left (1-i \sqrt {2}\right ) \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 [C]  time = 0.06, size = 122, normalized size = 0.83 \begin {gather*} \frac {x \left (6 \sqrt [3]{x+1} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};-x\right )+\left (-1-i \sqrt {2}\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {2 x}{\left (2+i \sqrt {2}\right ) (x+1)}\right )+i \left (\sqrt {2}+i\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {2 i x}{\left (2 i+\sqrt {2}\right ) (x+1)}\right )\right )}{2 \sqrt [3]{x^2 (x+1)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(6*(1 + x)^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, -x] + (-1 - I*Sqrt[2])*Hypergeometric2F1[1/3, 1, 4/3, (2*
x)/((2 + I*Sqrt[2])*(1 + x))] + I*(I + Sqrt[2])*Hypergeometric2F1[1/3, 1, 4/3, ((2*I)*x)/((2*I + Sqrt[2])*(1 +
x))]))/(2*(x^2*(1 + x))^(1/3))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.00, size = 147, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 8.73, size = 2531, normalized size = 17.22

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*18^(5/6)*cos(2/3*arctan(2*sqrt(2) + 3))*log(36*(2*18^(2/3)*sqrt(2)*(x^3 + x^2)^(1/3)*x*cos(2/3*arctan(2*s
qrt(2) + 3))*sin(2/3*arctan(2*sqrt(2) + 3)) + 4*18^(2/3)*(x^3 + x^2)^(1/3)*x*cos(2/3*arctan(2*sqrt(2) + 3))^2
+ 3*18^(1/3)*x^2 - 2*18^(2/3)*(x^3 + x^2)^(1/3)*x + 9*(x^3 + x^2)^(2/3))/x^2) + 1/9*18^(5/6)*arctan(-1/6*(6*18
^(1/3)*sqrt(2)*(x^3 + x^2)^(1/3)*cos(2/3*arctan(2*sqrt(2) + 3))^2 + 12*(6*x*cos(2/3*arctan(2*sqrt(2) + 3))^3 -
(3*x - 18^(1/3)*(x^3 + x^2)^(1/3))*cos(2/3*arctan(2*sqrt(2) + 3)))*sin(2/3*arctan(2*sqrt(2) + 3)) + 6*sqrt(2)
*x - (2*18^(1/3)*sqrt(2)*x*cos(2/3*arctan(2*sqrt(2) + 3))^2 + 4*18^(1/3)*x*cos(2/3*arctan(2*sqrt(2) + 3))*sin(
2/3*arctan(2*sqrt(2) + 3)) - 18^(1/3)*sqrt(2)*x)*sqrt((2*18^(2/3)*sqrt(2)*(x^3 + x^2)^(1/3)*x*cos(2/3*arctan(2
*sqrt(2) + 3))*sin(2/3*arctan(2*sqrt(2) + 3)) + 4*18^(2/3)*(x^3 + x^2)^(1/3)*x*cos(2/3*arctan(2*sqrt(2) + 3))^
2 + 3*18^(1/3)*x^2 - 2*18^(2/3)*(x^3 + x^2)^(1/3)*x + 9*(x^3 + x^2)^(2/3))/x^2) - 3*18^(1/3)*sqrt(2)*(x^3 + x^
2)^(1/3))/(12*x*cos(2/3*arctan(2*sqrt(2) + 3))^4 - 12*x*cos(2/3*arctan(2*sqrt(2) + 3))^2 + x))*sin(2/3*arctan(
2*sqrt(2) + 3)) - 1/18*(18^(5/6)*sqrt(3)*cos(2/3*arctan(2*sqrt(2) + 3)) + 18^(5/6)*sin(2/3*arctan(2*sqrt(2) +
3)))*arctan(1/6*(576*18^(1/3)*(x^3 + x^2)^(1/3)*(2*sqrt(3) + sqrt(2))*cos(2/3*arctan(2*sqrt(2) + 3))^6 - 288*(
3*18^(1/3)*(x^3 + x^2)^(1/3)*(2*sqrt(3) + sqrt(2)) + sqrt(3)*x - 2*sqrt(2)*x)*cos(2/3*arctan(2*sqrt(2) + 3))^4
+ 12*(18^(1/3)*(x^3 + x^2)^(1/3)*(58*sqrt(3) + 19*sqrt(2)) + 24*sqrt(3)*x - 48*sqrt(2)*x)*cos(2/3*arctan(2*sq
rt(2) + 3))^2 - sqrt(2)*(96*18^(1/3)*(2*sqrt(3)*x + sqrt(2)*x)*cos(2/3*arctan(2*sqrt(2) + 3))^6 - 144*18^(1/3)
*(2*sqrt(3)*x + sqrt(2)*x)*cos(2/3*arctan(2*sqrt(2) + 3))^4 + 2*18^(1/3)*(58*sqrt(3)*x + 19*sqrt(2)*x)*cos(2/3
*arctan(2*sqrt(2) + 3))^2 - 2*(48*18^(1/3)*(sqrt(3)*sqrt(2)*x - 2*x)*cos(2/3*arctan(2*sqrt(2) + 3))^5 - 48*18^
(1/3)*(sqrt(3)*sqrt(2)*x - 2*x)*cos(2/3*arctan(2*sqrt(2) + 3))^3 + 18^(1/3)*(11*sqrt(3)*sqrt(2)*x - 26*x)*cos(
2/3*arctan(2*sqrt(2) + 3)))*sin(2/3*arctan(2*sqrt(2) + 3)) - 5*18^(1/3)*(2*sqrt(3)*x - sqrt(2)*x))*sqrt((2*18^
(2/3)*(x^3 + x^2)^(1/3)*(sqrt(3)*sqrt(2)*x - 2*x)*cos(2/3*arctan(2*sqrt(2) + 3))^2 - 2*18^(2/3)*(x^3 + x^2)^(1
/3)*(2*sqrt(3)*x + sqrt(2)*x)*cos(2/3*arctan(2*sqrt(2) + 3))*sin(2/3*arctan(2*sqrt(2) + 3)) + 6*18^(1/3)*x^2 -
18^(2/3)*(x^3 + x^2)^(1/3)*(sqrt(3)*sqrt(2)*x - 2*x) + 18*(x^3 + x^2)^(2/3))/x^2) - 12*(1152*x*cos(2/3*arctan
(2*sqrt(2) + 3))^7 + 48*(18^(1/3)*(x^3 + x^2)^(1/3)*(sqrt(3)*sqrt(2) - 2) - 36*x)*cos(2/3*arctan(2*sqrt(2) + 3
))^5 - 24*(2*sqrt(3)*sqrt(2)*x + 2*18^(1/3)*(x^3 + x^2)^(1/3)*(sqrt(3)*sqrt(2) - 2) - 31*x)*cos(2/3*arctan(2*s
qrt(2) + 3))^3 + (24*sqrt(3)*sqrt(2)*x + 18^(1/3)*(x^3 + x^2)^(1/3)*(11*sqrt(3)*sqrt(2) - 26) - 84*x)*cos(2/3*
arctan(2*sqrt(2) + 3)))*sin(2/3*arctan(2*sqrt(2) + 3)) - 30*18^(1/3)*(x^3 + x^2)^(1/3)*(2*sqrt(3) - sqrt(2)) -
90*sqrt(3)*x + 120*sqrt(2)*x)/(2304*x*cos(2/3*arctan(2*sqrt(2) + 3))^8 - 4608*x*cos(2/3*arctan(2*sqrt(2) + 3)
)^6 + 2976*x*cos(2/3*arctan(2*sqrt(2) + 3))^4 - 672*x*cos(2/3*arctan(2*sqrt(2) + 3))^2 + 25*x)) - 1/18*(18^(5/
6)*sqrt(3)*cos(2/3*arctan(2*sqrt(2) + 3)) - 18^(5/6)*sin(2/3*arctan(2*sqrt(2) + 3)))*arctan(1/6*(576*18^(1/3)*
(x^3 + x^2)^(1/3)*(2*sqrt(3) - sqrt(2))*cos(2/3*arctan(2*sqrt(2) + 3))^6 - 288*(3*18^(1/3)*(x^3 + x^2)^(1/3)*(
2*sqrt(3) - sqrt(2)) + sqrt(3)*x + 2*sqrt(2)*x)*cos(2/3*arctan(2*sqrt(2) + 3))^4 + 12*(18^(1/3)*(x^3 + x^2)^(1
/3)*(58*sqrt(3) - 19*sqrt(2)) + 24*sqrt(3)*x + 48*sqrt(2)*x)*cos(2/3*arctan(2*sqrt(2) + 3))^2 - sqrt(2)*(96*18
^(1/3)*(2*sqrt(3)*x - sqrt(2)*x)*cos(2/3*arctan(2*sqrt(2) + 3))^6 - 144*18^(1/3)*(2*sqrt(3)*x - sqrt(2)*x)*cos
(2/3*arctan(2*sqrt(2) + 3))^4 + 2*18^(1/3)*(58*sqrt(3)*x - 19*sqrt(2)*x)*cos(2/3*arctan(2*sqrt(2) + 3))^2 - 2*
(48*18^(1/3)*(sqrt(3)*sqrt(2)*x + 2*x)*cos(2/3*arctan(2*sqrt(2) + 3))^5 - 48*18^(1/3)*(sqrt(3)*sqrt(2)*x + 2*x
)*cos(2/3*arctan(2*sqrt(2) + 3))^3 + 18^(1/3)*(11*sqrt(3)*sqrt(2)*x + 26*x)*cos(2/3*arctan(2*sqrt(2) + 3)))*si
n(2/3*arctan(2*sqrt(2) + 3)) - 5*18^(1/3)*(2*sqrt(3)*x + sqrt(2)*x))*sqrt(-(2*18^(2/3)*(x^3 + x^2)^(1/3)*(sqrt
(3)*sqrt(2)*x + 2*x)*cos(2/3*arctan(2*sqrt(2) + 3))^2 - 2*18^(2/3)*(x^3 + x^2)^(1/3)*(2*sqrt(3)*x - sqrt(2)*x)
*cos(2/3*arctan(2*sqrt(2) + 3))*sin(2/3*arctan(2*sqrt(2) + 3)) - 6*18^(1/3)*x^2 - 18^(2/3)*(x^3 + x^2)^(1/3)*(
sqrt(3)*sqrt(2)*x + 2*x) - 18*(x^3 + x^2)^(2/3))/x^2) + 12*(1152*x*cos(2/3*arctan(2*sqrt(2) + 3))^7 - 48*(18^(
1/3)*(x^3 + x^2)^(1/3)*(sqrt(3)*sqrt(2) + 2) + 36*x)*cos(2/3*arctan(2*sqrt(2) + 3))^5 + 24*(2*sqrt(3)*sqrt(2)*
x + 2*18^(1/3)*(x^3 + x^2)^(1/3)*(sqrt(3)*sqrt(2) + 2) + 31*x)*cos(2/3*arctan(2*sqrt(2) + 3))^3 - (24*sqrt(3)*
sqrt(2)*x + 18^(1/3)*(x^3 + x^2)^(1/3)*(11*sqrt(3)*sqrt(2) + 26) + 84*x)*cos(2/3*arctan(2*sqrt(2) + 3)))*sin(2
/3*arctan(2*sqrt(2) + 3)) - 30*18^(1/3)*(x^3 + x^2)^(1/3)*(2*sqrt(3) + sqrt(2)) - 90*sqrt(3)*x - 120*sqrt(2)*x
)/(2304*x*cos(2/3*arctan(2*sqrt(2) + 3))^8 - 4608*x*cos(2/3*arctan(2*sqrt(2) + 3))^6 + 2976*x*cos(2/3*arctan(2
*sqrt(2) + 3))^4 - 672*x*cos(2/3*arctan(2*sqrt(2) + 3))^2 + 25*x)) - 1/72*(18^(5/6)*sqrt(3)*sin(2/3*arctan(2*s
qrt(2) + 3)) + 18^(5/6)*cos(2/3*arctan(2*sqrt(2) + 3)))*log(-72*(2*18^(2/3)*(x^3 + x^2)^(1/3)*(sqrt(3)*sqrt(2)
*x + 2*x)*cos(2/3*arctan(2*sqrt(2) + 3))^2 - 2*18^(2/3)*(x^3 + x^2)^(1/3)*(2*sqrt(3)*x - sqrt(2)*x)*cos(2/3*ar
ctan(2*sqrt(2) + 3))*sin(2/3*arctan(2*sqrt(2) + 3)) - 6*18^(1/3)*x^2 - 18^(2/3)*(x^3 + x^2)^(1/3)*(sqrt(3)*sqr
t(2)*x + 2*x) - 18*(x^3 + x^2)^(2/3))/x^2) + 1/72*(18^(5/6)*sqrt(3)*sin(2/3*arctan(2*sqrt(2) + 3)) - 18^(5/6)*
cos(2/3*arctan(2*sqrt(2) + 3)))*log(72*(2*18^(2/3)*(x^3 + x^2)^(1/3)*(sqrt(3)*sqrt(2)*x - 2*x)*cos(2/3*arctan(
2*sqrt(2) + 3))^2 - 2*18^(2/3)*(x^3 + x^2)^(1/3)*(2*sqrt(3)*x + sqrt(2)*x)*cos(2/3*arctan(2*sqrt(2) + 3))*sin(
2/3*arctan(2*sqrt(2) + 3)) + 6*18^(1/3)*x^2 - 18^(2/3)*(x^3 + x^2)^(1/3)*(sqrt(3)*sqrt(2)*x - 2*x) + 18*(x^3 +
x^2)^(2/3))/x^2) - sqrt(3)*arctan(1/3*(sqrt(3)*x + 2*sqrt(3)*(x^3 + x^2)^(1/3))/x) - log(-(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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + x + 2}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 2 \, x + 3\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 21.42, size = 10276, normalized size = 69.90

method	result	size
trager	\(\text {Expression too large to display}\)	\(10276\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + x + 2}{{\left (x^{3} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 2 \, x + 3\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2+x+2}{{\left (x^3+x^2\right )}^{1/3}\,\left (x^2+2\,x+3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + x + 2}{\sqrt [3]{x^{2} \left (x + 1\right )} \left (x^{2} + 2 x + 3\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________