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

Optimal. Leaf size=140 \[ -\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^6-2 \text {$\#$1}^3+2\& ,\frac {\log \left (\sqrt [3]{x^3-x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]-\log \left (\sqrt [3]{x^3-x^2}-x\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^3-x^2} x+\left (x^3-x^2\right )^{2/3}\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-x^2}+x}\right ) \]

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Rubi [C]  time = 0.51, antiderivative size = 489, normalized size of antiderivative = 3.49, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2056, 6725, 59, 912, 91} \begin {gather*} -\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{x}}-1\right )}{2 \sqrt [3]{x^3-x^2}}-\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{x-1}}{\sqrt [3]{1-i}}\right )}{4 \sqrt [3]{1-i} \sqrt [3]{x^3-x^2}}-\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{x-1}}{\sqrt [3]{1+i}}\right )}{4 \sqrt [3]{1+i} \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} x^{2/3} \log (-x+i)}{4 \sqrt [3]{1+i} \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (x)}{2 \sqrt [3]{x^3-x^2}}+\frac {\sqrt [3]{x-1} x^{2/3} \log (x+i)}{4 \sqrt [3]{1-i} \sqrt [3]{x^3-x^2}}-\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \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 {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1-i} \sqrt {3} \sqrt [3]{x}}\right )}{2 \sqrt [3]{1-i} \sqrt [3]{x^3-x^2}}-\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x-1}}{\sqrt [3]{1+i} \sqrt {3} \sqrt [3]{x}}\right )}{2 \sqrt [3]{1+i} \sqrt [3]{x^3-x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)*ArcTan[1/Sqrt[3] + (2*(-1 + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(-x^2 + x^3)^(1/3)
) - (Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)*ArcTan[1/Sqrt[3] + (2*(-1 + x)^(1/3))/((1 - I)^(1/3)*Sqrt[3]*x^(1/3))])/(2
*(1 - I)^(1/3)*(-x^2 + x^3)^(1/3)) - (Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)*ArcTan[1/Sqrt[3] + (2*(-1 + x)^(1/3))/((1
+ I)^(1/3)*Sqrt[3]*x^(1/3))])/(2*(1 + I)^(1/3)*(-x^2 + x^3)^(1/3)) - (3*(-1 + x)^(1/3)*x^(2/3)*Log[-1 + (-1 +
x)^(1/3)/x^(1/3)])/(2*(-x^2 + x^3)^(1/3)) - (3*(-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3)/(1 - I)^(1/3) - x^(1
/3)])/(4*(1 - I)^(1/3)*(-x^2 + x^3)^(1/3)) - (3*(-1 + x)^(1/3)*x^(2/3)*Log[(-1 + x)^(1/3)/(1 + I)^(1/3) - x^(1
/3)])/(4*(1 + I)^(1/3)*(-x^2 + x^3)^(1/3)) + ((-1 + x)^(1/3)*x^(2/3)*Log[I - x])/(4*(1 + I)^(1/3)*(-x^2 + x^3)
^(1/3)) - ((-1 + x)^(1/3)*x^(2/3)*Log[x])/(2*(-x^2 + x^3)^(1/3)) + ((-1 + x)^(1/3)*x^(2/3)*Log[I + x])/(4*(1 -
I)^(1/3)*(-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 912

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)^n, 1/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[c*d^2 + 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 6725

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

Rubi steps

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

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Mathematica [C]  time = 0.08, size = 95, normalized size = 0.68 \begin {gather*} \frac {\left (\frac {3}{8}+\frac {3 i}{8}\right ) \left ((x-1) x^2\right )^{2/3} \left ((2-2 i) x^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};1-x\right )-i \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) (x-1)}{x}\right )+\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) (x-1)}{x}\right )\right )}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3/8 + (3*I)/8)*((-1 + x)*x^2)^(2/3)*((2 - 2*I)*x^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, 1 - x] - I*Hypergeom
etric2F1[2/3, 1, 5/3, ((1/2 - I/2)*(-1 + x))/x] + Hypergeometric2F1[2/3, 1, 5/3, ((1/2 + I/2)*(-1 + x))/x]))/x
^2

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IntegrateAlgebraic [A]  time = 0.35, size = 140, 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 [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2 + x^2)/((1 + 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 - 2*#1^3 + #1^6 & , (-Log[x] + Log[(-x^2 + x^3)^(1/3) - x*#1])/#1
& ]/2

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fricas [B]  time = 0.74, size = 1652, normalized size = 11.80

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*2^(5/6)*cos(2/3*arctan(sqrt(2) + 1))*log(-4*(2*2^(2/3)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(2) + 1))^2
- 2*2^(2/3)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(2) + 1))*sin(2/3*arctan(sqrt(2) + 1)) - 2^(1/3)*x^2 - 2^(2
/3)*(x^3 - x^2)^(1/3)*x - (x^3 - x^2)^(2/3))/x^2) + 2^(5/6)*arctan((2*x*cos(2/3*arctan(sqrt(2) + 1))^2 - 2*x*c
os(2/3*arctan(sqrt(2) + 1))*sin(2/3*arctan(sqrt(2) + 1)) + 2^(1/3)*x*sqrt(-(2*2^(2/3)*(x^3 - x^2)^(1/3)*x*cos(
2/3*arctan(sqrt(2) + 1))^2 - 2*2^(2/3)*(x^3 - x^2)^(1/3)*x*cos(2/3*arctan(sqrt(2) + 1))*sin(2/3*arctan(sqrt(2)
+ 1)) - 2^(1/3)*x^2 - 2^(2/3)*(x^3 - x^2)^(1/3)*x - (x^3 - x^2)^(2/3))/x^2) - x - 2^(1/3)*(x^3 - x^2)^(1/3))/
(2*x*cos(2/3*arctan(sqrt(2) + 1))^2 + 2*x*cos(2/3*arctan(sqrt(2) + 1))*sin(2/3*arctan(sqrt(2) + 1)) - x))*sin(
2/3*arctan(sqrt(2) + 1)) - 1/2*(sqrt(3)*2^(5/6)*cos(2/3*arctan(sqrt(2) + 1)) - 2^(5/6)*sin(2/3*arctan(sqrt(2)
+ 1)))*arctan(-1/2*(32*x*cos(2/3*arctan(sqrt(2) + 1))^4 + 4*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(1/3) + 2^(1/3))*cos(
2/3*arctan(sqrt(2) + 1))*sin(2/3*arctan(sqrt(2) + 1)) - 4*((x^3 - x^2)^(1/3)*(sqrt(3)*2^(1/3) - 2^(1/3)) + 8*x
)*cos(2/3*arctan(sqrt(2) + 1))^2 + sqrt(2)*(2*(sqrt(3)*2^(1/3)*x - 2^(1/3)*x)*cos(2/3*arctan(sqrt(2) + 1))^2 -
2*(sqrt(3)*2^(1/3)*x + 2^(1/3)*x)*cos(2/3*arctan(sqrt(2) + 1))*sin(2/3*arctan(sqrt(2) + 1)) - sqrt(3)*2^(1/3)
*x + 2^(1/3)*x)*sqrt(-(2*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x - 2^(2/3)*x)*cos(2/3*arctan(sqrt(2) + 1))^2 + 2*
(x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x + 2^(2/3)*x)*cos(2/3*arctan(sqrt(2) + 1))*sin(2/3*arctan(sqrt(2) + 1)) -
2*2^(1/3)*x^2 - (x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x - 2^(2/3)*x) - 2*(x^3 - x^2)^(2/3))/x^2) - 2*sqrt(3)*x +
2*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(1/3) - 2^(1/3)) + 4*x)/(8*(2*x*cos(2/3*arctan(sqrt(2) + 1))^3 - x*cos(2/3*arct
an(sqrt(2) + 1)))*sin(2/3*arctan(sqrt(2) + 1)) - x)) - 1/2*(sqrt(3)*2^(5/6)*cos(2/3*arctan(sqrt(2) + 1)) + 2^(
5/6)*sin(2/3*arctan(sqrt(2) + 1)))*arctan(1/2*(32*x*cos(2/3*arctan(sqrt(2) + 1))^4 - 4*(x^3 - x^2)^(1/3)*(sqrt
(3)*2^(1/3) - 2^(1/3))*cos(2/3*arctan(sqrt(2) + 1))*sin(2/3*arctan(sqrt(2) + 1)) + 4*((x^3 - x^2)^(1/3)*(sqrt(
3)*2^(1/3) + 2^(1/3)) - 8*x)*cos(2/3*arctan(sqrt(2) + 1))^2 - sqrt(2)*(2*(sqrt(3)*2^(1/3)*x + 2^(1/3)*x)*cos(2
/3*arctan(sqrt(2) + 1))^2 - 2*(sqrt(3)*2^(1/3)*x - 2^(1/3)*x)*cos(2/3*arctan(sqrt(2) + 1))*sin(2/3*arctan(sqrt
(2) + 1)) - sqrt(3)*2^(1/3)*x - 2^(1/3)*x)*sqrt((2*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x + 2^(2/3)*x)*cos(2/3*a
rctan(sqrt(2) + 1))^2 + 2*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x - 2^(2/3)*x)*cos(2/3*arctan(sqrt(2) + 1))*sin(2
/3*arctan(sqrt(2) + 1)) + 2*2^(1/3)*x^2 - (x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x + 2^(2/3)*x) + 2*(x^3 - x^2)^(2
/3))/x^2) + 2*sqrt(3)*x - 2*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(1/3) + 2^(1/3)) + 4*x)/(8*(2*x*cos(2/3*arctan(sqrt(2
) + 1))^3 - x*cos(2/3*arctan(sqrt(2) + 1)))*sin(2/3*arctan(sqrt(2) + 1)) - x)) + 1/8*(sqrt(3)*2^(5/6)*sin(2/3*
arctan(sqrt(2) + 1)) - 2^(5/6)*cos(2/3*arctan(sqrt(2) + 1)))*log(8*(2*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x + 2
^(2/3)*x)*cos(2/3*arctan(sqrt(2) + 1))^2 + 2*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x - 2^(2/3)*x)*cos(2/3*arctan(
sqrt(2) + 1))*sin(2/3*arctan(sqrt(2) + 1)) + 2*2^(1/3)*x^2 - (x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x + 2^(2/3)*x)
+ 2*(x^3 - x^2)^(2/3))/x^2) - 1/8*(sqrt(3)*2^(5/6)*sin(2/3*arctan(sqrt(2) + 1)) + 2^(5/6)*cos(2/3*arctan(sqrt
(2) + 1)))*log(-8*(2*(x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x - 2^(2/3)*x)*cos(2/3*arctan(sqrt(2) + 1))^2 + 2*(x^3
- x^2)^(1/3)*(sqrt(3)*2^(2/3)*x + 2^(2/3)*x)*cos(2/3*arctan(sqrt(2) + 1))*sin(2/3*arctan(sqrt(2) + 1)) - 2*2^
(1/3)*x^2 - (x^3 - x^2)^(1/3)*(sqrt(3)*2^(2/3)*x - 2^(2/3)*x) - 2*(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 25.66, size = 12421, normalized size = 88.72

method	result	size
trager	\(\text {Expression too large to display}\)	\(12421\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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