3.27.43 \(\int \frac {(1+x^3) \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx\) [2643]

Optimal. Leaf size=235 \[ \frac {1}{3} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\& ,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}^2}\& \right ]-\frac {1}{6} \text {RootSum}\left [1-2 \text {$\#$1}^3+5 \text {$\#$1}^6-4 \text {$\#$1}^9+\text {$\#$1}^{12}\& ,\frac {\log (x)-\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3+3 \log (x) \text {$\#$1}^6-3 \log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^6-2 \log (x) \text {$\#$1}^9+2 \log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^9}{-\text {$\#$1}^2+5 \text {$\#$1}^5-6 \text {$\#$1}^8+2 \text {$\#$1}^{11}}\& \right ] \]

[Out]

Unintegrable

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Rubi [C] Result contains complex when optimal does not.
time = 1.59, antiderivative size = 2197, normalized size of antiderivative = 9.35, number of steps used = 31, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2081, 6857, 103, 163, 61, 93} \begin {gather*} \text {Too large to display} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((1 + x^3)*(-x^2 + x^3)^(1/3))/(1 + x^6),x]

[Out]

-1/3*(-x^2 + x^3)^(1/3) + ((-1)^(1/3)*(-x^2 + x^3)^(1/3))/3 - ((-1)^(2/3)*(-x^2 + x^3)^(1/3))/3 - (4*(-x^2 + x
^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/(3*Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)) - ((I/1
8)*((-3 + 6*I) + (1 + 2*I)*Sqrt[3])*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))
])/((-1 + x)^(1/3)*x^(2/3)) + (((-6 - 3*I) + (2 - I)*Sqrt[3])*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3)
)/(Sqrt[3]*(-1 + x)^(1/3))])/(18*(-1 + x)^(1/3)*x^(2/3)) + (((-6 + 3*I) + (2 + I)*Sqrt[3])*(-x^2 + x^3)^(1/3)*
ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/(18*(-1 + x)^(1/3)*x^(2/3)) + (((6 - 3*I) + (2 + I)*
Sqrt[3])*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/(18*(-1 + x)^(1/3)*x^(2/
3)) + ((-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*(1 - I)^(1/3)*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/((1 - I)^(2/
3)*Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)) + ((-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*(1 + I)^(1/3)*x^(1/3))/(Sqrt[3]
*(-1 + x)^(1/3))])/((1 + I)^(2/3)*Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)) + ((3 - Sqrt[3])*(-x^2 + x^3)^(1/3)*ArcTan[1
/Sqrt[3] + (2*(1 - (-1)^(1/6))^(1/3)*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/(6*(1 - (-1)^(1/6))^(2/3)*(-1 + x)^(1
/3)*x^(2/3)) - ((1/2 - I/2)*(-1)^(1/6)*(1 + (-1)^(1/6))^(1/3)*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*(1 + (-
1)^(1/6))^(1/3)*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/(Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)) - ((1/2 - I/2)*(-1 + (-1)
^(5/6))^(1/3)*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*(1 - (-1)^(5/6))^(1/3)*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3)
)])/(Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)) - ((1/2 - I/2)*(-1)^(5/6)*(1 + (-1)^(5/6))^(1/3)*(-x^2 + x^3)^(1/3)*ArcTa
n[1/Sqrt[3] + (2*(1 + (-1)^(5/6))^(1/3)*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/(Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)) +
((-x^2 + x^3)^(1/3)*Log[-(-1 + x)^(1/3) + (1 - I)^(1/3)*x^(1/3)])/(2*(1 - I)^(2/3)*(-1 + x)^(1/3)*x^(2/3)) +
((-x^2 + x^3)^(1/3)*Log[-(-1 + x)^(1/3) + (1 + I)^(1/3)*x^(1/3)])/(2*(1 + I)^(2/3)*(-1 + x)^(1/3)*x^(2/3)) + (
(1/4 + I/4)*(-1)^(1/6)*(1 - (-1)^(1/6))^(1/3)*(-x^2 + x^3)^(1/3)*Log[-(-1 + x)^(1/3) + (1 - (-1)^(1/6))^(1/3)*
x^(1/3)])/((-1 + x)^(1/3)*x^(2/3)) - ((1/4 - I/4)*(-1)^(1/6)*(1 + (-1)^(1/6))^(1/3)*(-x^2 + x^3)^(1/3)*Log[-(-
1 + x)^(1/3) + (1 + (-1)^(1/6))^(1/3)*x^(1/3)])/((-1 + x)^(1/3)*x^(2/3)) - ((1/4 - I/4)*(-1 + (-1)^(5/6))^(1/3
)*(-x^2 + x^3)^(1/3)*Log[-(-1 + x)^(1/3) + (1 - (-1)^(5/6))^(1/3)*x^(1/3)])/((-1 + x)^(1/3)*x^(2/3)) - ((1/4 -
I/4)*(-1)^(5/6)*(1 + (-1)^(5/6))^(1/3)*(-x^2 + x^3)^(1/3)*Log[-(-1 + x)^(1/3) + (1 + (-1)^(5/6))^(1/3)*x^(1/3
)])/((-1 + x)^(1/3)*x^(2/3)) - (2*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1/3)/(-1 + x)^(1/3)])/(3*(-1 + x)^(1/3)*x^(2/
3)) - ((1/12 + I/12)*(-1)^(1/6)*(3 - (-1)^(1/6))*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1/3)/(-1 + x)^(1/3)])/((-1 + x
)^(1/3)*x^(2/3)) + ((1/12 - I/12)*(-1)^(1/6)*(3 + (-1)^(1/6))*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1/3)/(-1 + x)^(1/
3)])/((-1 + x)^(1/3)*x^(2/3)) + ((1/12 - I/12)*(-1)^(5/6)*(3 + (-1)^(5/6))*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1/3)
/(-1 + x)^(1/3)])/((-1 + x)^(1/3)*x^(2/3)) + (((2 - I) + (2 + I)*Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1/3)/
(-1 + x)^(1/3)])/(12*(-1 + x)^(1/3)*x^(2/3)) + ((1/12 - I/12)*(-1)^(5/6)*(1 + (-1)^(5/6))^(1/3)*(-x^2 + x^3)^(
1/3)*Log[(-1)^(1/6) - x])/((-1 + x)^(1/3)*x^(2/3)) - ((1/12 + I/12)*(-1)^(1/6)*(1 - (-1)^(1/6))^(1/3)*(-x^2 +
x^3)^(1/3)*Log[-(-1)^(5/6) - x])/((-1 + x)^(1/3)*x^(2/3)) - (2*(-x^2 + x^3)^(1/3)*Log[-1 + x])/(9*(-1 + x)^(1/
3)*x^(2/3)) - ((1/36 + I/36)*(-1)^(1/6)*(3 - (-1)^(1/6))*(-x^2 + x^3)^(1/3)*Log[-1 + x])/((-1 + x)^(1/3)*x^(2/
3)) + ((1/36 - I/36)*(-1)^(1/6)*(3 + (-1)^(1/6))*(-x^2 + x^3)^(1/3)*Log[-1 + x])/((-1 + x)^(1/3)*x^(2/3)) - ((
1/36 + I/36)*(-1)^(1/3)*(3*I + (-1)^(1/3))*(-x^2 + x^3)^(1/3)*Log[-1 + x])/((-1 + x)^(1/3)*x^(2/3)) + ((1/36 -
I/36)*(-1)^(5/6)*(3 + (-1)^(5/6))*(-x^2 + x^3)^(1/3)*Log[-1 + x])/((-1 + x)^(1/3)*x^(2/3)) + ((1/12 - I/12)*(
-1)^(1/6)*(1 + (-1)^(1/6))^(1/3)*(-x^2 + x^3)^(1/3)*Log[(-1)^(1/6) + (-1)^(1/3)*x])/((-1 + x)^(1/3)*x^(2/3)) -
((-x^2 + x^3)^(1/3)*Log[-(-1)^(5/6) + (-1)^(1/3)*x])/(6*(1 + I)^(2/3)*(-1 + x)^(1/3)*x^(2/3)) - ((-x^2 + x^3)
^(1/3)*Log[(-1)^(1/6) - (-1)^(2/3)*x])/(6*(1 - I)^(2/3)*(-1 + x)^(1/3)*x^(2/3)) + ((1/12 - I/12)*(-1 + (-1)^(5
/6))^(1/3)*(-x^2 + x^3)^(1/3)*Log[-(-1)^(5/6) - (-1)^(2/3)*x])/((-1 + x)^(1/3)*x^(2/3))

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 93

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 103

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b
*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))

Rule 163

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

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 6857

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 {\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx &=\frac {\sqrt [3]{-x^2+x^3} \int \frac {\sqrt [3]{-1+x} x^{2/3} \left (1+x^3\right )}{1+x^6} \, dx}{\sqrt [3]{-1+x} x^{2/3}}\\ &=\frac {\sqrt [3]{-x^2+x^3} \int \left (-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [3]{-1+x} x^{2/3}}{i-x^3}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [3]{-1+x} x^{2/3}}{i+x^3}\right ) \, dx}{\sqrt [3]{-1+x} x^{2/3}}\\ &=-\frac {\left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [3]{-x^2+x^3}\right ) \int \frac {\sqrt [3]{-1+x} x^{2/3}}{i-x^3} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [3]{-x^2+x^3}\right ) \int \frac {\sqrt [3]{-1+x} x^{2/3}}{i+x^3} \, dx}{\sqrt [3]{-1+x} x^{2/3}}\\ &=-\frac {\left (\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [3]{-x^2+x^3}\right ) \int \left (-\frac {(-1)^{2/3} \sqrt [3]{-1+x} x^{2/3}}{3 \left (\sqrt [6]{-1}-x\right )}-\frac {(-1)^{2/3} \sqrt [3]{-1+x} x^{2/3}}{3 \left (\sqrt [6]{-1}+\sqrt [3]{-1} x\right )}-\frac {(-1)^{2/3} \sqrt [3]{-1+x} x^{2/3}}{3 \left (\sqrt [6]{-1}-(-1)^{2/3} x\right )}\right ) \, dx}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [3]{-x^2+x^3}\right ) \int \left (-\frac {\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3}}{3 \left (-(-1)^{5/6}-x\right )}-\frac {\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3}}{3 \left (-(-1)^{5/6}+\sqrt [3]{-1} x\right )}-\frac {\sqrt [3]{-1} \sqrt [3]{-1+x} x^{2/3}}{3 \left (-(-1)^{5/6}-(-1)^{2/3} x\right )}\right ) \, dx}{\sqrt [3]{-1+x} x^{2/3}}\\ &=-\frac {\left (\left (\frac {1}{6}+\frac {i}{6}\right ) \sqrt [3]{-1} \sqrt [3]{-x^2+x^3}\right ) \int \frac {\sqrt [3]{-1+x} x^{2/3}}{-(-1)^{5/6}-x} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+-\frac {\left (\left (\frac {1}{6}+\frac {i}{6}\right ) \sqrt [3]{-1} \sqrt [3]{-x^2+x^3}\right ) \int \frac {\sqrt [3]{-1+x} x^{2/3}}{-(-1)^{5/6}+\sqrt [3]{-1} x} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+-\frac {\left (\left (\frac {1}{6}+\frac {i}{6}\right ) \sqrt [3]{-1} \sqrt [3]{-x^2+x^3}\right ) \int \frac {\sqrt [3]{-1+x} x^{2/3}}{-(-1)^{5/6}-(-1)^{2/3} x} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\left (\frac {1}{6}-\frac {i}{6}\right ) (-1)^{2/3} \sqrt [3]{-x^2+x^3}\right ) \int \frac {\sqrt [3]{-1+x} x^{2/3}}{\sqrt [6]{-1}-x} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\left (\frac {1}{6}-\frac {i}{6}\right ) (-1)^{2/3} \sqrt [3]{-x^2+x^3}\right ) \int \frac {\sqrt [3]{-1+x} x^{2/3}}{\sqrt [6]{-1}+\sqrt [3]{-1} x} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\left (\frac {1}{6}-\frac {i}{6}\right ) (-1)^{2/3} \sqrt [3]{-x^2+x^3}\right ) \int \frac {\sqrt [3]{-1+x} x^{2/3}}{\sqrt [6]{-1}-(-1)^{2/3} x} \, dx}{\sqrt [3]{-1+x} x^{2/3}}\\ &=-\frac {1}{3} \sqrt [3]{-x^2+x^3}+\frac {1}{3} \sqrt [3]{-1} \sqrt [3]{-x^2+x^3}-\frac {1}{3} (-1)^{2/3} \sqrt [3]{-x^2+x^3}+\frac {\left (\left (\frac {1}{6}-\frac {i}{6}\right ) \sqrt [3]{-x^2+x^3}\right ) \int \frac {-\frac {2 \sqrt [6]{-1}}{3}+\left (1-\frac {i}{3}\right ) \sqrt [6]{-1} x}{(-1+x)^{2/3} \sqrt [3]{x} \left (\sqrt [6]{-1}-(-1)^{2/3} x\right )} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\left (\frac {1}{6}+\frac {i}{6}\right ) \sqrt [3]{-x^2+x^3}\right ) \int \frac {\frac {2}{3} (-1)^{5/6}+\left (\frac {1}{3}-i\right ) \sqrt [3]{-1} x}{(-1+x)^{2/3} \sqrt [3]{x} \left (-(-1)^{5/6}+\sqrt [3]{-1} x\right )} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+-\frac {\left (\left (\frac {1}{6}+\frac {i}{6}\right ) \sqrt [3]{-1} \sqrt [3]{-x^2+x^3}\right ) \int \frac {\frac {2}{3} (-1)^{5/6}-\frac {1}{3} (-1)^{5/6} \left (3-\sqrt [6]{-1}\right ) x}{\left (-(-1)^{5/6}-x\right ) (-1+x)^{2/3} \sqrt [3]{x}} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+-\frac {\left (\left (\frac {1}{6}-\frac {i}{6}\right ) \sqrt [3]{-1} \sqrt [3]{-x^2+x^3}\right ) \int \frac {-\frac {2 \sqrt [6]{-1}}{3}+\frac {1}{3} \sqrt [6]{-1} \left (3+\sqrt [6]{-1}\right ) x}{(-1+x)^{2/3} \sqrt [3]{x} \left (\sqrt [6]{-1}+\sqrt [3]{-1} x\right )} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\left (\frac {1}{6}-\frac {i}{6}\right ) (-1)^{2/3} \sqrt [3]{-x^2+x^3}\right ) \int \frac {-\frac {2 \sqrt [6]{-1}}{3}+\frac {1}{3} \left (-1+3 \sqrt [6]{-1}\right ) x}{\left (\sqrt [6]{-1}-x\right ) (-1+x)^{2/3} \sqrt [3]{x}} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\left (\frac {1}{6}+\frac {i}{6}\right ) (-1)^{2/3} \sqrt [3]{-x^2+x^3}\right ) \int \frac {\frac {2}{3} (-1)^{5/6}-\frac {1}{3} (-1)^{5/6} \left (3-(-1)^{5/6}\right ) x}{(-1+x)^{2/3} \sqrt [3]{x} \left (-(-1)^{5/6}-(-1)^{2/3} x\right )} \, dx}{\sqrt [3]{-1+x} x^{2/3}}\\ &=-\frac {1}{3} \sqrt [3]{-x^2+x^3}+\frac {1}{3} \sqrt [3]{-1} \sqrt [3]{-x^2+x^3}-\frac {1}{3} (-1)^{2/3} \sqrt [3]{-x^2+x^3}+\frac {\left (\left (\frac {2}{9}-\frac {i}{9}\right ) \sqrt [3]{-x^2+x^3}\right ) \int \frac {1}{(-1+x)^{2/3} \sqrt [3]{x}} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\left (\frac {2}{9}+\frac {i}{9}\right ) \sqrt [3]{-x^2+x^3}\right ) \int \frac {1}{(-1+x)^{2/3} \sqrt [3]{x}} \, dx}{\sqrt [3]{-1+x} x^{2/3}}-\frac {\left (\sqrt [6]{-1} \sqrt [3]{-x^2+x^3}\right ) \int \frac {1}{(-1+x)^{2/3} \sqrt [3]{x} \left (\sqrt [6]{-1}-(-1)^{2/3} x\right )} \, dx}{3 \sqrt [3]{-1+x} x^{2/3}}+\frac {\left ((-1)^{5/6} \sqrt [3]{-x^2+x^3}\right ) \int \frac {1}{(-1+x)^{2/3} \sqrt [3]{x} \left (-(-1)^{5/6}+\sqrt [3]{-1} x\right )} \, dx}{3 \sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\left (\frac {1}{18}-\frac {i}{18}\right ) (-1)^{2/3} \left (1-3 \sqrt [6]{-1}\right ) \sqrt [3]{-x^2+x^3}\right ) \int \frac {1}{(-1+x)^{2/3} \sqrt [3]{x}} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+-\frac {\left (\left (\frac {1}{6}+\frac {i}{6}\right ) \left (1-\sqrt [6]{-1}\right ) \sqrt [3]{-x^2+x^3}\right ) \int \frac {1}{\left (-(-1)^{5/6}-x\right ) (-1+x)^{2/3} \sqrt [3]{x}} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+-\frac {\left (\left (\frac {1}{6}-\frac {i}{6}\right ) (-1)^{5/6} \left (1-\sqrt [6]{-1}\right ) \sqrt [3]{-x^2+x^3}\right ) \int \frac {1}{\left (\sqrt [6]{-1}-x\right ) (-1+x)^{2/3} \sqrt [3]{x}} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\left (\frac {1}{6}-\frac {i}{6}\right ) \sqrt [3]{-1} \left (1+\sqrt [6]{-1}\right ) \sqrt [3]{-x^2+x^3}\right ) \int \frac {1}{(-1+x)^{2/3} \sqrt [3]{x} \left (\sqrt [6]{-1}+\sqrt [3]{-1} x\right )} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+-\frac {\left (\left (\frac {1}{18}-\frac {i}{18}\right ) \sqrt [6]{-1} \left (3+\sqrt [6]{-1}\right ) \sqrt [3]{-x^2+x^3}\right ) \int \frac {1}{(-1+x)^{2/3} \sqrt [3]{x}} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\left (\frac {1}{18}+\frac {i}{18}\right ) (-1)^{2/3} \left (1+3 \sqrt [6]{-1}\right ) \sqrt [3]{-x^2+x^3}\right ) \int \frac {1}{(-1+x)^{2/3} \sqrt [3]{x}} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\left (\frac {1}{6}+\frac {i}{6}\right ) (-1)^{2/3} \left (-1+(-1)^{5/6}\right ) \sqrt [3]{-x^2+x^3}\right ) \int \frac {1}{(-1+x)^{2/3} \sqrt [3]{x} \left (-(-1)^{5/6}-(-1)^{2/3} x\right )} \, dx}{\sqrt [3]{-1+x} x^{2/3}}+-\frac {\left (\left (\frac {1}{18}+\frac {i}{18}\right ) \sqrt [3]{-1} \left (1+3 (-1)^{5/6}\right ) \sqrt [3]{-x^2+x^3}\right ) \int \frac {1}{(-1+x)^{2/3} \sqrt [3]{x}} \, dx}{\sqrt [3]{-1+x} x^{2/3}}\\ &=-\frac {1}{3} \sqrt [3]{-x^2+x^3}+\frac {1}{3} \sqrt [3]{-1} \sqrt [3]{-x^2+x^3}-\frac {1}{3} (-1)^{2/3} \sqrt [3]{-x^2+x^3}-\frac {4 \sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{3 \sqrt {3} \sqrt [3]{-1+x} x^{2/3}}+\frac {\left ((-2-i)+\frac {2-i}{\sqrt {3}}\right ) \sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{6 \sqrt [3]{-1+x} x^{2/3}}+\frac {\left ((6+3 i)+(2-i) \sqrt {3}\right ) \sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{18 \sqrt [3]{-1+x} x^{2/3}}+\frac {\left ((-6+3 i)+(2+i) \sqrt {3}\right ) \sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{18 \sqrt [3]{-1+x} x^{2/3}}+\frac {\left ((6-3 i)+(2+i) \sqrt {3}\right ) \sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{18 \sqrt [3]{-1+x} x^{2/3}}+\frac {\sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-i} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{(1-i)^{2/3} \sqrt {3} \sqrt [3]{-1+x} x^{2/3}}+\frac {\sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+i} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{(1+i)^{2/3} \sqrt {3} \sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [6]{-1} \sqrt [3]{1-\sqrt [6]{-1}} \sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-\sqrt [6]{-1}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{\sqrt {3} \sqrt [3]{-1+x} x^{2/3}}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [6]{-1} \sqrt [3]{1+\sqrt [6]{-1}} \sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+\sqrt [6]{-1}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{\sqrt {3} \sqrt [3]{-1+x} x^{2/3}}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [3]{-1+(-1)^{5/6}} \sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-(-1)^{5/6}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{\sqrt {3} \sqrt [3]{-1+x} x^{2/3}}+\frac {\left (3-\sqrt {3}\right ) \sqrt [3]{-x^2+x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+(-1)^{5/6}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+x}}\right )}{6 \left (1+(-1)^{5/6}\right )^{2/3} \sqrt [3]{-1+x} x^{2/3}}+\frac {\sqrt [3]{-x^2+x^3} \log \left (-\sqrt [3]{-1+x}+\sqrt [3]{1-i} \sqrt [3]{x}\right )}{2 (1-i)^{2/3} \sqrt [3]{-1+x} x^{2/3}}+\frac {\sqrt [3]{-x^2+x^3} \log \left (-\sqrt [3]{-1+x}+\sqrt [3]{1+i} \sqrt [3]{x}\right )}{2 (1+i)^{2/3} \sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt [6]{-1} \sqrt [3]{1-\sqrt [6]{-1}} \sqrt [3]{-x^2+x^3} \log \left (-\sqrt [3]{-1+x}+\sqrt [3]{1-\sqrt [6]{-1}} \sqrt [3]{x}\right )}{\sqrt [3]{-1+x} x^{2/3}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt [6]{-1} \sqrt [3]{1+\sqrt [6]{-1}} \sqrt [3]{-x^2+x^3} \log \left (-\sqrt [3]{-1+x}+\sqrt [3]{1+\sqrt [6]{-1}} \sqrt [3]{x}\right )}{\sqrt [3]{-1+x} x^{2/3}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt [3]{-1+(-1)^{5/6}} \sqrt [3]{-x^2+x^3} \log \left (-\sqrt [3]{-1+x}+\sqrt [3]{1-(-1)^{5/6}} \sqrt [3]{x}\right )}{\sqrt [3]{-1+x} x^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{-x^2+x^3} \log \left (-\sqrt [3]{-1+x}+\sqrt [3]{1+(-1)^{5/6}} \sqrt [3]{x}\right )}{4 \left (1+(-1)^{5/6}\right )^{2/3} \sqrt [3]{-1+x} x^{2/3}}-\frac {2 \sqrt [3]{-x^2+x^3} \log \left (-1+\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{3 \sqrt [3]{-1+x} x^{2/3}}-\frac {\left (\frac {1}{12}+\frac {i}{12}\right ) \sqrt [6]{-1} \left (3-\sqrt [6]{-1}\right ) \sqrt [3]{-x^2+x^3} \log \left (-1+\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) \sqrt [6]{-1} \left (3+\sqrt [6]{-1}\right ) \sqrt [3]{-x^2+x^3} \log \left (-1+\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left ((2+i)-(2-i) \sqrt {3}\right ) \sqrt [3]{-x^2+x^3} \log \left (-1+\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{12 \sqrt [3]{-1+x} x^{2/3}}+\frac {\left ((2-i)+(2+i) \sqrt {3}\right ) \sqrt [3]{-x^2+x^3} \log \left (-1+\frac {\sqrt [3]{x}}{\sqrt [3]{-1+x}}\right )}{12 \sqrt [3]{-1+x} x^{2/3}}+\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{-x^2+x^3} \log \left (\sqrt [6]{-1}-x\right )}{12 \left (1+(-1)^{5/6}\right )^{2/3} \sqrt [3]{-1+x} x^{2/3}}-\frac {\left (\frac {1}{12}+\frac {i}{12}\right ) \sqrt [6]{-1} \sqrt [3]{1-\sqrt [6]{-1}} \sqrt [3]{-x^2+x^3} \log \left (-(-1)^{5/6}-x\right )}{\sqrt [3]{-1+x} x^{2/3}}-\frac {2 \sqrt [3]{-x^2+x^3} \log (-1+x)}{9 \sqrt [3]{-1+x} x^{2/3}}-\frac {\left (\frac {1}{36}-\frac {i}{36}\right ) (-1)^{2/3} \left (1-3 \sqrt [6]{-1}\right ) \sqrt [3]{-x^2+x^3} \log (-1+x)}{\sqrt [3]{-1+x} x^{2/3}}-\frac {\left (\frac {1}{36}+\frac {i}{36}\right ) \sqrt [6]{-1} \left (3-\sqrt [6]{-1}\right ) \sqrt [3]{-x^2+x^3} \log (-1+x)}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\frac {1}{36}-\frac {i}{36}\right ) \sqrt [6]{-1} \left (3+\sqrt [6]{-1}\right ) \sqrt [3]{-x^2+x^3} \log (-1+x)}{\sqrt [3]{-1+x} x^{2/3}}-\frac {\left (\frac {1}{36}+\frac {i}{36}\right ) (-1)^{5/6} \left (3-(-1)^{5/6}\right ) \sqrt [3]{-x^2+x^3} \log (-1+x)}{\sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) \sqrt [6]{-1} \sqrt [3]{1+\sqrt [6]{-1}} \sqrt [3]{-x^2+x^3} \log \left (\sqrt [6]{-1}+\sqrt [3]{-1} x\right )}{\sqrt [3]{-1+x} x^{2/3}}-\frac {\sqrt [3]{-x^2+x^3} \log \left (-(-1)^{5/6}+\sqrt [3]{-1} x\right )}{6 (1+i)^{2/3} \sqrt [3]{-1+x} x^{2/3}}-\frac {\sqrt [3]{-x^2+x^3} \log \left (\sqrt [6]{-1}-(-1)^{2/3} x\right )}{6 (1-i)^{2/3} \sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) \sqrt [3]{-1+(-1)^{5/6}} \sqrt [3]{-x^2+x^3} \log \left (-(-1)^{5/6}-(-1)^{2/3} x\right )}{\sqrt [3]{-1+x} x^{2/3}}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 253, normalized size = 1.08 \begin {gather*} \frac {(-1+x)^{2/3} x^{4/3} \left (6 \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}^2}\&\right ]+\text {RootSum}\left [1-2 \text {$\#$1}^3+5 \text {$\#$1}^6-4 \text {$\#$1}^9+\text {$\#$1}^{12}\&,\frac {-\log (x)+3 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )-\log (x) \text {$\#$1}^3+3 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^3-3 \log (x) \text {$\#$1}^6+9 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^6+2 \log (x) \text {$\#$1}^9-6 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^9}{-\text {$\#$1}^2+5 \text {$\#$1}^5-6 \text {$\#$1}^8+2 \text {$\#$1}^{11}}\&\right ]\right )}{18 \left ((-1+x) x^2\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x^3)*(-x^2 + x^3)^(1/3))/(1 + x^6),x]

[Out]

((-1 + x)^(2/3)*x^(4/3)*(6*RootSum[2 - 2*#1^3 + #1^6 & , (-Log[x^(1/3)] + Log[(-1 + x)^(1/3) - x^(1/3)*#1])/#1
^2 & ] + RootSum[1 - 2*#1^3 + 5*#1^6 - 4*#1^9 + #1^12 & , (-Log[x] + 3*Log[(-1 + x)^(1/3) - x^(1/3)*#1] - Log[
x]*#1^3 + 3*Log[(-1 + x)^(1/3) - x^(1/3)*#1]*#1^3 - 3*Log[x]*#1^6 + 9*Log[(-1 + x)^(1/3) - x^(1/3)*#1]*#1^6 +
2*Log[x]*#1^9 - 6*Log[(-1 + x)^(1/3) - x^(1/3)*#1]*#1^9)/(-#1^2 + 5*#1^5 - 6*#1^8 + 2*#1^11) & ]))/(18*((-1 +
x)*x^2)^(2/3))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 57.42, size = 70173, normalized size = 298.61

method	result	size
trager	\(\text {Expression too large to display}\)	\(70173\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+1)*(x^3-x^2)^(1/3)/(x^6+1),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*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)*(x^3-x^2)^(1/3)/(x^6+1),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 2.26, size = 7128, normalized size = 30.33 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)*(x^3-x^2)^(1/3)/(x^6+1),x, algorithm="fricas")

[Out]

-1/6*4^(2/3)*sqrt(3)*(27*sqrt(1/1944*I) + 27*sqrt(-1/1944*I) + sqrt(-3/16*(108*sqrt(1/1944*I) + I + 3)^2 - 1/8
*(108*sqrt(1/1944*I) + I - 9)*(108*sqrt(-1/1944*I) - I + 3) - 3/16*(108*sqrt(-1/1944*I) - I + 3)^2 + 162*sqrt(
1/1944*I) + 3/2*I - 1/2) - 3/2)^(1/3)*arctan(1/384*((4^(1/3)*sqrt(3)*x*(108*sqrt(1/1944*I) + I + 3)^2 - (4^(1/
3)*sqrt(3)*x*(108*sqrt(1/1944*I) + I + 3) - 4^(1/3)*sqrt(3)*x)*(108*sqrt(-1/1944*I) - I + 3)^2 - 12*4^(1/3)*sq
rt(3)*x*(108*sqrt(1/1944*I) + I + 3) - (4^(1/3)*sqrt(3)*x*(108*sqrt(1/1944*I) + I + 3)^2 - 12*4^(1/3)*sqrt(3)*
x*(108*sqrt(1/1944*I) + I + 3) + 12*4^(1/3)*sqrt(3)*x)*(108*sqrt(-1/1944*I) - I + 3) - 4*(4^(1/3)*sqrt(3)*x*(1
08*sqrt(1/1944*I) + I + 3) - (4^(1/3)*sqrt(3)*x*(108*sqrt(1/1944*I) + I + 3) - 4^(1/3)*sqrt(3)*x)*(108*sqrt(-1
/1944*I) - I + 3))*sqrt(-3/16*(108*sqrt(1/1944*I) + I + 3)^2 - 1/8*(108*sqrt(1/1944*I) + I - 9)*(108*sqrt(-1/1
944*I) - I + 3) - 3/16*(108*sqrt(-1/1944*I) - I + 3)^2 + 162*sqrt(1/1944*I) + 3/2*I - 1/2))*(27*sqrt(1/1944*I)
+ 27*sqrt(-1/1944*I) + sqrt(-3/16*(108*sqrt(1/1944*I) + I + 3)^2 - 1/8*(108*sqrt(1/1944*I) + I - 9)*(108*sqrt
(-1/1944*I) - I + 3) - 3/16*(108*sqrt(-1/1944*I) - I + 3)^2 + 162*sqrt(1/1944*I) + 3/2*I - 1/2) - 3/2)^(2/3)*s
qrt((2*(4^(1/3)*x^2*(108*sqrt(1/1944*I) + I + 3)^2 + 4^(1/3)*x^2*(108*sqrt(-1/1944*I) - I + 3)^2 - 8*4^(1/3)*x
^2*(108*sqrt(1/1944*I) + I + 3) - 8*4^(1/3)*x^2*(108*sqrt(-1/1944*I) - I + 3) + 32*4^(1/3)*x^2 - 4*(4^(1/3)*x^
2*(108*sqrt(1/1944*I) + I + 3) + 4^(1/3)*x^2*(108*sqrt(-1/1944*I) - I + 3) - 4*4^(1/3)*x^2)*sqrt(-3/16*(108*sq
rt(1/1944*I) + I + 3)^2 - 1/8*(108*sqrt(1/1944*I) + I - 9)*(108*sqrt(-1/1944*I) - I + 3) - 3/16*(108*sqrt(-1/1
944*I) - I + 3)^2 + 162*sqrt(1/1944*I) + 3/2*I - 1/2))*(27*sqrt(1/1944*I) + 27*sqrt(-1/1944*I) + sqrt(-3/16*(1
08*sqrt(1/1944*I) + I + 3)^2 - 1/8*(108*sqrt(1/1944*I) + I - 9)*(108*sqrt(-1/1944*I) - I + 3) - 3/16*(108*sqrt
(-1/1944*I) - I + 3)^2 + 162*sqrt(1/1944*I) + 3/2*I - 1/2) - 3/2)^(2/3) - (2*4^(2/3)*(x^3 - x^2)^(1/3)*x*(108*
sqrt(1/1944*I) + I + 3)^2 - 20*4^(2/3)*(x^3 - x^2)^(1/3)*x*(108*sqrt(1/1944*I) + I + 3) - (4^(2/3)*(x^3 - x^2)
^(1/3)*x*(108*sqrt(1/1944*I) + I + 3) - 2*4^(2/3)*(x^3 - x^2)^(1/3)*x)*(108*sqrt(-1/1944*I) - I + 3)^2 + 48*4^
(2/3)*(x^3 - x^2)^(1/3)*x - (4^(2/3)*(x^3 - x^2)^(1/3)*x*(108*sqrt(1/1944*I) + I + 3)^2 - 12*4^(2/3)*(x^3 - x^
2)^(1/3)*x*(108*sqrt(1/1944*I) + I + 3) + 20*4^(2/3)*(x^3 - x^2)^(1/3)*x)*(108*sqrt(-1/1944*I) - I + 3) - 4*(2
*4^(2/3)*(x^3 - x^2)^(1/3)*x*(108*sqrt(1/1944*I) + I + 3) - 4*4^(2/3)*(x^3 - x^2)^(1/3)*x - (4^(2/3)*(x^3 - x^
2)^(1/3)*x*(108*sqrt(1/1944*I) + I + 3) - 2*4^(2/3)*(x^3 - x^2)^(1/3)*x)*(108*sqrt(-1/1944*I) - I + 3))*sqrt(-
3/16*(108*sqrt(1/1944*I) + I + 3)^2 - 1/8*(108*sqrt(1/1944*I) + I - 9)*(108*sqrt(-1/1944*I) - I + 3) - 3/16*(1
08*sqrt(-1/1944*I) - I + 3)^2 + 162*sqrt(1/1944*I) + 3/2*I - 1/2))*(27*sqrt(1/1944*I) + 27*sqrt(-1/1944*I) + s
qrt(-3/16*(108*sqrt(1/1944*I) + I + 3)^2 - 1/8*(108*sqrt(1/1944*I) + I - 9)*(108*sqrt(-1/1944*I) - I + 3) - 3/
16*(108*sqrt(-1/1944*I) - I + 3)^2 + 162*sqrt(1/1944*I) + 3/2*I - 1/2) - 3/2)^(1/3) + 64*(x^3 - x^2)^(2/3))/x^
2) - 8*(4^(1/3)*sqrt(3)*(x^3 - x^2)^(1/3)*(108*sqrt(1/1944*I) + I + 3)^2 - (4^(1/3)*sqrt(3)*(x^3 - x^2)^(1/3)*
(108*sqrt(1/1944*I) + I + 3) - 4^(1/3)*sqrt(3)*(x^3 - x^2)^(1/3))*(108*sqrt(-1/1944*I) - I + 3)^2 - 12*4^(1/3)
*sqrt(3)*(x^3 - x^2)^(1/3)*(108*sqrt(1/1944*I) + I + 3) - (4^(1/3)*sqrt(3)*(x^3 - x^2)^(1/3)*(108*sqrt(1/1944*
I) + I + 3)^2 - 12*4^(1/3)*sqrt(3)*(x^3 - x^2)^(1/3)*(108*sqrt(1/1944*I) + I + 3) + 12*4^(1/3)*sqrt(3)*(x^3 -
x^2)^(1/3))*(108*sqrt(-1/1944*I) - I + 3) - 4*(4^(1/3)*sqrt(3)*(x^3 - x^2)^(1/3)*(108*sqrt(1/1944*I) + I + 3)
- (4^(1/3)*sqrt(3)*(x^3 - x^2)^(1/3)*(108*sqrt(1/1944*I) + I + 3) - 4^(1/3)*sqrt(3)*(x^3 - x^2)^(1/3))*(108*sq
rt(-1/1944*I) - I + 3))*sqrt(-3/16*(108*sqrt(1/1944*I) + I + 3)^2 - 1/8*(108*sqrt(1/1944*I) + I - 9)*(108*sqrt
(-1/1944*I) - I + 3) - 3/16*(108*sqrt(-1/1944*I) - I + 3)^2 + 162*sqrt(1/1944*I) + 3/2*I - 1/2))*(27*sqrt(1/19
44*I) + 27*sqrt(-1/1944*I) + sqrt(-3/16*(108*sqrt(1/1944*I) + I + 3)^2 - 1/8*(108*sqrt(1/1944*I) + I - 9)*(108
*sqrt(-1/1944*I) - I + 3) - 3/16*(108*sqrt(-1/1944*I) - I + 3)^2 + 162*sqrt(1/1944*I) + 3/2*I - 1/2) - 3/2)^(2
/3) + 128*sqrt(3)*x)/x) + 1/6*4^(2/3)*sqrt(3)*(27*sqrt(1/1944*I) + 27*sqrt(-1/1944*I) - sqrt(-3/16*(108*sqrt(1
/1944*I) + I + 3)^2 - 1/8*(108*sqrt(1/1944*I) + I - 9)*(108*sqrt(-1/1944*I) - I + 3) - 3/16*(108*sqrt(-1/1944*
I) - I + 3)^2 + 162*sqrt(1/1944*I) + 3/2*I - 1/2) - 3/2)^(1/3)*arctan(-1/384*((4^(1/3)*sqrt(3)*x*(108*sqrt(1/1
944*I) + I + 3)^2 - (4^(1/3)*sqrt(3)*x*(108*sqrt(1/1944*I) + I + 3) - 4^(1/3)*sqrt(3)*x)*(108*sqrt(-1/1944*I)
- I + 3)^2 - 12*4^(1/3)*sqrt(3)*x*(108*sqrt(1/1944*I) + I + 3) - (4^(1/3)*sqrt(3)*x*(108*sqrt(1/1944*I) + I +
3)^2 - 12*4^(1/3)*sqrt(3)*x*(108*sqrt(1/1944*I) + I + 3) + 12*4^(1/3)*sqrt(3)*x)*(108*sqrt(-1/1944*I) - I + 3)
+ 4*(4^(1/3)*sqrt(3)*x*(108*sqrt(1/1944*I) + I + 3) - (4^(1/3)*sqrt(3)*x*(108*sqrt(1/1944*I) + I + 3) - 4^(1/
3)*sqrt(3)*x)*(108*sqrt(-1/1944*I) - I + 3))*sq...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{x^{2} \left (x - 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}{\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+1)*(x**3-x**2)**(1/3)/(x**6+1),x)

[Out]

Integral((x**2*(x - 1))**(1/3)*(x + 1)*(x**2 - x + 1)/((x**2 + 1)*(x**4 - x**2 + 1)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)*(x^3-x^2)^(1/3)/(x^6+1),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (x^3+1\right )\,{\left (x^3-x^2\right )}^{1/3}}{x^6+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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