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

Optimal result

Integrand size = 26, antiderivative size = 235 \[ \int \frac {\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\frac {1}{3} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}^2}\&\right ]-\frac {1}{6} \text {RootSum}\left [1-2 \text {$\#$1}^3+5 \text {$\#$1}^6-4 \text {$\#$1}^9+\text {$\#$1}^{12}\&,\frac {\log (x)-\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3+3 \log (x) \text {$\#$1}^6-3 \log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^6-2 \log (x) \text {$\#$1}^9+2 \log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^9}{-\text {$\#$1}^2+5 \text {$\#$1}^5-6 \text {$\#$1}^8+2 \text {$\#$1}^{11}}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.55 (sec) , antiderivative size = 2197, normalized size of antiderivative = 9.35, number of steps used = 31, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2081, 6857, 103, 163, 61, 93} \[ \int \frac {\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\frac {\left ((6-3 i)+(2+i) \sqrt {3}\right ) \sqrt [3]{x^3-x^2} \arctan \left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{18 \sqrt [3]{x-1} x^{2/3}}+\frac {\left ((-6+3 i)+(2+i) \sqrt {3}\right ) \sqrt [3]{x^3-x^2} \arctan \left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{18 \sqrt [3]{x-1} x^{2/3}}+\frac {\left ((-6-3 i)+(2-i) \sqrt {3}\right ) \sqrt [3]{x^3-x^2} \arctan \left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{18 \sqrt [3]{x-1} x^{2/3}}-\frac {i \left ((-3+6 i)+(1+2 i) \sqrt {3}\right ) \sqrt [3]{x^3-x^2} \arctan \left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{18 \sqrt [3]{x-1} x^{2/3}}-\frac {4 \sqrt [3]{x^3-x^2} \arctan \left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{x-1} x^{2/3}}+\frac {\sqrt [3]{x^3-x^2} \arctan \left (\frac {2 \sqrt [3]{1-i} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{(1-i)^{2/3} \sqrt {3} \sqrt [3]{x-1} x^{2/3}}+\frac {\sqrt [3]{x^3-x^2} \arctan \left (\frac {2 \sqrt [3]{1+i} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{(1+i)^{2/3} \sqrt {3} \sqrt [3]{x-1} x^{2/3}}+\frac {\left (3-\sqrt {3}\right ) \sqrt [3]{x^3-x^2} \arctan \left (\frac {2 \sqrt [3]{1-\sqrt [6]{-1}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{6 \left (1-\sqrt [6]{-1}\right )^{2/3} \sqrt [3]{x-1} x^{2/3}}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [6]{-1} \sqrt [3]{1+\sqrt [6]{-1}} \sqrt [3]{x^3-x^2} \arctan \left (\frac {2 \sqrt [3]{1+\sqrt [6]{-1}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{x-1} x^{2/3}}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [3]{-1+(-1)^{5/6}} \sqrt [3]{x^3-x^2} \arctan \left (\frac {2 \sqrt [3]{1-(-1)^{5/6}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{x-1} x^{2/3}}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) (-1)^{5/6} \sqrt [3]{1+(-1)^{5/6}} \sqrt [3]{x^3-x^2} \arctan \left (\frac {2 \sqrt [3]{1+(-1)^{5/6}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{x-1} x^{2/3}}+\frac {\sqrt [3]{x^3-x^2} \log \left (\sqrt [3]{1-i} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{2 (1-i)^{2/3} \sqrt [3]{x-1} x^{2/3}}+\frac {\sqrt [3]{x^3-x^2} \log \left (\sqrt [3]{1+i} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{2 (1+i)^{2/3} \sqrt [3]{x-1} x^{2/3}}+\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt [6]{-1} \sqrt [3]{1-\sqrt [6]{-1}} \sqrt [3]{x^3-x^2} \log \left (\sqrt [3]{1-\sqrt [6]{-1}} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{\sqrt [3]{x-1} x^{2/3}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt [6]{-1} \sqrt [3]{1+\sqrt [6]{-1}} \sqrt [3]{x^3-x^2} \log \left (\sqrt [3]{1+\sqrt [6]{-1}} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{\sqrt [3]{x-1} x^{2/3}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \sqrt [3]{-1+(-1)^{5/6}} \sqrt [3]{x^3-x^2} \log \left (\sqrt [3]{1-(-1)^{5/6}} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{\sqrt [3]{x-1} x^{2/3}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) (-1)^{5/6} \sqrt [3]{1+(-1)^{5/6}} \sqrt [3]{x^3-x^2} \log \left (\sqrt [3]{1+(-1)^{5/6}} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{\sqrt [3]{x-1} x^{2/3}}+\frac {\left ((2-i)+(2+i) \sqrt {3}\right ) \sqrt [3]{x^3-x^2} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )}{12 \sqrt [3]{x-1} x^{2/3}}+\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) (-1)^{5/6} \left (3+(-1)^{5/6}\right ) \sqrt [3]{x^3-x^2} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )}{\sqrt [3]{x-1} x^{2/3}}+\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) \sqrt [6]{-1} \left (3+\sqrt [6]{-1}\right ) \sqrt [3]{x^3-x^2} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )}{\sqrt [3]{x-1} x^{2/3}}-\frac {\left (\frac {1}{12}+\frac {i}{12}\right ) \sqrt [6]{-1} \left (3-\sqrt [6]{-1}\right ) \sqrt [3]{x^3-x^2} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )}{\sqrt [3]{x-1} x^{2/3}}-\frac {2 \sqrt [3]{x^3-x^2} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )}{3 \sqrt [3]{x-1} x^{2/3}}+\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) (-1)^{5/6} \sqrt [3]{1+(-1)^{5/6}} \sqrt [3]{x^3-x^2} \log \left (\sqrt [6]{-1}-x\right )}{\sqrt [3]{x-1} x^{2/3}}-\frac {\left (\frac {1}{12}+\frac {i}{12}\right ) \sqrt [6]{-1} \sqrt [3]{1-\sqrt [6]{-1}} \sqrt [3]{x^3-x^2} \log \left (-x-(-1)^{5/6}\right )}{\sqrt [3]{x-1} x^{2/3}}+\frac {\left (\frac {1}{36}-\frac {i}{36}\right ) (-1)^{5/6} \left (3+(-1)^{5/6}\right ) \sqrt [3]{x^3-x^2} \log (x-1)}{\sqrt [3]{x-1} x^{2/3}}-\frac {\left (\frac {1}{36}+\frac {i}{36}\right ) \sqrt [3]{-1} \left (3 i+\sqrt [3]{-1}\right ) \sqrt [3]{x^3-x^2} \log (x-1)}{\sqrt [3]{x-1} x^{2/3}}+\frac {\left (\frac {1}{36}-\frac {i}{36}\right ) \sqrt [6]{-1} \left (3+\sqrt [6]{-1}\right ) \sqrt [3]{x^3-x^2} \log (x-1)}{\sqrt [3]{x-1} x^{2/3}}-\frac {\left (\frac {1}{36}+\frac {i}{36}\right ) \sqrt [6]{-1} \left (3-\sqrt [6]{-1}\right ) \sqrt [3]{x^3-x^2} \log (x-1)}{\sqrt [3]{x-1} x^{2/3}}-\frac {2 \sqrt [3]{x^3-x^2} \log (x-1)}{9 \sqrt [3]{x-1} x^{2/3}}+\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) \sqrt [6]{-1} \sqrt [3]{1+\sqrt [6]{-1}} \sqrt [3]{x^3-x^2} \log \left (\sqrt [3]{-1} x+\sqrt [6]{-1}\right )}{\sqrt [3]{x-1} x^{2/3}}-\frac {\sqrt [3]{x^3-x^2} \log \left (\sqrt [3]{-1} x-(-1)^{5/6}\right )}{6 (1+i)^{2/3} \sqrt [3]{x-1} x^{2/3}}-\frac {\sqrt [3]{x^3-x^2} \log \left (\sqrt [6]{-1}-(-1)^{2/3} x\right )}{6 (1-i)^{2/3} \sqrt [3]{x-1} x^{2/3}}+\frac {\left (\frac {1}{12}-\frac {i}{12}\right ) \sqrt [3]{-1+(-1)^{5/6}} \sqrt [3]{x^3-x^2} \log \left (-(-1)^{2/3} x-(-1)^{5/6}\right )}{\sqrt [3]{x-1} x^{2/3}}-\frac {1}{3} (-1)^{2/3} \sqrt [3]{x^3-x^2}+\frac {1}{3} \sqrt [3]{-1} \sqrt [3]{x^3-x^2}-\frac {1}{3} \sqrt [3]{x^3-x^2} \]

[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*} \text {integral}& = \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}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.08 \[ \int \frac {\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\frac {(-1+x)^{2/3} x^{4/3} \left (6 \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}^2}\&\right ]+\text {RootSum}\left [1-2 \text {$\#$1}^3+5 \text {$\#$1}^6-4 \text {$\#$1}^9+\text {$\#$1}^{12}\&,\frac {-\log (x)+3 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right )-\log (x) \text {$\#$1}^3+3 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^3-3 \log (x) \text {$\#$1}^6+9 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^6+2 \log (x) \text {$\#$1}^9-6 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^9}{-\text {$\#$1}^2+5 \text {$\#$1}^5-6 \text {$\#$1}^8+2 \text {$\#$1}^{11}}\&\right ]\right )}{18 \left ((-1+x) x^2\right )^{2/3}} \]

[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))

Maple [N/A] (verified)

Time = 65.63 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.54

method	result	size
pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 \textit {\_Z}^{3}+2\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{2}}\right )}{3}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{12}-4 \textit {\_Z}^{9}+5 \textit {\_Z}^{6}-2 \textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (2 \textit {\_R}^{9}-3 \textit {\_R}^{6}-\textit {\_R}^{3}-1\right ) \ln \left (\frac {-\textit {\_R} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{2} \left (2 \textit {\_R}^{9}-6 \textit {\_R}^{6}+5 \textit {\_R}^{3}-1\right )}\right )}{6}\)	\(126\)
trager	\(\text {Expression too large to display}\)	\(59856\)

[In]

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

[Out]

1/3*sum(ln((-_R*x+((-1+x)*x^2)^(1/3))/x)/_R^2,_R=RootOf(_Z^6-2*_Z^3+2))-1/6*sum((2*_R^9-3*_R^6-_R^3-1)*ln((-_R
*x+((-1+x)*x^2)^(1/3))/x)/_R^2/(2*_R^9-6*_R^6+5*_R^3-1),_R=RootOf(_Z^12-4*_Z^9+5*_Z^6-2*_Z^3+1))

Fricas [C] (verification not implemented)

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

Time = 1.08 (sec) , antiderivative size = 3658, normalized size of antiderivative = 15.57 \[ \int \frac {\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/24*4^(2/3)*(27*sqrt(1/1944*I) + 27*sqrt(-1/1944*I) + sqrt(-3/16*(108*sqrt(1/1944*I) + I + 3)^2 - 1/8*(108*s
qrt(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)*(sqrt(-3) + 1)*log(-1/16*((2*4^(2/3)*(sqrt(-3)*x + x)*(108*sqrt(1/1944*I) + I +
3)^2 - (4^(2/3)*(sqrt(-3)*x + x)*(108*sqrt(1/1944*I) + I + 3) - 2*4^(2/3)*(sqrt(-3)*x + x))*(108*sqrt(-1/1944
*I) - I + 3)^2 - 20*4^(2/3)*(sqrt(-3)*x + x)*(108*sqrt(1/1944*I) + I + 3) - (4^(2/3)*(sqrt(-3)*x + x)*(108*sqr
t(1/1944*I) + I + 3)^2 - 12*4^(2/3)*(sqrt(-3)*x + x)*(108*sqrt(1/1944*I) + I + 3) + 20*4^(2/3)*(sqrt(-3)*x + x
))*(108*sqrt(-1/1944*I) - I + 3) + 48*4^(2/3)*(sqrt(-3)*x + x) - 4*(2*4^(2/3)*(sqrt(-3)*x + x)*(108*sqrt(1/194
4*I) + I + 3) - (4^(2/3)*(sqrt(-3)*x + x)*(108*sqrt(1/1944*I) + I + 3) - 2*4^(2/3)*(sqrt(-3)*x + x))*(108*sqrt
(-1/1944*I) - I + 3) - 4*4^(2/3)*(sqrt(-3)*x + x))*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(1/1944*I) + 27*sqrt(-1/1944*I) + sqrt(-3/16*(108*sqrt(1/1944*I) + I + 3)^2 - 1/8*(108*s
qrt(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) - 128*(x^3 - x^2)^(1/3))/x) + 1/24*4^(2/3)*(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)*(sqrt(-3) - 1)*log(
1/16*((2*4^(2/3)*(sqrt(-3)*x - x)*(108*sqrt(1/1944*I) + I + 3)^2 - (4^(2/3)*(sqrt(-3)*x - x)*(108*sqrt(1/1944*
I) + I + 3) - 2*4^(2/3)*(sqrt(-3)*x - x))*(108*sqrt(-1/1944*I) - I + 3)^2 - 20*4^(2/3)*(sqrt(-3)*x - x)*(108*s
qrt(1/1944*I) + I + 3) - (4^(2/3)*(sqrt(-3)*x - x)*(108*sqrt(1/1944*I) + I + 3)^2 - 12*4^(2/3)*(sqrt(-3)*x - x
)*(108*sqrt(1/1944*I) + I + 3) + 20*4^(2/3)*(sqrt(-3)*x - x))*(108*sqrt(-1/1944*I) - I + 3) + 48*4^(2/3)*(sqrt
(-3)*x - x) - 4*(2*4^(2/3)*(sqrt(-3)*x - x)*(108*sqrt(1/1944*I) + I + 3) - (4^(2/3)*(sqrt(-3)*x - x)*(108*sqrt
(1/1944*I) + I + 3) - 2*4^(2/3)*(sqrt(-3)*x - x))*(108*sqrt(-1/1944*I) - I + 3) - 4*4^(2/3)*(sqrt(-3)*x - x))*
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(1/1944*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) + 128*(x^3 - x^2)^(1
/3))/x) - 1/24*4^(2/3)*(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*sqr
t(1/1944*I) + 3/2*I - 1/2) - 3/2)^(1/3)*(sqrt(-3) + 1)*log(-1/16*((2*4^(2/3)*(sqrt(-3)*x + x)*(108*sqrt(1/1944
*I) + I + 3)^2 - (4^(2/3)*(sqrt(-3)*x + x)*(108*sqrt(1/1944*I) + I + 3) - 2*4^(2/3)*(sqrt(-3)*x + x))*(108*sqr
t(-1/1944*I) - I + 3)^2 - 20*4^(2/3)*(sqrt(-3)*x + x)*(108*sqrt(1/1944*I) + I + 3) - (4^(2/3)*(sqrt(-3)*x + x)
*(108*sqrt(1/1944*I) + I + 3)^2 - 12*4^(2/3)*(sqrt(-3)*x + x)*(108*sqrt(1/1944*I) + I + 3) + 20*4^(2/3)*(sqrt(
-3)*x + x))*(108*sqrt(-1/1944*I) - I + 3) + 48*4^(2/3)*(sqrt(-3)*x + x) + 4*(2*4^(2/3)*(sqrt(-3)*x + x)*(108*s
qrt(1/1944*I) + I + 3) - (4^(2/3)*(sqrt(-3)*x + x)*(108*sqrt(1/1944*I) + I + 3) - 2*4^(2/3)*(sqrt(-3)*x + x))*
(108*sqrt(-1/1944*I) - I + 3) - 4*4^(2/3)*(sqrt(-3)*x + x))*sqrt(-3/16*(108*sqrt(1/1944*I) + I + 3)^2 - 1/8*(1
08*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(1/1
944*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*sqr
t(1/1944*I) + 3/2*I - 1/2) - 3/2)^(1/3) - 128*(x^3 - x^2)^(1/3))/x) + 1/24*4^(2/3)*(27*sqrt(1/1944*I) + 27*sqr
t(-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)*(sqrt(-3)
- 1)*log(1/16*((2*4^(2/3)*(sqrt(-3)*x - x)*(108*sqrt(1/1944*I) + I + 3)^2 - (4^(2/3)*(sqrt(-3)*x - x)*(108*sqr
t(1/1944*I) + I + 3) - 2*4^(2/3)*(sqrt(-3)*x - x))*(108*sqrt(-1/1944*I) - I + 3)^2 - 20*4^(2/3)*(sqrt(-3)*x -
x)*(108*sqrt(1/1944*I) + I + 3) - (4^(2/3)*(sqrt(-3)*x - x)*(108*sqrt(1/1944*I) + I + 3)^2 - 12*4^(2/3)*(sqrt(
-3)*x - x)*(108*sqrt(1/1944*I) + I + 3) + 20*4^(2/3)*(sqrt(-3)*x - x))*(108*sqrt(-1/1944*I) - I + 3) + 48*4^(2
/3)*(sqrt(-3)*x - x) + 4*(2*4^(2/3)*(sqrt(-3)*x - x)*(108*sqrt(1/1944*I) + I + 3) - (4^(2/3)*(sqrt(-3)*x - x)*
(108*sqrt(1/1944*I) + I + 3) - 2*4^(2/3)*(sqrt(-3)*x - x))*(108*sqrt(-1/1944*I) - I + 3) - 4*4^(2/3)*(sqrt(-3)
*x - x))*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(1/1944*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) + 128*(x^3
- x^2)^(1/3))/x) + 1/12*2^(2/3)*(-1)^(1/6)*(sqrt(-3) + 1)*log(-2*(2^(2/3)*((-1)^(2/3)*(sqrt(-3)*x + x) + (-1)^
(1/6)*(sqrt(-3)*x + x)) - 4*(x^3 - x^2)^(1/3))/x) - 1/12*2^(2/3)*(-1)^(1/6)*(sqrt(-3) + 1)*log(-2*(2^(2/3)*((-
1)^(2/3)*(sqrt(-3)*x + x) - (-1)^(1/6)*(sqrt(-3)*x + x)) - 4*(x^3 - x^2)^(1/3))/x) - 1/12*2^(2/3)*(-1)^(1/6)*(
sqrt(-3) - 1)*log(2*(2^(2/3)*((-1)^(2/3)*(sqrt(-3)*x - x) + (-1)^(1/6)*(sqrt(-3)*x - x)) + 4*(x^3 - x^2)^(1/3)
)/x) + 1/12*2^(2/3)*(-1)^(1/6)*(sqrt(-3) - 1)*log(2*(2^(2/3)*((-1)^(2/3)*(sqrt(-3)*x - x) - (-1)^(1/6)*(sqrt(-
3)*x - x)) + 4*(x^3 - x^2)^(1/3))/x) - 1/2*(-1/2*sqrt(1/1944*I) - 1/216*I - 1/72)^(1/3)*(sqrt(-3) + 1)*log(1/1
6*(3*((sqrt(-3)*x + x)*(108*sqrt(1/1944*I) + I + 3)^3 - 10*(sqrt(-3)*x + x)*(108*sqrt(1/1944*I) + I + 3)^2 + 3
6*(sqrt(-3)*x + x)*(108*sqrt(1/1944*I) + I + 3) - 32*sqrt(-3)*x - 32*x)*(-1/2*sqrt(1/1944*I) - 1/216*I - 1/72)
^(1/3) + 16*(x^3 - x^2)^(1/3))/x) + 1/2*(-1/2*sqrt(1/1944*I) - 1/216*I - 1/72)^(1/3)*(sqrt(-3) - 1)*log(-1/16*
(3*((sqrt(-3)*x - x)*(108*sqrt(1/1944*I) + I + 3)^3 - 10*(sqrt(-3)*x - x)*(108*sqrt(1/1944*I) + I + 3)^2 + 36*
(sqrt(-3)*x - x)*(108*sqrt(1/1944*I) + I + 3) - 32*sqrt(-3)*x + 32*x)*(-1/2*sqrt(1/1944*I) - 1/216*I - 1/72)^(
1/3) - 16*(x^3 - x^2)^(1/3))/x) - 1/2*(-1/2*sqrt(-1/1944*I) + 1/216*I - 1/72)^(1/3)*(sqrt(-3) + 1)*log(-1/16*(
3*((sqrt(-3)*x + x)*(108*sqrt(1/1944*I) + I + 3)^3 - 12*(sqrt(-3)*x + x)*(108*sqrt(1/1944*I) + I + 3)^2 + ((sq
rt(-3)*x + x)*(108*sqrt(1/1944*I) + I + 3) - 2*sqrt(-3)*x - 2*x)*(108*sqrt(-1/1944*I) - I + 3)^2 + 56*(sqrt(-3
)*x + x)*(108*sqrt(1/1944*I) + I + 3) + ((sqrt(-3)*x + x)*(108*sqrt(1/1944*I) + I + 3)^2 - 12*(sqrt(-3)*x + x)
*(108*sqrt(1/1944*I) + I + 3) + 20*sqrt(-3)*x + 20*x)*(108*sqrt(-1/1944*I) - I + 3) - 64*sqrt(-3)*x - 64*x)*(-
1/2*sqrt(-1/1944*I) + 1/216*I - 1/72)^(1/3) - 16*(x^3 - x^2)^(1/3))/x) + 1/2*(-1/2*sqrt(-1/1944*I) + 1/216*I -
1/72)^(1/3)*(sqrt(-3) - 1)*log(1/16*(3*((sqrt(-3)*x - x)*(108*sqrt(1/1944*I) + I + 3)^3 - 12*(sqrt(-3)*x - x)
*(108*sqrt(1/1944*I) + I + 3)^2 + ((sqrt(-3)*x - x)*(108*sqrt(1/1944*I) + I + 3) - 2*sqrt(-3)*x + 2*x)*(108*sq
rt(-1/1944*I) - I + 3)^2 + 56*(sqrt(-3)*x - x)*(108*sqrt(1/1944*I) + I + 3) + ((sqrt(-3)*x - x)*(108*sqrt(1/19
44*I) + I + 3)^2 - 12*(sqrt(-3)*x - x)*(108*sqrt(1/1944*I) + I + 3) + 20*sqrt(-3)*x - 20*x)*(108*sqrt(-1/1944*
I) - I + 3) - 64*sqrt(-3)*x + 64*x)*(-1/2*sqrt(-1/1944*I) + 1/216*I - 1/72)^(1/3) + 16*(x^3 - x^2)^(1/3))/x) +
1/12*4^(2/3)*(27*sqrt(1/1944*I) + 27*sqrt(-1/1944*I) + sqrt(-3/16*(108*sqrt(1/1944*I) + I + 3)^2 - 1/8*(108*s
qrt(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)*log(1/16*((2*4^(2/3)*x*(108*sqrt(1/1944*I) + I + 3)^2 - (4^(2/3)*x*(108*sqrt(1/
1944*I) + I + 3) - 2*4^(2/3)*x)*(108*sqrt(-1/1944*I) - I + 3)^2 - 20*4^(2/3)*x*(108*sqrt(1/1944*I) + I + 3) -
(4^(2/3)*x*(108*sqrt(1/1944*I) + I + 3)^2 - 12*4^(2/3)*x*(108*sqrt(1/1944*I) + I + 3) + 20*4^(2/3)*x)*(108*sqr
t(-1/1944*I) - I + 3) + 48*4^(2/3)*x - 4*(2*4^(2/3)*x*(108*sqrt(1/1944*I) + I + 3) - (4^(2/3)*x*(108*sqrt(1/19
44*I) + I + 3) - 2*4^(2/3)*x)*(108*sqrt(-1/1944*I) - I + 3) - 4*4^(2/3)*x)*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(1/1944*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) + 64*(x^3 - x^2)^(1/3))/x) + 1/12*4^(2/3)*(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)^(1
/3)*log(1/16*((2*4^(2/3)*x*(108*sqrt(1/1944*I) + I + 3)^2 - (4^(2/3)*x*(108*sqrt(1/1944*I) + I + 3) - 2*4^(2/3
)*x)*(108*sqrt(-1/1944*I) - I + 3)^2 - 20*4^(2/3)*x*(108*sqrt(1/1944*I) + I + 3) - (4^(2/3)*x*(108*sqrt(1/1944
*I) + I + 3)^2 - 12*4^(2/3)*x*(108*sqrt(1/1944*I) + I + 3) + 20*4^(2/3)*x)*(108*sqrt(-1/1944*I) - I + 3) + 48*
4^(2/3)*x + 4*(2*4^(2/3)*x*(108*sqrt(1/1944*I) + I + 3) - (4^(2/3)*x*(108*sqrt(1/1944*I) + I + 3) - 2*4^(2/3)*
x)*(108*sqrt(-1/1944*I) - I + 3) - 4*4^(2/3)*x)*sqrt(-3/16*(108*sqrt(1/1944*I) + I + 3)^2 - 1/8*(108*sqrt(1/19
44*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(1/1944*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) + 64*(x^3 - x^2)^(1/3))/x) - 1/6*2^(2/3)*(-1)^(1/6)*log(2*(2^(2/3)*((-1)^(2/3)*x +
(-1)^(1/6)*x) + 2*(x^3 - x^2)^(1/3))/x) + 1/6*2^(2/3)*(-1)^(1/6)*log(2*(2^(2/3)*((-1)^(2/3)*x - (-1)^(1/6)*x)
+ 2*(x^3 - x^2)^(1/3))/x) + (-1/2*sqrt(1/1944*I) - 1/216*I - 1/72)^(1/3)*log(-1/8*(3*(x*(108*sqrt(1/1944*I) +
I + 3)^3 - 10*x*(108*sqrt(1/1944*I) + I + 3)^2 + 36*x*(108*sqrt(1/1944*I) + I + 3) - 32*x)*(-1/2*sqrt(1/1944*
I) - 1/216*I - 1/72)^(1/3) - 8*(x^3 - x^2)^(1/3))/x) + (-1/2*sqrt(-1/1944*I) + 1/216*I - 1/72)^(1/3)*log(1/8*(
3*(x*(108*sqrt(1/1944*I) + I + 3)^3 - 12*x*(108*sqrt(1/1944*I) + I + 3)^2 + (x*(108*sqrt(1/1944*I) + I + 3) -
2*x)*(108*sqrt(-1/1944*I) - I + 3)^2 + 56*x*(108*sqrt(1/1944*I) + I + 3) + (x*(108*sqrt(1/1944*I) + I + 3)^2 -
12*x*(108*sqrt(1/1944*I) + I + 3) + 20*x)*(108*sqrt(-1/1944*I) - I + 3) - 64*x)*(-1/2*sqrt(-1/1944*I) + 1/216
*I - 1/72)^(1/3) + 8*(x^3 - x^2)^(1/3))/x)

Sympy [N/A]

Not integrable

Time = 1.45 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.14 \[ \int \frac {\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\int \frac {\sqrt [3]{x^{2} \left (x - 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}{\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \]

[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)

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.11 \[ \int \frac {\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\int { \frac {{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + 1\right )}}{x^{6} + 1} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.67 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.11 \[ \int \frac {\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\int { \frac {{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + 1\right )}}{x^{6} + 1} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 6.67 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.11 \[ \int \frac {\left (1+x^3\right ) \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\int \frac {\left (x^3+1\right )\,{\left (x^3-x^2\right )}^{1/3}}{x^6+1} \,d x \]

[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)