\(\int \frac {x^3 \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx\) [2519]

Optimal result

Integrand size = 24, antiderivative size = 210 \[ \int \frac {x^3 \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\frac {1}{6} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ]-\frac {1}{6} \text {RootSum}\left [1-2 \text {$\#$1}^3+5 \text {$\#$1}^6-4 \text {$\#$1}^9+\text {$\#$1}^{12}\&,\frac {\log (x) \text {$\#$1}-\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}+2 \log (x) \text {$\#$1}^4-2 \log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^4-\log (x) \text {$\#$1}^7+\log \left (\sqrt [3]{-x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^7}{-1+5 \text {$\#$1}^3-6 \text {$\#$1}^6+2 \text {$\#$1}^9}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 5.92 (sec) , antiderivative size = 2680, normalized size of antiderivative = 12.76, number of steps used = 66, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2081, 6857, 103, 161, 93, 6820, 12, 1600, 1637, 965, 81, 61} \[ \int \frac {x^3 \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=-\frac {1}{216} \sqrt [6]{-1} \left ((1+6 i)+6 \sqrt {3}\right ) \sqrt [3]{x^3-x^2} (1-x)^2+\frac {1}{216} (-1)^{5/6} \left ((1-6 i)+6 \sqrt {3}\right ) \sqrt [3]{x^3-x^2} (1-x)^2+\frac {1}{216} \sqrt [6]{-1} \left ((1-6 i)-6 \sqrt {3}\right ) \sqrt [3]{x^3-x^2} (1-x)^2+\frac {1}{216} (-1)^{2/3} \left (12-\sqrt [6]{-1}\right ) \sqrt [3]{x^3-x^2} (1-x)^2+\frac {1}{9} \sqrt [3]{x^3-x^2} (1-x)^2+\frac {1}{324} (-1)^{2/3} \left ((-33-16 i)+11 \sqrt {3}\right ) \sqrt [3]{x^3-x^2} (1-x)-\left (\frac {1}{648}+\frac {i}{648}\right ) (-1)^{2/3} \left ((-8+25 i)+(3+30 i) \sqrt {3}\right ) \sqrt [3]{x^3-x^2} (1-x)-\frac {1}{324} (-1)^{5/6} \left ((16+33 i)-11 i \sqrt {3}\right ) \sqrt [3]{x^3-x^2} (1-x)+\frac {1}{324} (-1)^{5/6} \left (5+27 (-1)^{2/3}+33 (-1)^{5/6}\right ) \sqrt [3]{x^3-x^2} (1-x)-\frac {11}{54} \sqrt [3]{x^3-x^2} (1-x)-\frac {i \left ((288+759 i)+(179+66 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 )}{2916 \sqrt [3]{x-1} x^{2/3}}+\frac {\left ((759+288 i)+(66+179 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 )}{2916 \sqrt [3]{x-1} x^{2/3}}+\frac {\left ((-759-288 i)+(66+179 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 )}{2916 \sqrt [3]{x-1} x^{2/3}}+\frac {i \left ((288+759 i)-(179+66 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 )}{2916 \sqrt [3]{x-1} x^{2/3}}-\frac {22 \sqrt [3]{x^3-x^2} \arctan \left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-1}}+\frac {1}{\sqrt {3}}\right )}{81 \sqrt {3} \sqrt [3]{x-1} x^{2/3}}+\frac {\sqrt [6]{-1} \sqrt [3]{-1+i} \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 )}{2 \sqrt {3} \sqrt [3]{x-1} x^{2/3}}-\frac {i \sqrt [3]{1+i} \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 )}{2 \sqrt {3} \sqrt [3]{x-1} x^{2/3}}-\frac {(-1)^{5/6} \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 )}{2 \sqrt {3} \sqrt [3]{x-1} x^{2/3}}-\frac {\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 )}{2 \sqrt {3} \sqrt [3]{x-1} x^{2/3}}+\frac {i \sqrt [3]{\frac {1}{2} \left ((-2+i)-\sqrt {3}\right )} \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 )}{2 \sqrt {3} \sqrt [3]{x-1} x^{2/3}}+\frac {\left (\frac {1}{12}+\frac {i}{12}\right ) \left (3-\sqrt {3}\right ) \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 )}{\left (1+(-1)^{5/6}\right )^{2/3} \sqrt [3]{x-1} x^{2/3}}+\frac {\sqrt [6]{-1} \sqrt [3]{-1+i} \sqrt [3]{x^3-x^2} \log \left (\sqrt [3]{1-i} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{4 \sqrt [3]{x-1} x^{2/3}}-\frac {i \sqrt [3]{1+i} \sqrt [3]{x^3-x^2} \log \left (\sqrt [3]{1+i} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{4 \sqrt [3]{x-1} x^{2/3}}-\frac {(-1)^{5/6} \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 )}{4 \sqrt [3]{x-1} x^{2/3}}-\frac {\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 )}{4 \sqrt [3]{x-1} x^{2/3}}-\frac {\sqrt [3]{-1} \left ((2+i)+\sqrt {3}\right ) \sqrt [3]{x^3-x^2} \log \left (\sqrt [3]{1-(-1)^{5/6}} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{8 \left (-1+(-1)^{5/6}\right )^{2/3} \sqrt [3]{x-1} x^{2/3}}-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (1-\sqrt {3}\right ) \sqrt [3]{x^3-x^2} \log \left (\sqrt [3]{1+(-1)^{5/6}} \sqrt [3]{x}-\sqrt [3]{x-1}\right )}{\left (1+(-1)^{5/6}\right )^{2/3} \sqrt [3]{x-1} x^{2/3}}+\frac {\left ((66+179 i)+(253+96 i) \sqrt {3}\right ) \sqrt [3]{x^3-x^2} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )}{1944 \sqrt [3]{x-1} x^{2/3}}-\frac {i \left ((179+66 i)+(96+253 i) \sqrt {3}\right ) \sqrt [3]{x^3-x^2} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )}{1944 \sqrt [3]{x-1} x^{2/3}}+\frac {i \left ((-179-66 i)+(96+253 i) \sqrt {3}\right ) \sqrt [3]{x^3-x^2} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )}{1944 \sqrt [3]{x-1} x^{2/3}}+\frac {\left ((66+179 i)-(253+96 i) \sqrt {3}\right ) \sqrt [3]{x^3-x^2} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )}{1944 \sqrt [3]{x-1} x^{2/3}}-\frac {11 \sqrt [3]{x^3-x^2} \log \left (\frac {\sqrt [3]{x}}{\sqrt [3]{x-1}}-1\right )}{81 \sqrt [3]{x-1} x^{2/3}}+\frac {\left (\frac {1}{24}+\frac {i}{24}\right ) \left (1-\sqrt {3}\right ) \sqrt [3]{x^3-x^2} \log \left (\sqrt [6]{-1}-x\right )}{\left (1+(-1)^{5/6}\right )^{2/3} \sqrt [3]{x-1} x^{2/3}}+\frac {(-1)^{5/6} \sqrt [3]{-1+\sqrt [6]{-1}} \sqrt [3]{x^3-x^2} \log \left (-x-(-1)^{5/6}\right )}{12 \sqrt [3]{x-1} x^{2/3}}+\frac {\left ((66+179 i)+(253+96 i) \sqrt {3}\right ) \sqrt [3]{x^3-x^2} \log (x-1)}{5832 \sqrt [3]{x-1} x^{2/3}}-\frac {i \left ((179+66 i)+(96+253 i) \sqrt {3}\right ) \sqrt [3]{x^3-x^2} \log (x-1)}{5832 \sqrt [3]{x-1} x^{2/3}}+\frac {i \left ((-179-66 i)+(96+253 i) \sqrt {3}\right ) \sqrt [3]{x^3-x^2} \log (x-1)}{5832 \sqrt [3]{x-1} x^{2/3}}+\frac {\left ((66+179 i)-(253+96 i) \sqrt {3}\right ) \sqrt [3]{x^3-x^2} \log (x-1)}{5832 \sqrt [3]{x-1} x^{2/3}}-\frac {11 \sqrt [3]{x^3-x^2} \log (x-1)}{243 \sqrt [3]{x-1} x^{2/3}}+\frac {\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 )}{12 \sqrt [3]{x-1} x^{2/3}}+\frac {i \sqrt [3]{1+i} \sqrt [3]{x^3-x^2} \log \left (\sqrt [3]{-1} x-(-1)^{5/6}\right )}{12 \sqrt [3]{x-1} x^{2/3}}-\frac {\sqrt [6]{-1} \sqrt [3]{-1+i} \sqrt [3]{x^3-x^2} \log \left (\sqrt [6]{-1}-(-1)^{2/3} x\right )}{12 \sqrt [3]{x-1} x^{2/3}}+\frac {\sqrt [3]{-1} \left ((2+i)+\sqrt {3}\right ) \sqrt [3]{x^3-x^2} \log \left (-(-1)^{2/3} x-(-1)^{5/6}\right )}{24 \left (-1+(-1)^{5/6}\right )^{2/3} \sqrt [3]{x-1} x^{2/3}}+\frac {\sqrt [3]{-1} \left ((354+13 i)+(54+30 i) \sqrt {3}\right ) \sqrt [3]{x^3-x^2}}{1944}+\frac {(-1)^{2/3} \left ((-354-13 i)+(54+30 i) \sqrt {3}\right ) \sqrt [3]{x^3-x^2}}{1944}-\frac {\sqrt [6]{-1} \left ((13+354 i)+(30+54 i) \sqrt {3}\right ) \sqrt [3]{x^3-x^2}}{1944}+\frac {(-1)^{5/6} \left ((-13-354 i)+(30+54 i) \sqrt {3}\right ) \sqrt [3]{x^3-x^2}}{1944}-\frac {22}{81} \sqrt [3]{x^3-x^2} \]

[In]

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

[Out]

(-22*(-x^2 + x^3)^(1/3))/81 + ((-1)^(5/6)*((-13 - 354*I) + (30 + 54*I)*Sqrt[3])*(-x^2 + x^3)^(1/3))/1944 - ((-
1)^(1/6)*((13 + 354*I) + (30 + 54*I)*Sqrt[3])*(-x^2 + x^3)^(1/3))/1944 + ((-1)^(2/3)*((-354 - 13*I) + (54 + 30
*I)*Sqrt[3])*(-x^2 + x^3)^(1/3))/1944 + ((-1)^(1/3)*((354 + 13*I) + (54 + 30*I)*Sqrt[3])*(-x^2 + x^3)^(1/3))/1
944 - (11*(1 - x)*(-x^2 + x^3)^(1/3))/54 + ((-1)^(5/6)*(5 + 27*(-1)^(2/3) + 33*(-1)^(5/6))*(1 - x)*(-x^2 + x^3
)^(1/3))/324 - ((-1)^(5/6)*((16 + 33*I) - (11*I)*Sqrt[3])*(1 - x)*(-x^2 + x^3)^(1/3))/324 - (1/648 + I/648)*(-
1)^(2/3)*((-8 + 25*I) + (3 + 30*I)*Sqrt[3])*(1 - x)*(-x^2 + x^3)^(1/3) + ((-1)^(2/3)*((-33 - 16*I) + 11*Sqrt[3
])*(1 - x)*(-x^2 + x^3)^(1/3))/324 + ((1 - x)^2*(-x^2 + x^3)^(1/3))/9 + ((-1)^(2/3)*(12 - (-1)^(1/6))*(1 - x)^
2*(-x^2 + x^3)^(1/3))/216 + ((-1)^(1/6)*((1 - 6*I) - 6*Sqrt[3])*(1 - x)^2*(-x^2 + x^3)^(1/3))/216 + ((-1)^(5/6
)*((1 - 6*I) + 6*Sqrt[3])*(1 - x)^2*(-x^2 + x^3)^(1/3))/216 - ((-1)^(1/6)*((1 + 6*I) + 6*Sqrt[3])*(1 - x)^2*(-
x^2 + x^3)^(1/3))/216 - (22*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/(81*S
qrt[3]*(-1 + x)^(1/3)*x^(2/3)) + ((I/2916)*((288 + 759*I) - (179 + 66*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)) + (((-759 - 288*I) + (66 + 179*I)*Sq
rt[3])*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/(2916*(-1 + x)^(1/3)*x^(2/
3)) + (((759 + 288*I) + (66 + 179*I)*Sqrt[3])*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*x^(1/3))/(Sqrt[3]*(-1 +
x)^(1/3))])/(2916*(-1 + x)^(1/3)*x^(2/3)) - ((I/2916)*((288 + 759*I) + (179 + 66*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)) + ((-1)^(1/6)*(-1 + I)^(
1/3)*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*(1 - I)^(1/3)*x^(1/3))/(Sqrt[3]*(-1 + x)^(1/3))])/(2*Sqrt[3]*(-1
+ x)^(1/3)*x^(2/3)) - ((I/2)*(1 + I)^(1/3)*(-x^2 + x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*(1 + I)^(1/3)*x^(1/3))/(S
qrt[3]*(-1 + x)^(1/3))])/(Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)) - ((-1)^(5/6)*(-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))])/(2*Sqrt[3]*(-1 + x)^(1/3
)*x^(2/3)) - ((-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))])/(2*Sqrt[3]*(-1 + x)^(1/3)*x^(2/3)) + ((I/2)*(((-2 + I) - Sqrt[3])/2)^(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)) + ((1/12 + I/12)*(3 - Sqrt[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))])/((1 + (-1)^(5/6))^(2/3)*(-1 + x)^(1/3)*x^(2/3)) + ((-1)^(1/6)*(
-1 + I)^(1/3)*(-x^2 + x^3)^(1/3)*Log[-(-1 + x)^(1/3) + (1 - I)^(1/3)*x^(1/3)])/(4*(-1 + x)^(1/3)*x^(2/3)) - ((
I/4)*(1 + I)^(1/3)*(-x^2 + x^3)^(1/3)*Log[-(-1 + x)^(1/3) + (1 + I)^(1/3)*x^(1/3)])/((-1 + x)^(1/3)*x^(2/3)) -
((-1)^(5/6)*(-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)])
/(4*(-1 + x)^(1/3)*x^(2/3)) - ((-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)])/(4*(-1 + x)^(1/3)*x^(2/3)) - ((-1)^(1/3)*((2 + I) + Sqrt[3])*(-x^2 + x^3)^(1/3)*L
og[-(-1 + x)^(1/3) + (1 - (-1)^(5/6))^(1/3)*x^(1/3)])/(8*(-1 + (-1)^(5/6))^(2/3)*(-1 + x)^(1/3)*x^(2/3)) - ((1
/8 + I/8)*(1 - Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-(-1 + x)^(1/3) + (1 + (-1)^(5/6))^(1/3)*x^(1/3)])/((1 + (-1)^(
5/6))^(2/3)*(-1 + x)^(1/3)*x^(2/3)) - (11*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1/3)/(-1 + x)^(1/3)])/(81*(-1 + x)^(1
/3)*x^(2/3)) + (((66 + 179*I) - (253 + 96*I)*Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1/3)/(-1 + x)^(1/3)])/(19
44*(-1 + x)^(1/3)*x^(2/3)) + ((I/1944)*((-179 - 66*I) + (96 + 253*I)*Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-1 + x^(1
/3)/(-1 + x)^(1/3)])/((-1 + x)^(1/3)*x^(2/3)) - ((I/1944)*((179 + 66*I) + (96 + 253*I)*Sqrt[3])*(-x^2 + x^3)^(
1/3)*Log[-1 + x^(1/3)/(-1 + x)^(1/3)])/((-1 + x)^(1/3)*x^(2/3)) + (((66 + 179*I) + (253 + 96*I)*Sqrt[3])*(-x^2
+ x^3)^(1/3)*Log[-1 + x^(1/3)/(-1 + x)^(1/3)])/(1944*(-1 + x)^(1/3)*x^(2/3)) + ((1/24 + I/24)*(1 - Sqrt[3])*(
-x^2 + x^3)^(1/3)*Log[(-1)^(1/6) - x])/((1 + (-1)^(5/6))^(2/3)*(-1 + x)^(1/3)*x^(2/3)) + ((-1)^(5/6)*(-1 + (-1
)^(1/6))^(1/3)*(-x^2 + x^3)^(1/3)*Log[-(-1)^(5/6) - x])/(12*(-1 + x)^(1/3)*x^(2/3)) - (11*(-x^2 + x^3)^(1/3)*L
og[-1 + x])/(243*(-1 + x)^(1/3)*x^(2/3)) + (((66 + 179*I) - (253 + 96*I)*Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-1 +
x])/(5832*(-1 + x)^(1/3)*x^(2/3)) + ((I/5832)*((-179 - 66*I) + (96 + 253*I)*Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-1
+ x])/((-1 + x)^(1/3)*x^(2/3)) - ((I/5832)*((179 + 66*I) + (96 + 253*I)*Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-1 +
x])/((-1 + x)^(1/3)*x^(2/3)) + (((66 + 179*I) + (253 + 96*I)*Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-1 + x])/(5832*(-
1 + x)^(1/3)*x^(2/3)) + ((-1)^(1/6)*(1 + (-1)^(1/6))^(1/3)*(-x^2 + x^3)^(1/3)*Log[(-1)^(1/6) + (-1)^(1/3)*x])/
(12*(-1 + x)^(1/3)*x^(2/3)) + ((I/12)*(1 + I)^(1/3)*(-x^2 + x^3)^(1/3)*Log[-(-1)^(5/6) + (-1)^(1/3)*x])/((-1 +
x)^(1/3)*x^(2/3)) - ((-1)^(1/6)*(-1 + I)^(1/3)*(-x^2 + x^3)^(1/3)*Log[(-1)^(1/6) - (-1)^(2/3)*x])/(12*(-1 + x
)^(1/3)*x^(2/3)) + ((-1)^(1/3)*((2 + I) + Sqrt[3])*(-x^2 + x^3)^(1/3)*Log[-(-1)^(5/6) - (-1)^(2/3)*x])/(24*(-1
+ (-1)^(5/6))^(2/3)*(-1 + x)^(1/3)*x^(2/3))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 81

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

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 161

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((g_.) + (h_.)*(x_)))/((e_.) + (f_.)*(x_)), x_Symbol]
:> Dist[(f*g - e*h)*((c*f - d*e)^(m + n + 1)/f^(m + n + 2)), Int[(a + b*x)^m/((c + d*x)^(m + 1)*(e + f*x)), x
], x] + Dist[1/f^(m + n + 2), Int[((a + b*x)^m/(c + d*x)^(m + 1))*ExpandToSum[(f^(m + n + 2)*(c + d*x)^(m + n
+ 1)*(g + h*x) - (f*g - e*h)*(c*f - d*e)^(m + n + 1))/(e + f*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h},
x] && IGtQ[m + n + 1, 0] && (LtQ[m, 0] || SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 965

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

Rule 1600

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 1637

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coef
f[Px, x, Expon[Px, x]]}, Simp[k*(a + b*x)^(m + q)*((c + d*x)^(n + 1)/(d*b^q*(m + n + q + 1))), x] + Dist[1/(d*
b^q*(m + n + q + 1)), Int[(a + b*x)^m*(c + d*x)^n*ExpandToSum[d*b^q*(m + n + q + 1)*Px - d*k*(m + n + q + 1)*(
a + b*x)^q - k*(b*c - a*d)*(m + q)*(a + b*x)^(q - 1), x], x], x] /; NeQ[m + n + q + 1, 0]] /; FreeQ[{a, b, c,
d, m, n}, x] && PolyQ[Px, x] && GtQ[Expon[Px, x], 2]

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 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

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^{11/3}}{1+x^6} \, dx}{\sqrt [3]{-1+x} x^{2/3}} \\ & = \frac {\sqrt [3]{-x^2+x^3} \int \left (\frac {i \sqrt [3]{-1+x} x^{11/3}}{2 \left (i-x^3\right )}+\frac {i \sqrt [3]{-1+x} x^{11/3}}{2 \left (i+x^3\right )}\right ) \, dx}{\sqrt [3]{-1+x} x^{2/3}} \\ & = \frac {\left (i \sqrt [3]{-x^2+x^3}\right ) \int \frac {\sqrt [3]{-1+x} x^{11/3}}{i-x^3} \, dx}{2 \sqrt [3]{-1+x} x^{2/3}}+\frac {\left (i \sqrt [3]{-x^2+x^3}\right ) \int \frac {\sqrt [3]{-1+x} x^{11/3}}{i+x^3} \, dx}{2 \sqrt [3]{-1+x} x^{2/3}} \\ & = \frac {\left (i \sqrt [3]{-x^2+x^3}\right ) \int \left (-\frac {(-1)^{2/3} \sqrt [3]{-1+x} x^{11/3}}{3 \left (\sqrt [6]{-1}-x\right )}-\frac {(-1)^{2/3} \sqrt [3]{-1+x} x^{11/3}}{3 \left (\sqrt [6]{-1}+\sqrt [3]{-1} x\right )}-\frac {(-1)^{2/3} \sqrt [3]{-1+x} x^{11/3}}{3 \left (\sqrt [6]{-1}-(-1)^{2/3} x\right )}\right ) \, dx}{2 \sqrt [3]{-1+x} x^{2/3}}+\frac {\left (i \sqrt [3]{-x^2+x^3}\right ) \int \left (-\frac {\sqrt [3]{-1} \sqrt [3]{-1+x} x^{11/3}}{3 \left (-(-1)^{5/6}-x\right )}-\frac {\sqrt [3]{-1} \sqrt [3]{-1+x} x^{11/3}}{3 \left (-(-1)^{5/6}+\sqrt [3]{-1} x\right )}-\frac {\sqrt [3]{-1} \sqrt [3]{-1+x} x^{11/3}}{3 \left (-(-1)^{5/6}-(-1)^{2/3} x\right )}\right ) \, dx}{2 \sqrt [3]{-1+x} x^{2/3}} \\ & = \frac {\left (\sqrt [6]{-1} \sqrt [3]{-x^2+x^3}\right ) \int \frac {\sqrt [3]{-1+x} x^{11/3}}{\sqrt [6]{-1}-x} \, dx}{6 \sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\sqrt [6]{-1} \sqrt [3]{-x^2+x^3}\right ) \int \frac {\sqrt [3]{-1+x} x^{11/3}}{\sqrt [6]{-1}+\sqrt [3]{-1} x} \, dx}{6 \sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\sqrt [6]{-1} \sqrt [3]{-x^2+x^3}\right ) \int \frac {\sqrt [3]{-1+x} x^{11/3}}{\sqrt [6]{-1}-(-1)^{2/3} x} \, dx}{6 \sqrt [3]{-1+x} x^{2/3}}-\frac {\left ((-1)^{5/6} \sqrt [3]{-x^2+x^3}\right ) \int \frac {\sqrt [3]{-1+x} x^{11/3}}{-(-1)^{5/6}-x} \, dx}{6 \sqrt [3]{-1+x} x^{2/3}}-\frac {\left ((-1)^{5/6} \sqrt [3]{-x^2+x^3}\right ) \int \frac {\sqrt [3]{-1+x} x^{11/3}}{-(-1)^{5/6}+\sqrt [3]{-1} x} \, dx}{6 \sqrt [3]{-1+x} x^{2/3}}-\frac {\left ((-1)^{5/6} \sqrt [3]{-x^2+x^3}\right ) \int \frac {\sqrt [3]{-1+x} x^{11/3}}{-(-1)^{5/6}-(-1)^{2/3} x} \, dx}{6 \sqrt [3]{-1+x} x^{2/3}} \\ & = \frac {\left (i \sqrt [3]{-x^2+x^3}\right ) \int \frac {x^{8/3} \left (\frac {11}{3} (-1)^{5/6}+\left (\frac {1}{3}-4 i\right ) \sqrt [3]{-1} x\right )}{(-1+x)^{2/3} \left (-(-1)^{5/6}+\sqrt [3]{-1} x\right )} \, dx}{24 \sqrt [3]{-1+x} x^{2/3}}-\frac {\left (i \sqrt [3]{-x^2+x^3}\right ) \int \frac {x^{8/3} \left (-\frac {11 \sqrt [6]{-1}}{3}+\left (4-\frac {i}{3}\right ) \sqrt [6]{-1} x\right )}{(-1+x)^{2/3} \left (\sqrt [6]{-1}-(-1)^{2/3} x\right )} \, dx}{24 \sqrt [3]{-1+x} x^{2/3}}+\frac {\left (\sqrt [6]{-1} \sqrt [3]{-x^2+x^3}\right ) \int \frac {x^{8/3} \left (-\frac {11 \sqrt [6]{-1}}{3}+\frac {1}{3} \left ((-1+6 i)+6 \sqrt {3}\right ) x\right )}{\left (\sqrt [6]{-1}-x\right ) (-1+x)^{2/3}} \, dx}{24 \sqrt [3]{-1+x} x^{2/3}}-\frac {\left (\sqrt [6]{-1} \sqrt [3]{-x^2+x^3}\right ) \int \frac {x^{8/3} \left (\frac {11}{3} (-1)^{5/6}-\frac {1}{3} (-1)^{2/3} \left ((1+6 i)+6 \sqrt {3}\right ) x\right )}{(-1+x)^{2/3} \left (-(-1)^{5/6}-(-1)^{2/3} x\right )} \, dx}{24 \sqrt [3]{-1+x} x^{2/3}}-\frac {\left ((-1)^{5/6} \sqrt [3]{-x^2+x^3}\right ) \int \frac {x^{8/3} \left (\frac {11}{3} (-1)^{5/6}-\frac {1}{3} (-1)^{5/6} \left (12-\sqrt [6]{-1}\right ) x\right )}{\left (-(-1)^{5/6}-x\right ) (-1+x)^{2/3}} \, dx}{24 \sqrt [3]{-1+x} x^{2/3}}+\frac {\left ((-1)^{5/6} \sqrt [3]{-x^2+x^3}\right ) \int \frac {x^{8/3} \left (-\frac {11 \sqrt [6]{-1}}{3}+\frac {1}{3} \sqrt [6]{-1} \left (12+\sqrt [6]{-1}\right ) x\right )}{(-1+x)^{2/3} \left (\sqrt [6]{-1}+\sqrt [3]{-1} x\right )} \, dx}{24 \sqrt [3]{-1+x} x^{2/3}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.09 \[ \int \frac {x^3 \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\frac {(-1+x)^{2/3} x^{4/3} \left (3 \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right ) \text {$\#$1}+\log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ]+\text {RootSum}\left [1-2 \text {$\#$1}^3+5 \text {$\#$1}^6-4 \text {$\#$1}^9+\text {$\#$1}^{12}\&,\frac {-\log (x) \text {$\#$1}+3 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}-2 \log (x) \text {$\#$1}^4+6 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^4+\log (x) \text {$\#$1}^7-3 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^7}{-1+5 \text {$\#$1}^3-6 \text {$\#$1}^6+2 \text {$\#$1}^9}\&\right ]\right )}{18 \left ((-1+x) x^2\right )^{2/3}} \]

[In]

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

[Out]

((-1 + x)^(2/3)*x^(4/3)*(3*RootSum[2 - 2*#1^3 + #1^6 & , (-(Log[x^(1/3)]*#1) + Log[(-1 + x)^(1/3) - x^(1/3)*#1
]*#1)/(-1 + #1^3) & ] + RootSum[1 - 2*#1^3 + 5*#1^6 - 4*#1^9 + #1^12 & , (-(Log[x]*#1) + 3*Log[(-1 + x)^(1/3)
- x^(1/3)*#1]*#1 - 2*Log[x]*#1^4 + 6*Log[(-1 + x)^(1/3) - x^(1/3)*#1]*#1^4 + Log[x]*#1^7 - 3*Log[(-1 + x)^(1/3
) - x^(1/3)*#1]*#1^7)/(-1 + 5*#1^3 - 6*#1^6 + 2*#1^9) & ]))/(18*((-1 + x)*x^2)^(2/3))

Maple [N/A] (verified)

Time = 83.20 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.58

method	result	size
pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 \textit {\_Z}^{3}+2\right )}{\sum }\frac {\textit {\_R} \ln \left (\frac {-\textit {\_R} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{3}-1}\right )}{6}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{12}-4 \textit {\_Z}^{9}+5 \textit {\_Z}^{6}-2 \textit {\_Z}^{3}+1\right )}{\sum }\frac {\textit {\_R} \left (\textit {\_R}^{6}-2 \textit {\_R}^{3}-1\right ) \ln \left (\frac {-\textit {\_R} x +\left (\left (-1+x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{2 \textit {\_R}^{9}-6 \textit {\_R}^{6}+5 \textit {\_R}^{3}-1}\right )}{6}\)	\(122\)
trager	\(\text {Expression too large to display}\)	\(78775\)

[In]

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

[Out]

1/6*sum(_R*ln((-_R*x+((-1+x)*x^2)^(1/3))/x)/(_R^3-1),_R=RootOf(_Z^6-2*_Z^3+2))-1/6*sum(_R*(_R^6-2*_R^3-1)*ln((
-_R*x+((-1+x)*x^2)^(1/3))/x)/(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 = 0.30 (sec) , antiderivative size = 1493, normalized size of antiderivative = 7.11 \[ \int \frac {x^3 \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/24*2^(2/3)*(sqrt(-3) + 1)*(sqrt(4*sqrt(3) - 7) + 1)^(1/3)*log(((sqrt(3)*2^(2/3)*(sqrt(-3)*x + x) - (sqrt(3)
*2^(2/3)*(sqrt(-3)*x + x) + 2*2^(2/3)*(sqrt(-3)*x + x))*sqrt(4*sqrt(3) - 7))*(sqrt(4*sqrt(3) - 7) + 1)^(1/3) +
8*(x^3 - x^2)^(1/3))/x) + 1/24*2^(2/3)*(sqrt(-3) - 1)*(sqrt(4*sqrt(3) - 7) + 1)^(1/3)*log(-((sqrt(3)*2^(2/3)*
(sqrt(-3)*x - x) - (sqrt(3)*2^(2/3)*(sqrt(-3)*x - x) + 2*2^(2/3)*(sqrt(-3)*x - x))*sqrt(4*sqrt(3) - 7))*(sqrt(
4*sqrt(3) - 7) + 1)^(1/3) - 8*(x^3 - x^2)^(1/3))/x) - 1/24*2^(2/3)*(sqrt(-3) + 1)*(-sqrt(4*sqrt(3) - 7) + 1)^(
1/3)*log(((sqrt(3)*2^(2/3)*(sqrt(-3)*x + x) + (sqrt(3)*2^(2/3)*(sqrt(-3)*x + x) + 2*2^(2/3)*(sqrt(-3)*x + x))*
sqrt(4*sqrt(3) - 7))*(-sqrt(4*sqrt(3) - 7) + 1)^(1/3) + 8*(x^3 - x^2)^(1/3))/x) + 1/24*2^(2/3)*(sqrt(-3) - 1)*
(-sqrt(4*sqrt(3) - 7) + 1)^(1/3)*log(-((sqrt(3)*2^(2/3)*(sqrt(-3)*x - x) + (sqrt(3)*2^(2/3)*(sqrt(-3)*x - x) +
2*2^(2/3)*(sqrt(-3)*x - x))*sqrt(4*sqrt(3) - 7))*(-sqrt(4*sqrt(3) - 7) + 1)^(1/3) - 8*(x^3 - x^2)^(1/3))/x) -
1/24*2^(2/3)*(sqrt(-3) + 1)*(sqrt(-4*sqrt(3) - 7) + 1)^(1/3)*log(-((sqrt(3)*2^(2/3)*(sqrt(-3)*x + x) - (sqrt(
3)*2^(2/3)*(sqrt(-3)*x + x) - 2*2^(2/3)*(sqrt(-3)*x + x))*sqrt(-4*sqrt(3) - 7))*(sqrt(-4*sqrt(3) - 7) + 1)^(1/
3) - 8*(x^3 - x^2)^(1/3))/x) + 1/24*2^(2/3)*(sqrt(-3) - 1)*(sqrt(-4*sqrt(3) - 7) + 1)^(1/3)*log(((sqrt(3)*2^(2
/3)*(sqrt(-3)*x - x) - (sqrt(3)*2^(2/3)*(sqrt(-3)*x - x) - 2*2^(2/3)*(sqrt(-3)*x - x))*sqrt(-4*sqrt(3) - 7))*(
sqrt(-4*sqrt(3) - 7) + 1)^(1/3) + 8*(x^3 - x^2)^(1/3))/x) - 1/24*2^(2/3)*(sqrt(-3) + 1)*(-sqrt(-4*sqrt(3) - 7)
+ 1)^(1/3)*log(-((sqrt(3)*2^(2/3)*(sqrt(-3)*x + x) + (sqrt(3)*2^(2/3)*(sqrt(-3)*x + x) - 2*2^(2/3)*(sqrt(-3)*
x + x))*sqrt(-4*sqrt(3) - 7))*(-sqrt(-4*sqrt(3) - 7) + 1)^(1/3) - 8*(x^3 - x^2)^(1/3))/x) + 1/24*2^(2/3)*(sqrt
(-3) - 1)*(-sqrt(-4*sqrt(3) - 7) + 1)^(1/3)*log(((sqrt(3)*2^(2/3)*(sqrt(-3)*x - x) + (sqrt(3)*2^(2/3)*(sqrt(-3
)*x - x) - 2*2^(2/3)*(sqrt(-3)*x - x))*sqrt(-4*sqrt(3) - 7))*(-sqrt(-4*sqrt(3) - 7) + 1)^(1/3) + 8*(x^3 - x^2)
^(1/3))/x) - 1/12*(I - 1)^(1/3)*(sqrt(-3) + 1)*log(((I - 1)^(1/3)*(I*sqrt(-3)*x + I*x) + 2*(x^3 - x^2)^(1/3))/
x) + 1/12*(-I - 1)^(1/3)*(sqrt(-3) - 1)*log(((-I - 1)^(1/3)*(I*sqrt(-3)*x - I*x) + 2*(x^3 - x^2)^(1/3))/x) + 1
/12*(I - 1)^(1/3)*(sqrt(-3) - 1)*log(((I - 1)^(1/3)*(-I*sqrt(-3)*x + I*x) + 2*(x^3 - x^2)^(1/3))/x) - 1/12*(-I
- 1)^(1/3)*(sqrt(-3) + 1)*log(((-I - 1)^(1/3)*(-I*sqrt(-3)*x - I*x) + 2*(x^3 - x^2)^(1/3))/x) + 1/12*2^(2/3)*
(sqrt(4*sqrt(3) - 7) + 1)^(1/3)*log(-((sqrt(3)*2^(2/3)*x - (sqrt(3)*2^(2/3)*x + 2*2^(2/3)*x)*sqrt(4*sqrt(3) -
7))*(sqrt(4*sqrt(3) - 7) + 1)^(1/3) - 4*(x^3 - x^2)^(1/3))/x) + 1/12*2^(2/3)*(-sqrt(4*sqrt(3) - 7) + 1)^(1/3)*
log(-((sqrt(3)*2^(2/3)*x + (sqrt(3)*2^(2/3)*x + 2*2^(2/3)*x)*sqrt(4*sqrt(3) - 7))*(-sqrt(4*sqrt(3) - 7) + 1)^(
1/3) - 4*(x^3 - x^2)^(1/3))/x) + 1/12*2^(2/3)*(sqrt(-4*sqrt(3) - 7) + 1)^(1/3)*log(((sqrt(3)*2^(2/3)*x - (sqrt
(3)*2^(2/3)*x - 2*2^(2/3)*x)*sqrt(-4*sqrt(3) - 7))*(sqrt(-4*sqrt(3) - 7) + 1)^(1/3) + 4*(x^3 - x^2)^(1/3))/x)
+ 1/12*2^(2/3)*(-sqrt(-4*sqrt(3) - 7) + 1)^(1/3)*log(((sqrt(3)*2^(2/3)*x + (sqrt(3)*2^(2/3)*x - 2*2^(2/3)*x)*s
qrt(-4*sqrt(3) - 7))*(-sqrt(-4*sqrt(3) - 7) + 1)^(1/3) + 4*(x^3 - x^2)^(1/3))/x) + 1/6*(I - 1)^(1/3)*log((-I*(
I - 1)^(1/3)*x + (x^3 - x^2)^(1/3))/x) + 1/6*(-I - 1)^(1/3)*log((I*(-I - 1)^(1/3)*x + (x^3 - x^2)^(1/3))/x)

Sympy [N/A]

Not integrable

Time = 0.58 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.13 \[ \int \frac {x^3 \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\int \frac {x^{3} \sqrt [3]{x^{2} \left (x - 1\right )}}{\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.11 \[ \int \frac {x^3 \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\int { \frac {{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x^{3}}{x^{6} + 1} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.64 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.11 \[ \int \frac {x^3 \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\int { \frac {{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x^{3}}{x^{6} + 1} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 6.72 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.11 \[ \int \frac {x^3 \sqrt [3]{-x^2+x^3}}{1+x^6} \, dx=\int \frac {x^3\,{\left (x^3-x^2\right )}^{1/3}}{x^6+1} \,d x \]

[In]

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

[Out]

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