Optimal. Leaf size=43 \[ x^2 \left (-\, _2F_1\left (\frac {2}{3},\frac {4}{3};\frac {5}{3};x^3\right )\right )+\frac {x}{\sqrt [3]{1-x^3}}+\frac {1}{\sqrt [3]{1-x^3}} \]
[Out]
________________________________________________________________________________________
Rubi [F] time = 0.43, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx &=\int \left (-\frac {4 \left (1-x^3\right )^{2/3}}{3 \left (-1+i \sqrt {3}-2 x\right )^2}+\frac {4 i \left (1-x^3\right )^{2/3}}{3 \sqrt {3} \left (-1+i \sqrt {3}-2 x\right )}-\frac {4 \left (1-x^3\right )^{2/3}}{3 \left (1+i \sqrt {3}+2 x\right )^2}+\frac {4 i \left (1-x^3\right )^{2/3}}{3 \sqrt {3} \left (1+i \sqrt {3}+2 x\right )}\right ) \, dx\\ &=-\left (\frac {4}{3} \int \frac {\left (1-x^3\right )^{2/3}}{\left (-1+i \sqrt {3}-2 x\right )^2} \, dx\right )-\frac {4}{3} \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+i \sqrt {3}+2 x\right )^2} \, dx+\frac {(4 i) \int \frac {\left (1-x^3\right )^{2/3}}{-1+i \sqrt {3}-2 x} \, dx}{3 \sqrt {3}}+\frac {(4 i) \int \frac {\left (1-x^3\right )^{2/3}}{1+i \sqrt {3}+2 x} \, dx}{3 \sqrt {3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 43, normalized size = 1.00 \[ x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )+\frac {(2 x+1) \left (1-x^3\right )^{2/3}}{x^2+x+1} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{2} + x + 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.10, size = 34, normalized size = 0.79 \[ x^{2} \hypergeom \left (\left [\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )-\frac {\left (x -1\right ) \left (2 x +1\right )}{\left (-x^{3}+1\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{2} + x + 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (1-x^3\right )}^{2/3}}{{\left (x^2+x+1\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}}}{\left (x^{2} + x + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________