Optimal. Leaf size=45 \[ \frac {x \left (x^3+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-x^3\right )}{(x+1)^{2/3} \left (x^2-x+1\right )^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {713, 245} \[ \frac {x \left (x^3+1\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-x^3\right )}{(x+1)^{2/3} \left (x^2-x+1\right )^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 245
Rule 713
Rubi steps
\begin {align*} \int \frac {1}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}} \, dx &=\frac {\left (1+x^3\right )^{2/3} \int \frac {1}{\left (1+x^3\right )^{2/3}} \, dx}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}}\\ &=\frac {x \left (1+x^3\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-x^3\right )}{(1+x)^{2/3} \left (1-x+x^2\right )^{2/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.07, size = 143, normalized size = 3.18 \[ \frac {3 \left (2 i x+\sqrt {3}-i\right ) \sqrt [3]{x+1} \left (-\frac {\left (\sqrt {3}-3 i\right ) x+\sqrt {3}+3 i}{\left (\sqrt {3}+3 i\right ) x+\sqrt {3}-3 i}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {4 i \sqrt {3} (x+1)}{\left (3 i+\sqrt {3}\right ) \left (2 i x+\sqrt {3}-i\right )}\right )}{\left (\sqrt {3}-3 i\right ) \left (x^2-x+1\right )^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.20, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}^{\frac {1}{3}}}{x^{3} + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{2} - x + 1\right )}^{\frac {2}{3}} {\left (x + 1\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 2.39, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (x +1\right )^{\frac {2}{3}} \left (x^{2}-x +1\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{2} - x + 1\right )}^{\frac {2}{3}} {\left (x + 1\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (x+1\right )}^{2/3}\,{\left (x^2-x+1\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (x + 1\right )^{\frac {2}{3}} \left (x^{2} - x + 1\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________