Integrand size = 20, antiderivative size = 43 \[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx=\frac {1}{\sqrt [3]{1-x^3}}+\frac {x}{\sqrt [3]{1-x^3}}-x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {4}{3},\frac {5}{3},x^3\right ) \]
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Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2183, 197, 371, 267} \[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx=x^2 \left (-\operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {4}{3},\frac {5}{3},x^3\right )\right )+\frac {x}{\sqrt [3]{1-x^3}}+\frac {1}{\sqrt [3]{1-x^3}} \]
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Rule 197
Rule 267
Rule 371
Rule 2183
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\left (1-x^3\right )^{4/3}}-\frac {2 x}{\left (1-x^3\right )^{4/3}}+\frac {x^2}{\left (1-x^3\right )^{4/3}}\right ) \, dx \\ & = -\left (2 \int \frac {x}{\left (1-x^3\right )^{4/3}} \, dx\right )+\int \frac {1}{\left (1-x^3\right )^{4/3}} \, dx+\int \frac {x^2}{\left (1-x^3\right )^{4/3}} \, dx \\ & = \frac {1}{\sqrt [3]{1-x^3}}+\frac {x}{\sqrt [3]{1-x^3}}-x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {4}{3},\frac {5}{3},x^3\right ) \\ \end{align*}
Time = 10.18 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx=\frac {(1+2 x) \left (1-x^3\right )^{2/3}}{1+x+x^2}+x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},x^3\right ) \]
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Time = 0.86 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79
method | result | size |
risch | \(-\frac {\left (-1+x \right ) \left (1+2 x \right )}{\left (-x^{3}+1\right )^{\frac {1}{3}}}+x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^{3}\right )\) | \(34\) |
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\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{2} + x + 1\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx=\int \frac {\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}}}{\left (x^{2} + x + 1\right )^{2}}\, dx \]
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\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{2} + x + 1\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx=\int { \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{2} + x + 1\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (1-x^3\right )^{2/3}}{\left (1+x+x^2\right )^2} \, dx=\int \frac {{\left (1-x^3\right )}^{2/3}}{{\left (x^2+x+1\right )}^2} \,d x \]
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