Integrand size = 22, antiderivative size = 18 \[ \int \frac {-2+x^3}{\left (1+x^3\right ) \sqrt [3]{x+x^4}} \, dx=-\frac {3 \left (x+x^4\right )^{2/3}}{1+x^3} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2081, 460} \[ \int \frac {-2+x^3}{\left (1+x^3\right ) \sqrt [3]{x+x^4}} \, dx=-\frac {3 x}{\sqrt [3]{x^4+x}} \]
[In]
[Out]
Rule 460
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^3}\right ) \int \frac {-2+x^3}{\sqrt [3]{x} \left (1+x^3\right )^{4/3}} \, dx}{\sqrt [3]{x+x^4}} \\ & = -\frac {3 x}{\sqrt [3]{x+x^4}} \\ \end{align*}
Time = 5.86 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \frac {-2+x^3}{\left (1+x^3\right ) \sqrt [3]{x+x^4}} \, dx=-\frac {3 x}{\sqrt [3]{x+x^4}} \]
[In]
[Out]
Time = 0.96 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61
method | result | size |
gosper | \(-\frac {3 x}{\left (x^{4}+x \right )^{\frac {1}{3}}}\) | \(11\) |
pseudoelliptic | \(-\frac {3 x}{\left (x^{4}+x \right )^{\frac {1}{3}}}\) | \(11\) |
risch | \(-\frac {3 x}{{\left (x \left (x^{3}+1\right )\right )}^{\frac {1}{3}}}\) | \(13\) |
trager | \(-\frac {3 \left (x^{4}+x \right )^{\frac {2}{3}}}{x^{3}+1}\) | \(17\) |
meijerg | \(-3 x^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [\frac {2}{9}, \frac {4}{3}\right ], \left [\frac {11}{9}\right ], -x^{3}\right )+\frac {3 x^{\frac {11}{3}} \operatorname {hypergeom}\left (\left [\frac {11}{9}, \frac {4}{3}\right ], \left [\frac {20}{9}\right ], -x^{3}\right )}{11}\) | \(34\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-2+x^3}{\left (1+x^3\right ) \sqrt [3]{x+x^4}} \, dx=-\frac {3 \, {\left (x^{4} + x\right )}^{\frac {2}{3}}}{x^{3} + 1} \]
[In]
[Out]
\[ \int \frac {-2+x^3}{\left (1+x^3\right ) \sqrt [3]{x+x^4}} \, dx=\int \frac {x^{3} - 2}{\sqrt [3]{x \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
[In]
[Out]
\[ \int \frac {-2+x^3}{\left (1+x^3\right ) \sqrt [3]{x+x^4}} \, dx=\int { \frac {x^{3} - 2}{{\left (x^{4} + x\right )}^{\frac {1}{3}} {\left (x^{3} + 1\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {-2+x^3}{\left (1+x^3\right ) \sqrt [3]{x+x^4}} \, dx=\int { \frac {x^{3} - 2}{{\left (x^{4} + x\right )}^{\frac {1}{3}} {\left (x^{3} + 1\right )}} \,d x } \]
[In]
[Out]
Time = 4.98 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-2+x^3}{\left (1+x^3\right ) \sqrt [3]{x+x^4}} \, dx=-\frac {3\,{\left (x^4+x\right )}^{2/3}}{x^3+1} \]
[In]
[Out]