Integrand size = 22, antiderivative size = 18 \[ \int \frac {\left (-1+2 x^3\right ) \sqrt [3]{x^2+x^5}}{x^3} \, dx=\frac {3 \left (x^2+x^5\right )^{4/3}}{4 x^4} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1604} \[ \int \frac {\left (-1+2 x^3\right ) \sqrt [3]{x^2+x^5}}{x^3} \, dx=\frac {3 \left (x^5+x^2\right )^{4/3}}{4 x^4} \]
[In]
[Out]
Rule 1604
Rubi steps \begin{align*} \text {integral}& = \frac {3 \left (x^2+x^5\right )^{4/3}}{4 x^4} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {\left (-1+2 x^3\right ) \sqrt [3]{x^2+x^5}}{x^3} \, dx=\frac {3 \left (1+x^3\right ) \sqrt [3]{x^2+x^5}}{4 x^2} \]
[In]
[Out]
Time = 0.85 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11
method | result | size |
trager | \(\frac {3 \left (x^{3}+1\right ) \left (x^{5}+x^{2}\right )^{\frac {1}{3}}}{4 x^{2}}\) | \(20\) |
pseudoelliptic | \(\frac {3 \left (x^{3}+1\right ) \left (x^{5}+x^{2}\right )^{\frac {1}{3}}}{4 x^{2}}\) | \(20\) |
gosper | \(\frac {3 \left (1+x \right ) \left (x^{2}-x +1\right ) \left (x^{5}+x^{2}\right )^{\frac {1}{3}}}{4 x^{2}}\) | \(26\) |
meijerg | \(\frac {3 \operatorname {hypergeom}\left (\left [-\frac {4}{9}, -\frac {1}{3}\right ], \left [\frac {5}{9}\right ], -x^{3}\right )}{4 x^{\frac {4}{3}}}+\frac {6 x^{\frac {5}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {5}{9}\right ], \left [\frac {14}{9}\right ], -x^{3}\right )}{5}\) | \(34\) |
risch | \(\frac {3 \left (x^{2} \left (x^{3}+1\right )\right )^{\frac {1}{3}} \left (x^{6}+2 x^{3}+1\right )}{4 x^{2} \left (x^{3}+1\right )}\) | \(34\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\left (-1+2 x^3\right ) \sqrt [3]{x^2+x^5}}{x^3} \, dx=\frac {3 \, {\left (x^{5} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + 1\right )}}{4 \, x^{2}} \]
[In]
[Out]
\[ \int \frac {\left (-1+2 x^3\right ) \sqrt [3]{x^2+x^5}}{x^3} \, dx=\int \frac {\sqrt [3]{x^{2} \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (2 x^{3} - 1\right )}{x^{3}}\, dx \]
[In]
[Out]
\[ \int \frac {\left (-1+2 x^3\right ) \sqrt [3]{x^2+x^5}}{x^3} \, dx=\int { \frac {{\left (x^{5} + x^{2}\right )}^{\frac {1}{3}} {\left (2 \, x^{3} - 1\right )}}{x^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (-1+2 x^3\right ) \sqrt [3]{x^2+x^5}}{x^3} \, dx=\int { \frac {{\left (x^{5} + x^{2}\right )}^{\frac {1}{3}} {\left (2 \, x^{3} - 1\right )}}{x^{3}} \,d x } \]
[In]
[Out]
Time = 4.95 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\left (-1+2 x^3\right ) \sqrt [3]{x^2+x^5}}{x^3} \, dx=\frac {3\,{\left (x^5+x^2\right )}^{1/3}\,\left (x^3+1\right )}{4\,x^2} \]
[In]
[Out]