Integrand size = 22, antiderivative size = 14 \[ \int \frac {-2+x}{(-1+x) \sqrt [4]{-x^2+x^3}} \, dx=\frac {4 x}{\sqrt [4]{(-1+x) x^2}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2081, 75} \[ \int \frac {-2+x}{(-1+x) \sqrt [4]{-x^2+x^3}} \, dx=\frac {4 x}{\sqrt [4]{x^3-x^2}} \]
[In]
[Out]
Rule 75
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{-1+x} \sqrt {x}\right ) \int \frac {-2+x}{(-1+x)^{5/4} \sqrt {x}} \, dx}{\sqrt [4]{-x^2+x^3}} \\ & = \frac {4 x}{\sqrt [4]{-x^2+x^3}} \\ \end{align*}
Time = 1.97 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {-2+x}{(-1+x) \sqrt [4]{-x^2+x^3}} \, dx=\frac {4 x}{\sqrt [4]{(-1+x) x^2}} \]
[In]
[Out]
Time = 1.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
risch | \(\frac {4 x}{\left (\left (x -1\right ) x^{2}\right )^{\frac {1}{4}}}\) | \(13\) |
gosper | \(\frac {4 x}{\left (x^{3}-x^{2}\right )^{\frac {1}{4}}}\) | \(15\) |
trager | \(\frac {4 \left (x^{3}-x^{2}\right )^{\frac {3}{4}}}{x \left (x -1\right )}\) | \(22\) |
meijerg | \(\frac {4 \left (-\operatorname {signum}\left (x -1\right )\right )^{\frac {1}{4}} \sqrt {x}\, \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], x\right )}{\operatorname {signum}\left (x -1\right )^{\frac {1}{4}}}-\frac {2 \left (-\operatorname {signum}\left (x -1\right )\right )^{\frac {1}{4}} x^{\frac {3}{2}} \operatorname {hypergeom}\left (\left [\frac {5}{4}, \frac {3}{2}\right ], \left [\frac {5}{2}\right ], x\right )}{3 \operatorname {signum}\left (x -1\right )^{\frac {1}{4}}}\) | \(54\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int \frac {-2+x}{(-1+x) \sqrt [4]{-x^2+x^3}} \, dx=\frac {4 \, {\left (x^{3} - x^{2}\right )}^{\frac {3}{4}}}{x^{2} - x} \]
[In]
[Out]
\[ \int \frac {-2+x}{(-1+x) \sqrt [4]{-x^2+x^3}} \, dx=\int \frac {x - 2}{\sqrt [4]{x^{2} \left (x - 1\right )} \left (x - 1\right )}\, dx \]
[In]
[Out]
\[ \int \frac {-2+x}{(-1+x) \sqrt [4]{-x^2+x^3}} \, dx=\int { \frac {x - 2}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{4}} {\left (x - 1\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {-2+x}{(-1+x) \sqrt [4]{-x^2+x^3}} \, dx=\int { \frac {x - 2}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{4}} {\left (x - 1\right )}} \,d x } \]
[In]
[Out]
Time = 4.96 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.50 \[ \int \frac {-2+x}{(-1+x) \sqrt [4]{-x^2+x^3}} \, dx=\frac {4\,{\left (x^3-x^2\right )}^{3/4}}{x\,\left (x-1\right )} \]
[In]
[Out]