Integrand size = 31, antiderivative size = 33 \[ \int \frac {-1-2 x+x^2+3 x^3}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\frac {4 \left ((-1+x)^3\right )^{3/4} \left (47+158 x+245 x^2+135 x^3\right )}{585 (-1+x)^2} \]
[Out]
Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(33)=66\).
Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.12, number of steps used = 17, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {6874, 2092, 15, 30, 2106, 45} \[ \int \frac {-1-2 x+x^2+3 x^3}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\frac {12 (1-x)^4}{13 \sqrt [4]{(x-1)^3}}+\frac {36 (1-x)^2}{5 \sqrt [4]{(x-1)^3}}-\frac {4 (1-x)}{\sqrt [4]{(x-1)^3}}+\frac {40}{9} \left ((x-1)^3\right )^{3/4} \]
[In]
[Out]
Rule 15
Rule 30
Rule 45
Rule 2092
Rule 2106
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3}}-\frac {2 x}{\sqrt [4]{-1+3 x-3 x^2+x^3}}+\frac {x^2}{\sqrt [4]{-1+3 x-3 x^2+x^3}}+\frac {3 x^3}{\sqrt [4]{-1+3 x-3 x^2+x^3}}\right ) \, dx \\ & = -\left (2 \int \frac {x}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx\right )+3 \int \frac {x^3}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx-\int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx+\int \frac {x^2}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx \\ & = -\left (2 \text {Subst}\left (\int \frac {1+x}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )\right )+3 \text {Subst}\left (\int \frac {(1+x)^3}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )-\text {Subst}\left (\int \frac {1}{\sqrt [4]{x^3}} \, dx,x,-1+x\right )+\text {Subst}\left (\int \frac {(1+x)^2}{\sqrt [4]{x^3}} \, dx,x,-1+x\right ) \\ & = -\frac {(-1+x)^{3/4} \text {Subst}\left (\int \frac {1}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {(-1+x)^{3/4} \text {Subst}\left (\int \frac {(1+x)^2}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}-\frac {\left (2 (-1+x)^{3/4}\right ) \text {Subst}\left (\int \frac {1+x}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {\left (3 (-1+x)^{3/4}\right ) \text {Subst}\left (\int \frac {(1+x)^3}{x^{3/4}} \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}} \\ & = \frac {4 (1-x)}{\sqrt [4]{(-1+x)^3}}+\frac {(-1+x)^{3/4} \text {Subst}\left (\int \left (\frac {1}{x^{3/4}}+2 \sqrt [4]{x}+x^{5/4}\right ) \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}-\frac {\left (2 (-1+x)^{3/4}\right ) \text {Subst}\left (\int \left (\frac {1}{x^{3/4}}+\sqrt [4]{x}\right ) \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}}+\frac {\left (3 (-1+x)^{3/4}\right ) \text {Subst}\left (\int \left (\frac {1}{x^{3/4}}+3 \sqrt [4]{x}+3 x^{5/4}+x^{9/4}\right ) \, dx,x,-1+x\right )}{\sqrt [4]{(-1+x)^3}} \\ & = -\frac {4 (1-x)}{\sqrt [4]{(-1+x)^3}}+\frac {36 (1-x)^2}{5 \sqrt [4]{(-1+x)^3}}+\frac {12 (1-x)^4}{13 \sqrt [4]{(-1+x)^3}}+\frac {40}{9} \left ((-1+x)^3\right )^{3/4} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {-1-2 x+x^2+3 x^3}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\frac {4 (-1+x) \left (47+158 x+245 x^2+135 x^3\right )}{585 \sqrt [4]{(-1+x)^3}} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85
method | result | size |
risch | \(\frac {4 \left (x -1\right ) \left (135 x^{3}+245 x^{2}+158 x +47\right )}{585 \left (\left (x -1\right )^{3}\right )^{\frac {1}{4}}}\) | \(28\) |
gosper | \(\frac {4 \left (x -1\right ) \left (135 x^{3}+245 x^{2}+158 x +47\right )}{585 \left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}\) | \(36\) |
trager | \(\frac {4 \left (135 x^{3}+245 x^{2}+158 x +47\right ) \left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {3}{4}}}{585 \left (x -1\right )^{2}}\) | \(38\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {-1-2 x+x^2+3 x^3}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\frac {4 \, {\left (135 \, x^{3} + 245 \, x^{2} + 158 \, x + 47\right )} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}^{\frac {3}{4}}}{585 \, {\left (x^{2} - 2 \, x + 1\right )}} \]
[In]
[Out]
\[ \int \frac {-1-2 x+x^2+3 x^3}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\int \frac {3 x^{3} + x^{2} - 2 x - 1}{\sqrt [4]{\left (x - 1\right )^{3}}}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (29) = 58\).
Time = 0.22 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.18 \[ \int \frac {-1-2 x+x^2+3 x^3}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\frac {4 \, {\left (15 \, x^{4} + 5 \, x^{3} + 12 \, x^{2} + 96 \, x - 128\right )}}{65 \, {\left (x - 1\right )}^{\frac {3}{4}}} + \frac {4 \, {\left (5 \, x^{3} + 3 \, x^{2} + 24 \, x - 32\right )}}{45 \, {\left (x - 1\right )}^{\frac {3}{4}}} - \frac {8 \, {\left (x^{2} + 3 \, x - 4\right )}}{5 \, {\left (x - 1\right )}^{\frac {3}{4}}} - 4 \, {\left (x - 1\right )}^{\frac {1}{4}} \]
[In]
[Out]
\[ \int \frac {-1-2 x+x^2+3 x^3}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\int { \frac {3 \, x^{3} + x^{2} - 2 \, x - 1}{{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}^{\frac {1}{4}}} \,d x } \]
[In]
[Out]
Time = 5.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {-1-2 x+x^2+3 x^3}{\sqrt [4]{-1+3 x-3 x^2+x^3}} \, dx=\frac {{\left (x^3-3\,x^2+3\,x-1\right )}^{3/4}\,\left (\frac {12\,x^3}{13}+\frac {196\,x^2}{117}+\frac {632\,x}{585}+\frac {188}{585}\right )}{x^2-2\,x+1} \]
[In]
[Out]