Integrand size = 25, antiderivative size = 28 \[ \int \frac {2+x}{(-1+x) \sqrt {-x-x^2+x^3}} \, dx=-2 \arctan \left (\frac {\sqrt {-x-x^2+x^3}}{(-2+x) x}\right ) \]
[Out]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.66 (sec) , antiderivative size = 533, normalized size of antiderivative = 19.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {2081, 6865, 1724, 1112, 1228, 1470, 554, 432, 430, 552, 551} \[ \int \frac {2+x}{(-1+x) \sqrt {-x-x^2+x^3}} \, dx=\frac {3 \sqrt {2} \left (1+\sqrt {5}\right ) \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\left (1-\sqrt {5}\right ) \sqrt {x^3-x^2-x}}-\frac {6 \sqrt {x} \sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^3-x^2-x}}+\frac {\sqrt {x} \sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^3-x^2-x}}+\frac {6 \sqrt {2} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1+\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\left (1-\sqrt {5}\right ) \sqrt {x^3-x^2-x}} \]
[In]
[Out]
Rule 430
Rule 432
Rule 551
Rule 552
Rule 554
Rule 1112
Rule 1228
Rule 1470
Rule 1724
Rule 2081
Rule 6865
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {2+x}{(-1+x) \sqrt {x} \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-1-x+x^2}\right ) \text {Subst}\left (\int \frac {2+x^2}{\left (-1+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-1-x+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}+\frac {\left (6 \sqrt {x} \sqrt {-1-x+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}} \\ & = \frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\left (6 \sqrt {x} \sqrt {-1-x+x^2}\right ) \text {Subst}\left (\int \frac {-1-\sqrt {5}+2 x^2}{\left (-1+x^2\right ) \sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\left (-1+\sqrt {5}\right ) \sqrt {-x-x^2+x^3}}+\frac {\left (12 \sqrt {x} \sqrt {-1-x+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\left (-1+\sqrt {5}\right ) \sqrt {-x-x^2+x^3}} \\ & = \frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {6 \sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\left (3 \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \text {Subst}\left (\int \frac {\sqrt {-1-\sqrt {5}+2 x^2}}{\sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \left (-1+x^2\right )} \, dx,x,\sqrt {x}\right )}{\left (-1+\sqrt {5}\right ) \sqrt {-x-x^2+x^3}} \\ & = \frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {6 \sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\left (6 \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {-1-\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\left (-1+\sqrt {5}\right ) \sqrt {-x-x^2+x^3}}-\frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \left (-1+x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\left (-1+\sqrt {5}\right ) \sqrt {-x-x^2+x^3}} \\ & = \frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {6 \sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\left (6 \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\left (-1+\sqrt {5}\right ) \sqrt {-x-x^2+x^3}}-\frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-\frac {1}{-1-\sqrt {5}}+\frac {x^2}{2}} \left (-1+x^2\right ) \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\left (-1+\sqrt {5}\right ) \sqrt {-x-x^2+x^3}} \\ & = \frac {3 \sqrt {2} \left (1+\sqrt {5}\right ) \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\left (1-\sqrt {5}\right ) \sqrt {-x-x^2+x^3}}+\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {6 \sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right ),\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \left (1-\sqrt {5}\right ) \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {3 \left (1+\sqrt {5}\right ) \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1+\sqrt {5}\right ),\arcsin \left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right ),\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {2} \sqrt {-x-x^2+x^3}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(28)=56\).
Time = 0.76 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.04 \[ \int \frac {2+x}{(-1+x) \sqrt {-x-x^2+x^3}} \, dx=-\frac {2 \sqrt {x} \sqrt {-1-x+x^2} \arctan \left (\frac {\sqrt {-1-x+x^2}}{(-2+x) \sqrt {x}}\right )}{\sqrt {x \left (-1-x+x^2\right )}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.07
method | result | size |
trager | \(-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +2 \sqrt {x^{3}-x^{2}-x}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-4 \sqrt {x^{3}-x^{2}-x}}{\left (x -1\right )^{3}}\right )\) | \(86\) |
default | \(\frac {2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \sqrt {-5 \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}}+\frac {6 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \sqrt {-5 \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{-\frac {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}\, \left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}\) | \(250\) |
elliptic | \(\frac {2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \sqrt {-5 \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}}+\frac {6 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \sqrt {-5 \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{-\frac {1}{2}-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}\, \left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}\) | \(250\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {2+x}{(-1+x) \sqrt {-x-x^2+x^3}} \, dx=\arctan \left (\frac {\sqrt {x^{3} - x^{2} - x} {\left (x^{3} - 5 \, x^{2} + 5 \, x + 1\right )}}{2 \, {\left (x^{4} - 3 \, x^{3} + x^{2} + 2 \, x\right )}}\right ) \]
[In]
[Out]
\[ \int \frac {2+x}{(-1+x) \sqrt {-x-x^2+x^3}} \, dx=\int \frac {x + 2}{\sqrt {x \left (x^{2} - x - 1\right )} \left (x - 1\right )}\, dx \]
[In]
[Out]
\[ \int \frac {2+x}{(-1+x) \sqrt {-x-x^2+x^3}} \, dx=\int { \frac {x + 2}{\sqrt {x^{3} - x^{2} - x} {\left (x - 1\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {2+x}{(-1+x) \sqrt {-x-x^2+x^3}} \, dx=\int { \frac {x + 2}{\sqrt {x^{3} - x^{2} - x} {\left (x - 1\right )}} \,d x } \]
[In]
[Out]
Time = 5.30 (sec) , antiderivative size = 165, normalized size of antiderivative = 5.89 \[ \int \frac {2+x}{(-1+x) \sqrt {-x-x^2+x^3}} \, dx=\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\left (\sqrt {5}+1\right )\,\left (\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )-3\,\Pi \left (\frac {\sqrt {5}}{2}+\frac {1}{2};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )\right )\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}}{\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}} \]
[In]
[Out]