Integrand size = 20, antiderivative size = 144 \[ \int \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\frac {2}{5} x \sqrt {1+x} \sqrt {1-x+x^2}+\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} (1+x)^{3/2} \sqrt {1-x+x^2} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )} \]
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Time = 0.03 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {727, 201, 224} \[ \int \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} \sqrt {x^2-x+1} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} (x+1)^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \left (x^3+1\right )}+\frac {2}{5} x \sqrt {x^2-x+1} \sqrt {x+1} \]
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Rule 201
Rule 224
Rule 727
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \sqrt {1+x^3} \, dx}{\sqrt {1+x^3}} \\ & = \frac {2}{5} x \sqrt {1+x} \sqrt {1-x+x^2}+\frac {\left (3 \sqrt {1+x} \sqrt {1-x+x^2}\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx}{5 \sqrt {1+x^3}} \\ & = \frac {2}{5} x \sqrt {1+x} \sqrt {1-x+x^2}+\frac {2\ 3^{3/4} \sqrt {2+\sqrt {3}} (1+x)^{3/2} \sqrt {1-x+x^2} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{5 \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \left (1+x^3\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 20.35 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.17 \[ \int \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\frac {2 x \sqrt {1+x} \left (1-x+x^2\right )+\frac {i (1+x) \sqrt {1+\frac {6 i}{\left (-3 i+\sqrt {3}\right ) (1+x)}} \sqrt {6-\frac {36 i}{\left (3 i+\sqrt {3}\right ) (1+x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {1+x}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {i}{3 i+\sqrt {3}}}}}{5 \sqrt {1-x+x^2}} \]
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Time = 0.64 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.09
method | result | size |
elliptic | \(\frac {\sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}\, \left (\frac {2 x \sqrt {x^{3}+1}}{5}+\frac {6 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{5 \sqrt {x^{3}+1}}\right )}{\sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(157\) |
risch | \(\frac {2 x \sqrt {1+x}\, \sqrt {x^{2}-x +1}}{5}+\frac {6 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {\left (1+x \right ) \left (x^{2}-x +1\right )}}{5 \sqrt {x^{3}+1}\, \sqrt {1+x}\, \sqrt {x^{2}-x +1}}\) | \(164\) |
default | \(-\frac {\sqrt {1+x}\, \sqrt {x^{2}-x +1}\, \left (3 i \sqrt {3}\, \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, F\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right )-9 \sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}-2 x +1}{i \sqrt {3}+3}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, F\left (\sqrt {-\frac {2 \left (1+x \right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{i \sqrt {3}+3}}\right )-2 x^{4}-2 x \right )}{5 \left (x^{3}+1\right )}\) | \(252\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.17 \[ \int \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\frac {2}{5} \, \sqrt {x^{2} - x + 1} \sqrt {x + 1} x + \frac {6}{5} \, {\rm weierstrassPInverse}\left (0, -4, x\right ) \]
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\[ \int \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\int \sqrt {x + 1} \sqrt {x^{2} - x + 1}\, dx \]
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\[ \int \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\int { \sqrt {x^{2} - x + 1} \sqrt {x + 1} \,d x } \]
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\[ \int \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\int { \sqrt {x^{2} - x + 1} \sqrt {x + 1} \,d x } \]
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Timed out. \[ \int \sqrt {1+x} \sqrt {1-x+x^2} \, dx=\int \sqrt {x+1}\,\sqrt {x^2-x+1} \,d x \]
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