Integrand size = 30, antiderivative size = 33 \[ \int \frac {-2+2 x+x^2}{\left (3-x+2 x^2\right ) \sqrt {1+x^3}} \, dx=-\sqrt {2} \arctan \left (\frac {\sqrt {1+x^3}}{\sqrt {2} \left (1-x+x^2\right )}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2170, 209} \[ \int \frac {-2+2 x+x^2}{\left (3-x+2 x^2\right ) \sqrt {1+x^3}} \, dx=-\sqrt {2} \arctan \left (\frac {x+1}{\sqrt {2} \sqrt {x^3+1}}\right ) \]
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Rule 209
Rule 2170
Rubi steps \begin{align*} \text {integral}& = -\left (4 \text {Subst}\left (\int \frac {1}{4+2 x^2} \, dx,x,\frac {1+x}{\sqrt {1+x^3}}\right )\right ) \\ & = -\sqrt {2} \arctan \left (\frac {1+x}{\sqrt {2} \sqrt {1+x^3}}\right ) \\ \end{align*}
Time = 1.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {-2+2 x+x^2}{\left (3-x+2 x^2\right ) \sqrt {1+x^3}} \, dx=-\sqrt {2} \arctan \left (\frac {\sqrt {1+x^3}}{\sqrt {2} \left (1-x+x^2\right )}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.50 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.82
method | result | size |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x +4 \sqrt {x^{3}+1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{2 x^{2}-x +3}\right )}{2}\) | \(60\) |
default | \(\frac {\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}}}\, \operatorname {EllipticF}\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 {x^{3}+1}}+\frac {\left (\frac {5}{4}+\frac {i \sqrt {23}}{4}\right ) \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}}}\, \left (-\frac {5}{12}+\frac {i \sqrt {23}}{12}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{8}-\frac {i \sqrt {23}}{8}-\frac {i \sqrt {3}}{4}+\frac {i \sqrt {3}\, \left (\frac {1}{4}+\frac {i \sqrt {23}}{4}\right )}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {\left (\frac {5}{4}-\frac {i \sqrt {23}}{4}\right ) \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}}}\, \left (-\frac {5}{12}-\frac {i \sqrt {23}}{12}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{8}+\frac {i \sqrt {23}}{8}-\frac {i \sqrt {3}}{4}+\frac {i \sqrt {3}\, \left (\frac {1}{4}-\frac {i \sqrt {23}}{4}\right )}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\) | \(432\) |
elliptic | \(\frac {\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}}}\, \operatorname {EllipticF}\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 {x^{3}+1}}+\frac {2 \left (\frac {5}{8}+\frac {i \sqrt {23}}{8}\right ) \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}}}\, \left (-\frac {5}{12}+\frac {i \sqrt {23}}{12}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{8}-\frac {i \sqrt {23}}{8}-\frac {i \sqrt {3}}{4}+\frac {i \sqrt {3}\, \left (\frac {1}{4}+\frac {i \sqrt {23}}{4}\right )}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {2 \left (\frac {5}{8}-\frac {i \sqrt {23}}{8}\right ) \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}}}\, \left (-\frac {5}{12}-\frac {i \sqrt {23}}{12}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {5}{8}+\frac {i \sqrt {23}}{8}-\frac {i \sqrt {3}}{4}+\frac {i \sqrt {3}\, \left (\frac {1}{4}-\frac {i \sqrt {23}}{4}\right )}{6}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\) | \(434\) |
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Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {-2+2 x+x^2}{\left (3-x+2 x^2\right ) \sqrt {1+x^3}} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, x^{2} - 3 \, x + 1\right )}}{4 \, \sqrt {x^{3} + 1}}\right ) \]
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\[ \int \frac {-2+2 x+x^2}{\left (3-x+2 x^2\right ) \sqrt {1+x^3}} \, dx=\int \frac {x^{2} + 2 x - 2}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (2 x^{2} - x + 3\right )}\, dx \]
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\[ \int \frac {-2+2 x+x^2}{\left (3-x+2 x^2\right ) \sqrt {1+x^3}} \, dx=\int { \frac {x^{2} + 2 \, x - 2}{\sqrt {x^{3} + 1} {\left (2 \, x^{2} - x + 3\right )}} \,d x } \]
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\[ \int \frac {-2+2 x+x^2}{\left (3-x+2 x^2\right ) \sqrt {1+x^3}} \, dx=\int { \frac {x^{2} + 2 \, x - 2}{\sqrt {x^{3} + 1} {\left (2 \, x^{2} - x + 3\right )}} \,d x } \]
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Time = 5.21 (sec) , antiderivative size = 274, normalized size of antiderivative = 8.30 \[ \int \frac {-2+2 x+x^2}{\left (3-x+2 x^2\right ) \sqrt {1+x^3}} \, dx=-\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {5}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {5}{4}+\frac {\sqrt {23}\,1{}\mathrm {i}}{4}};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )}{2\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
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