Integrand size = 24, antiderivative size = 79 \[ \int \frac {-2+3 x}{(1+x)^3 \sqrt {2 x-x^2}} \, dx=-\frac {5 \sqrt {2 x-x^2}}{6 (1+x)^2}-\frac {2 \sqrt {2 x-x^2}}{3 (1+x)}+\frac {\arctan \left (\frac {1-2 x}{\sqrt {3} \sqrt {2 x-x^2}}\right )}{2 \sqrt {3}} \]
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Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {848, 820, 738, 210} \[ \int \frac {-2+3 x}{(1+x)^3 \sqrt {2 x-x^2}} \, dx=\frac {\arctan \left (\frac {1-2 x}{\sqrt {3} \sqrt {2 x-x^2}}\right )}{2 \sqrt {3}}-\frac {2 \sqrt {2 x-x^2}}{3 (x+1)}-\frac {5 \sqrt {2 x-x^2}}{6 (x+1)^2} \]
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Rule 210
Rule 738
Rule 820
Rule 848
Rubi steps \begin{align*} \text {integral}& = -\frac {5 \sqrt {2 x-x^2}}{6 (1+x)^2}+\frac {1}{6} \int \frac {-7+5 x}{(1+x)^2 \sqrt {2 x-x^2}} \, dx \\ & = -\frac {5 \sqrt {2 x-x^2}}{6 (1+x)^2}-\frac {2 \sqrt {2 x-x^2}}{3 (1+x)}-\frac {1}{2} \int \frac {1}{(1+x) \sqrt {2 x-x^2}} \, dx \\ & = -\frac {5 \sqrt {2 x-x^2}}{6 (1+x)^2}-\frac {2 \sqrt {2 x-x^2}}{3 (1+x)}+\text {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,\frac {-2+4 x}{\sqrt {2 x-x^2}}\right ) \\ & = -\frac {5 \sqrt {2 x-x^2}}{6 (1+x)^2}-\frac {2 \sqrt {2 x-x^2}}{3 (1+x)}-\frac {\arctan \left (\frac {-2+4 x}{2 \sqrt {3} \sqrt {2 x-x^2}}\right )}{2 \sqrt {3}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.99 \[ \int \frac {-2+3 x}{(1+x)^3 \sqrt {2 x-x^2}} \, dx=\frac {x \left (-18+x+4 x^2\right )-2 \sqrt {3} \sqrt {-2+x} \sqrt {x} (1+x)^2 \text {arctanh}\left (\frac {1-\sqrt {-2+x} \sqrt {x}+x}{\sqrt {3}}\right )}{6 \sqrt {-((-2+x) x)} (1+x)^2} \]
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Time = 0.56 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.63
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \left (1+x \right )^{2} \arctan \left (\frac {\sqrt {3}\, \sqrt {-x \left (-2+x \right )}}{3 x}\right )+\left (-4 x -9\right ) \sqrt {-x \left (-2+x \right )}}{6 \left (1+x \right )^{2}}\) | \(50\) |
risch | \(\frac {x \left (-2+x \right ) \left (4 x +9\right )}{6 \left (1+x \right )^{2} \sqrt {-x \left (-2+x \right )}}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (-2+4 x \right ) \sqrt {3}}{6 \sqrt {-\left (1+x \right )^{2}+1+4 x}}\right )}{6}\) | \(56\) |
trager | \(-\frac {\left (4 x +9\right ) \sqrt {-x^{2}+2 x}}{6 \left (1+x \right )^{2}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x +3 \sqrt {-x^{2}+2 x}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{1+x}\right )}{6}\) | \(69\) |
default | \(-\frac {5 \sqrt {-\left (1+x \right )^{2}+1+4 x}}{6 \left (1+x \right )^{2}}-\frac {2 \sqrt {-\left (1+x \right )^{2}+1+4 x}}{3 \left (1+x \right )}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (-2+4 x \right ) \sqrt {3}}{6 \sqrt {-\left (1+x \right )^{2}+1+4 x}}\right )}{6}\) | \(74\) |
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Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.81 \[ \int \frac {-2+3 x}{(1+x)^3 \sqrt {2 x-x^2}} \, dx=\frac {2 \, \sqrt {3} {\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {3} \sqrt {-x^{2} + 2 \, x}}{3 \, x}\right ) - \sqrt {-x^{2} + 2 \, x} {\left (4 \, x + 9\right )}}{6 \, {\left (x^{2} + 2 \, x + 1\right )}} \]
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\[ \int \frac {-2+3 x}{(1+x)^3 \sqrt {2 x-x^2}} \, dx=\int \frac {3 x - 2}{\sqrt {- x \left (x - 2\right )} \left (x + 1\right )^{3}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.84 \[ \int \frac {-2+3 x}{(1+x)^3 \sqrt {2 x-x^2}} \, dx=-\frac {1}{6} \, \sqrt {3} \arcsin \left (\frac {2 \, x}{{\left | x + 1 \right |}} - \frac {1}{{\left | x + 1 \right |}}\right ) - \frac {5 \, \sqrt {-x^{2} + 2 \, x}}{6 \, {\left (x^{2} + 2 \, x + 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 2 \, x}}{3 \, {\left (x + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (64) = 128\).
Time = 0.30 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.86 \[ \int \frac {-2+3 x}{(1+x)^3 \sqrt {2 x-x^2}} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (\sqrt {-x^{2} + 2 \, x} - 1\right )}}{x - 1} - 1\right )}\right ) + \frac {\frac {34 \, {\left (\sqrt {-x^{2} + 2 \, x} - 1\right )}}{x - 1} - \frac {39 \, {\left (\sqrt {-x^{2} + 2 \, x} - 1\right )}^{2}}{{\left (x - 1\right )}^{2}} + \frac {18 \, {\left (\sqrt {-x^{2} + 2 \, x} - 1\right )}^{3}}{{\left (x - 1\right )}^{3}} - 26}{24 \, {\left (\frac {\sqrt {-x^{2} + 2 \, x} - 1}{x - 1} - \frac {{\left (\sqrt {-x^{2} + 2 \, x} - 1\right )}^{2}}{{\left (x - 1\right )}^{2}} - 1\right )}^{2}} \]
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Timed out. \[ \int \frac {-2+3 x}{(1+x)^3 \sqrt {2 x-x^2}} \, dx=\int \frac {3\,x-2}{\sqrt {2\,x-x^2}\,{\left (x+1\right )}^3} \,d x \]
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