Integrand size = 24, antiderivative size = 39 \[ \int \frac {x \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {1}{6} \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {1}{12} \arcsin \left (3-6 x^2\right ) \]
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Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {455, 52, 55, 633, 222} \[ \int \frac {x \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {1}{12} \arcsin \left (3-6 x^2\right )-\frac {1}{6} \sqrt {2-3 x^2} \sqrt {3 x^2-1} \]
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Rule 52
Rule 55
Rule 222
Rule 455
Rule 633
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {-1+3 x}}{\sqrt {2-3 x}} \, dx,x,x^2\right ) \\ & = -\frac {1}{6} \sqrt {2-3 x^2} \sqrt {-1+3 x^2}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {2-3 x} \sqrt {-1+3 x}} \, dx,x,x^2\right ) \\ & = -\frac {1}{6} \sqrt {2-3 x^2} \sqrt {-1+3 x^2}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{\sqrt {-2+9 x-9 x^2}} \, dx,x,x^2\right ) \\ & = -\frac {1}{6} \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {1}{36} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{9}}} \, dx,x,9 \left (1-2 x^2\right )\right ) \\ & = -\frac {1}{6} \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {1}{12} \sin ^{-1}\left (3-6 x^2\right ) \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.36 \[ \int \frac {x \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\frac {1}{6} \left (-\sqrt {-2+9 x^2-9 x^4}+2 \arctan \left (\frac {\sqrt {-1+3 x^2}}{-1+\sqrt {2-3 x^2}}\right )\right ) \]
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Time = 3.14 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.54
method | result | size |
default | \(\frac {\sqrt {3 x^{2}-1}\, \sqrt {-3 x^{2}+2}\, \left (\arcsin \left (6 x^{2}-3\right )-2 \sqrt {-9 x^{4}+9 x^{2}-2}\right )}{12 \sqrt {-9 x^{4}+9 x^{2}-2}}\) | \(60\) |
elliptic | \(\frac {\sqrt {-\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )}\, \left (\frac {\arcsin \left (6 x^{2}-3\right )}{12}-\frac {\sqrt {-9 x^{4}+9 x^{2}-2}}{6}\right )}{\sqrt {-3 x^{2}+2}\, \sqrt {3 x^{2}-1}}\) | \(65\) |
risch | \(\frac {\left (3 x^{2}-2\right ) \sqrt {3 x^{2}-1}\, \sqrt {\left (3 x^{2}-1\right ) \left (-3 x^{2}+2\right )}}{6 \sqrt {-\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )}\, \sqrt {-3 x^{2}+2}}+\frac {\arcsin \left (6 x^{2}-3\right ) \sqrt {\left (3 x^{2}-1\right ) \left (-3 x^{2}+2\right )}}{12 \sqrt {-3 x^{2}+2}\, \sqrt {3 x^{2}-1}}\) | \(109\) |
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (31) = 62\).
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.67 \[ \int \frac {x \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {1}{6} \, \sqrt {3 \, x^{2} - 1} \sqrt {-3 \, x^{2} + 2} - \frac {1}{12} \, \arctan \left (\frac {3 \, \sqrt {3 \, x^{2} - 1} {\left (2 \, x^{2} - 1\right )} \sqrt {-3 \, x^{2} + 2}}{2 \, {\left (9 \, x^{4} - 9 \, x^{2} + 2\right )}}\right ) \]
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Time = 1.50 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.87 \[ \int \frac {x \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=- \frac {\sqrt {2 - 3 x^{2}} \sqrt {3 x^{2} - 1}}{6} + \frac {\operatorname {asin}{\left (\sqrt {3 x^{2} - 1} \right )}}{6} \]
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Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.69 \[ \int \frac {x \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {1}{6} \, \sqrt {-9 \, x^{4} + 9 \, x^{2} - 2} + \frac {1}{12} \, \arcsin \left (6 \, x^{2} - 3\right ) \]
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Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int \frac {x \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {1}{6} \, \sqrt {3 \, x^{2} - 1} \sqrt {-3 \, x^{2} + 2} + \frac {1}{6} \, \arcsin \left (\sqrt {3 \, x^{2} - 1}\right ) \]
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Time = 7.20 (sec) , antiderivative size = 206, normalized size of antiderivative = 5.28 \[ \int \frac {x \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {\mathrm {atan}\left (\frac {\sqrt {3\,x^2-1}-\mathrm {i}}{\sqrt {2}-\sqrt {2-3\,x^2}}\right )}{3}-\frac {-\frac {\sqrt {3\,x^2-1}-\mathrm {i}}{\sqrt {2}-\sqrt {2-3\,x^2}}+\frac {{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^3}+\frac {\sqrt {2}\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^2\,4{}\mathrm {i}}{3\,{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^2}}{\frac {2\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^2}+\frac {{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^4}+1} \]
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