Integrand size = 22, antiderivative size = 103 \[ \int \frac {\sqrt {3-2 x}}{\sqrt {1-3 x+x^2}} \, dx=-\frac {2 \sqrt [4]{5} \sqrt {-1+3 x-x^2} E\left (\left .\arcsin \left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt {1-3 x+x^2}}+\frac {2 \sqrt [4]{5} \sqrt {-1+3 x-x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right ),-1\right )}{\sqrt {1-3 x+x^2}} \]
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Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {705, 704, 313, 227, 1195, 21, 435} \[ \int \frac {\sqrt {3-2 x}}{\sqrt {1-3 x+x^2}} \, dx=\frac {2 \sqrt [4]{5} \sqrt {-x^2+3 x-1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right ),-1\right )}{\sqrt {x^2-3 x+1}}-\frac {2 \sqrt [4]{5} \sqrt {-x^2+3 x-1} E\left (\left .\arcsin \left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt {x^2-3 x+1}} \]
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Rule 21
Rule 227
Rule 313
Rule 435
Rule 704
Rule 705
Rule 1195
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-1+3 x-x^2} \int \frac {\sqrt {3-2 x}}{\sqrt {-\frac {1}{5}+\frac {3 x}{5}-\frac {x^2}{5}}} \, dx}{\sqrt {5} \sqrt {1-3 x+x^2}} \\ & = -\frac {\left (2 \sqrt {-1+3 x-x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^4}{5}}} \, dx,x,\sqrt {3-2 x}\right )}{\sqrt {5} \sqrt {1-3 x+x^2}} \\ & = \frac {\left (2 \sqrt {-1+3 x-x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^4}{5}}} \, dx,x,\sqrt {3-2 x}\right )}{\sqrt {1-3 x+x^2}}-\frac {\left (2 \sqrt {-1+3 x-x^2}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {5}}}{\sqrt {1-\frac {x^4}{5}}} \, dx,x,\sqrt {3-2 x}\right )}{\sqrt {1-3 x+x^2}} \\ & = \frac {2 \sqrt [4]{5} \sqrt {-1+3 x-x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt {1-3 x+x^2}}-\frac {\left (2 \sqrt {-1+3 x-x^2}\right ) \text {Subst}\left (\int \frac {1+\frac {x^2}{\sqrt {5}}}{\sqrt {\frac {1}{\sqrt {5}}-\frac {x^2}{5}} \sqrt {\frac {1}{\sqrt {5}}+\frac {x^2}{5}}} \, dx,x,\sqrt {3-2 x}\right )}{\sqrt {5} \sqrt {1-3 x+x^2}} \\ & = \frac {2 \sqrt [4]{5} \sqrt {-1+3 x-x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt {1-3 x+x^2}}-\frac {\left (2 \sqrt {-1+3 x-x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {\frac {1}{\sqrt {5}}+\frac {x^2}{5}}}{\sqrt {\frac {1}{\sqrt {5}}-\frac {x^2}{5}}} \, dx,x,\sqrt {3-2 x}\right )}{\sqrt {1-3 x+x^2}} \\ & = -\frac {2 \sqrt [4]{5} \sqrt {-1+3 x-x^2} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt {1-3 x+x^2}}+\frac {2 \sqrt [4]{5} \sqrt {-1+3 x-x^2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt {1-3 x+x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {3-2 x}}{\sqrt {1-3 x+x^2}} \, dx=-\frac {2 (3-2 x)^{3/2} \sqrt {-1+3 x-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {1}{5} (3-2 x)^2\right )}{3 \sqrt {5} \sqrt {1-3 x+x^2}} \]
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Time = 2.45 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.02
method | result | size |
default | \(-\frac {\sqrt {3-2 x}\, \sqrt {x^{2}-3 x +1}\, \sqrt {\left (-2 x +3+\sqrt {5}\right ) \sqrt {5}}\, \sqrt {\left (-3+2 x \right ) \sqrt {5}}\, \sqrt {\left (2 x -3+\sqrt {5}\right ) \sqrt {5}}\, \sqrt {5}\, E\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\left (-2 x +3+\sqrt {5}\right ) \sqrt {5}}}{10}, \sqrt {2}\right )}{5 \left (2 x^{3}-9 x^{2}+11 x -3\right )}\) | \(105\) |
elliptic | \(\frac {\sqrt {-\left (-3+2 x \right ) \left (x^{2}-3 x +1\right )}\, \left (-\frac {6 \sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {10}\, \sqrt {\left (x -\frac {3}{2}\right ) \sqrt {5}}\, \sqrt {\left (x -\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, F\left (\frac {\sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}, \sqrt {2}\right )}{25 \sqrt {-2 x^{3}+9 x^{2}-11 x +3}}+\frac {4 \sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {10}\, \sqrt {\left (x -\frac {3}{2}\right ) \sqrt {5}}\, \sqrt {\left (x -\frac {3}{2}+\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \left (\frac {\sqrt {5}\, E\left (\frac {\sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}, \sqrt {2}\right )}{2}+\frac {3 F\left (\frac {\sqrt {-5 \left (x -\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}}{5}, \sqrt {2}\right )}{2}\right )}{25 \sqrt {-2 x^{3}+9 x^{2}-11 x +3}}\right )}{\sqrt {3-2 x}\, \sqrt {x^{2}-3 x +1}}\) | \(228\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.14 \[ \int \frac {\sqrt {3-2 x}}{\sqrt {1-3 x+x^2}} \, dx=-2 \, \sqrt {-2} {\rm weierstrassZeta}\left (5, 0, {\rm weierstrassPInverse}\left (5, 0, x - \frac {3}{2}\right )\right ) \]
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\[ \int \frac {\sqrt {3-2 x}}{\sqrt {1-3 x+x^2}} \, dx=\int \frac {\sqrt {3 - 2 x}}{\sqrt {x^{2} - 3 x + 1}}\, dx \]
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\[ \int \frac {\sqrt {3-2 x}}{\sqrt {1-3 x+x^2}} \, dx=\int { \frac {\sqrt {-2 \, x + 3}}{\sqrt {x^{2} - 3 \, x + 1}} \,d x } \]
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\[ \int \frac {\sqrt {3-2 x}}{\sqrt {1-3 x+x^2}} \, dx=\int { \frac {\sqrt {-2 \, x + 3}}{\sqrt {x^{2} - 3 \, x + 1}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {3-2 x}}{\sqrt {1-3 x+x^2}} \, dx=\int \frac {\sqrt {3-2\,x}}{\sqrt {x^2-3\,x+1}} \,d x \]
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