Integrand size = 23, antiderivative size = 51 \[ \int \frac {\sqrt {d+e x}}{\sqrt {2 x-3 x^2}} \, dx=\frac {2 \sqrt {d+e x} E\left (\arcsin \left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {1+\frac {e x}{d}}} \]
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Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {728, 12, 113, 111} \[ \int \frac {\sqrt {d+e x}}{\sqrt {2 x-3 x^2}} \, dx=\frac {2 \sqrt {d+e x} E\left (\arcsin \left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {\frac {e x}{d}+1}} \]
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Rule 12
Rule 111
Rule 113
Rule 728
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {d+e x}}{\sqrt {2} \sqrt {1-\frac {3 x}{2}} \sqrt {x}} \, dx \\ & = \frac {\int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {3 x}{2}} \sqrt {x}} \, dx}{\sqrt {2}} \\ & = \frac {\sqrt {d+e x} \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {1-\frac {3 x}{2}} \sqrt {x}} \, dx}{\sqrt {2} \sqrt {1+\frac {e x}{d}}} \\ & = \frac {2 \sqrt {d+e x} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {1+\frac {e x}{d}}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(117\) vs. \(2(51)=102\).
Time = 4.85 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.29 \[ \int \frac {\sqrt {d+e x}}{\sqrt {2 x-3 x^2}} \, dx=\frac {2 \sqrt {-\frac {d}{e}} (-2+3 x) (d+e x)-2 d \sqrt {9-\frac {6}{x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (\arcsin \left (\frac {\sqrt {-\frac {d}{e}}}{\sqrt {x}}\right )|-\frac {2 e}{3 d}\right )}{3 \sqrt {-\frac {d}{e}} \sqrt {-x (-2+3 x)} \sqrt {d+e x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(214\) vs. \(2(41)=82\).
Time = 1.96 (sec) , antiderivative size = 215, normalized size of antiderivative = 4.22
method | result | size |
default | \(-\frac {2 \sqrt {e x +d}\, \sqrt {-x \left (-2+3 x \right )}\, d \sqrt {\frac {e x +d}{d}}\, \sqrt {-\frac {\left (-2+3 x \right ) e}{3 d +2 e}}\, \sqrt {-\frac {e x}{d}}\, \left (3 d F\left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right )+2 F\left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right ) e -3 E\left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right ) d -2 E\left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right ) e \right )}{3 e x \left (3 e \,x^{2}+3 d x -2 e x -2 d \right )}\) | \(215\) |
elliptic | \(\frac {\sqrt {-\left (-2+3 x \right ) x \left (e x +d \right )}\, \left (\frac {2 d^{2} \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {-\frac {2}{3}+x}{-\frac {d}{e}-\frac {2}{3}}}\, \sqrt {-\frac {e x}{d}}\, F\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {2}{3}\right )}}\right )}{e \sqrt {-3 e \,x^{3}-3 d \,x^{2}+2 e \,x^{2}+2 d x}}+\frac {2 d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {-\frac {2}{3}+x}{-\frac {d}{e}-\frac {2}{3}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}-\frac {2}{3}\right ) E\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {2}{3}\right )}}\right )+\frac {2 F\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {2}{3}\right )}}\right )}{3}\right )}{\sqrt {-3 e \,x^{3}-3 d \,x^{2}+2 e \,x^{2}+2 d x}}\right )}{\sqrt {-x \left (-2+3 x \right )}\, \sqrt {e x +d}}\) | \(285\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 211, normalized size of antiderivative = 4.14 \[ \int \frac {\sqrt {d+e x}}{\sqrt {2 x-3 x^2}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {3} {\left (3 \, d + e\right )} \sqrt {-e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (9 \, d^{2} + 6 \, d e + 4 \, e^{2}\right )}}{27 \, e^{2}}, -\frac {8 \, {\left (27 \, d^{3} + 27 \, d^{2} e - 18 \, d e^{2} - 8 \, e^{3}\right )}}{729 \, e^{3}}, \frac {9 \, e x + 3 \, d - 2 \, e}{9 \, e}\right ) - 9 \, \sqrt {3} \sqrt {-e} e {\rm weierstrassZeta}\left (\frac {4 \, {\left (9 \, d^{2} + 6 \, d e + 4 \, e^{2}\right )}}{27 \, e^{2}}, -\frac {8 \, {\left (27 \, d^{3} + 27 \, d^{2} e - 18 \, d e^{2} - 8 \, e^{3}\right )}}{729 \, e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (9 \, d^{2} + 6 \, d e + 4 \, e^{2}\right )}}{27 \, e^{2}}, -\frac {8 \, {\left (27 \, d^{3} + 27 \, d^{2} e - 18 \, d e^{2} - 8 \, e^{3}\right )}}{729 \, e^{3}}, \frac {9 \, e x + 3 \, d - 2 \, e}{9 \, e}\right )\right )\right )}}{27 \, e} \]
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\[ \int \frac {\sqrt {d+e x}}{\sqrt {2 x-3 x^2}} \, dx=\int \frac {\sqrt {d + e x}}{\sqrt {- x \left (3 x - 2\right )}}\, dx \]
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\[ \int \frac {\sqrt {d+e x}}{\sqrt {2 x-3 x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {-3 \, x^{2} + 2 \, x}} \,d x } \]
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\[ \int \frac {\sqrt {d+e x}}{\sqrt {2 x-3 x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {-3 \, x^{2} + 2 \, x}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {d+e x}}{\sqrt {2 x-3 x^2}} \, dx=\int \frac {\sqrt {d+e\,x}}{\sqrt {2\,x-3\,x^2}} \,d x \]
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