Integrand size = 23, antiderivative size = 51 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {2 x-3 x^2}} \, dx=\frac {2 \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} \sqrt {x}\right ),-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {d+e x}} \]
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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, 118, 116} \[ \int \frac {1}{\sqrt {d+e x} \sqrt {2 x-3 x^2}} \, dx=\frac {2 \sqrt {\frac {e x}{d}+1} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} \sqrt {x}\right ),-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {d+e x}} \]
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Rule 12
Rule 116
Rule 118
Rule 728
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {2} \sqrt {1-\frac {3 x}{2}} \sqrt {x} \sqrt {d+e x}} \, dx \\ & = \frac {\int \frac {1}{\sqrt {1-\frac {3 x}{2}} \sqrt {x} \sqrt {d+e x}} \, dx}{\sqrt {2}} \\ & = \frac {\sqrt {1+\frac {e x}{d}} \int \frac {1}{\sqrt {1-\frac {3 x}{2}} \sqrt {x} \sqrt {1+\frac {e x}{d}}} \, dx}{\sqrt {2} \sqrt {d+e x}} \\ & = \frac {2 \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {d+e x}} \\ \end{align*}
Time = 4.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.49 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {2 x-3 x^2}} \, dx=-\frac {\sqrt {6-\frac {4}{x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),-\frac {3 d}{2 e}\right )}{\sqrt {-x (-2+3 x)} \sqrt {d+e x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(114\) vs. \(2(41)=82\).
Time = 2.12 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.25
method | result | size |
default | \(-\frac {2 F\left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right ) \sqrt {-\frac {e x}{d}}\, \sqrt {-\frac {\left (-2+3 x \right ) e}{3 d +2 e}}\, \sqrt {\frac {e x +d}{d}}\, d \sqrt {e x +d}\, \sqrt {-x \left (-2+3 x \right )}}{e x \left (3 e \,x^{2}+3 d x -2 e x -2 d \right )}\) | \(115\) |
elliptic | \(\frac {2 \sqrt {-\left (-2+3 x \right ) x \left (e x +d \right )}\, d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {-\frac {2}{3}+x}{-\frac {d}{e}-\frac {2}{3}}}\, \sqrt {-\frac {e x}{d}}\, F\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {2}{3}\right )}}\right )}{\sqrt {-x \left (-2+3 x \right )}\, \sqrt {e x +d}\, e \sqrt {-3 e \,x^{3}-3 d \,x^{2}+2 e \,x^{2}+2 d x}}\) | \(136\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.53 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {2 x-3 x^2}} \, dx=-\frac {2 \, \sqrt {3} \sqrt {-e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (9 \, d^{2} + 6 \, d e + 4 \, e^{2}\right )}}{27 \, e^{2}}, -\frac {8 \, {\left (27 \, d^{3} + 27 \, d^{2} e - 18 \, d e^{2} - 8 \, e^{3}\right )}}{729 \, e^{3}}, \frac {9 \, e x + 3 \, d - 2 \, e}{9 \, e}\right )}{3 \, e} \]
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\[ \int \frac {1}{\sqrt {d+e x} \sqrt {2 x-3 x^2}} \, dx=\int \frac {1}{\sqrt {- x \left (3 x - 2\right )} \sqrt {d + e x}}\, dx \]
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\[ \int \frac {1}{\sqrt {d+e x} \sqrt {2 x-3 x^2}} \, dx=\int { \frac {1}{\sqrt {e x + d} \sqrt {-3 \, x^{2} + 2 \, x}} \,d x } \]
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\[ \int \frac {1}{\sqrt {d+e x} \sqrt {2 x-3 x^2}} \, dx=\int { \frac {1}{\sqrt {e x + d} \sqrt {-3 \, x^{2} + 2 \, x}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {d+e x} \sqrt {2 x-3 x^2}} \, dx=\int \frac {1}{\sqrt {2\,x-3\,x^2}\,\sqrt {d+e\,x}} \,d x \]
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