Integrand size = 27, antiderivative size = 87 \[ \int \frac {1}{\sqrt {e x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\frac {2 \sqrt {a} \sqrt {1-\frac {b x}{a}} \sqrt {1+\frac {b x}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),-1\right )}{\sqrt {b} \sqrt {e} \sqrt {a-b x} \sqrt {a+b x}} \]
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Time = 0.02 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {118, 117} \[ \int \frac {1}{\sqrt {e x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\frac {2 \sqrt {a} \sqrt {1-\frac {b x}{a}} \sqrt {\frac {b x}{a}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right ),-1\right )}{\sqrt {b} \sqrt {e} \sqrt {a-b x} \sqrt {a+b x}} \]
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Rule 117
Rule 118
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {1-\frac {b x}{a}} \sqrt {1+\frac {b x}{a}}\right ) \int \frac {1}{\sqrt {e x} \sqrt {1-\frac {b x}{a}} \sqrt {1+\frac {b x}{a}}} \, dx}{\sqrt {a-b x} \sqrt {a+b x}} \\ & = \frac {2 \sqrt {a} \sqrt {1-\frac {b x}{a}} \sqrt {1+\frac {b x}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )\right |-1\right )}{\sqrt {b} \sqrt {e} \sqrt {a-b x} \sqrt {a+b x}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\sqrt {e x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\frac {2 x \sqrt {1-\frac {b^2 x^2}{a^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b^2 x^2}{a^2}\right )}{\sqrt {e x} \sqrt {a-b x} \sqrt {a+b x}} \]
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Time = 1.50 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.06
method | result | size |
default | \(\frac {\sqrt {-b x +a}\, \sqrt {b x +a}\, a \sqrt {\frac {b x +a}{a}}\, \sqrt {2}\, \sqrt {\frac {-b x +a}{a}}\, \sqrt {-\frac {b x}{a}}\, F\left (\sqrt {\frac {b x +a}{a}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {e x}\, \left (-b^{2} x^{2}+a^{2}\right )}\) | \(92\) |
elliptic | \(\frac {\sqrt {e x \left (-b^{2} x^{2}+a^{2}\right )}\, a \sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}\, \sqrt {-\frac {2 \left (-\frac {a}{b}+x \right ) b}{a}}\, \sqrt {-\frac {b x}{a}}\, F\left (\sqrt {\frac {\left (x +\frac {a}{b}\right ) b}{a}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {e x}\, \sqrt {-b x +a}\, \sqrt {b x +a}\, b \sqrt {-b^{2} e \,x^{3}+a^{2} e x}}\) | \(120\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt {e x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=-\frac {2 \, \sqrt {-b^{2} e} {\rm weierstrassPInverse}\left (\frac {4 \, a^{2}}{b^{2}}, 0, x\right )}{b^{2} e} \]
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Time = 11.41 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\sqrt {e x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {a^{2}}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {a} \sqrt {b} \sqrt {e}} - \frac {i {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {a^{2} e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} \sqrt {a} \sqrt {b} \sqrt {e}} \]
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\[ \int \frac {1}{\sqrt {e x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {-b x + a} \sqrt {e x}} \,d x } \]
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\[ \int \frac {1}{\sqrt {e x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\int { \frac {1}{\sqrt {b x + a} \sqrt {-b x + a} \sqrt {e x}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {e x} \sqrt {a-b x} \sqrt {a+b x}} \, dx=\int \frac {1}{\sqrt {e\,x}\,\sqrt {a+b\,x}\,\sqrt {a-b\,x}} \,d x \]
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