Optimal. Leaf size=42 \[ \frac{\sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{e x}}{\sqrt{2} \sqrt{e}}\right ),-1\right )}{\sqrt{b} \sqrt{e}} \]
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Rubi [A] time = 0.0122834, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {116} \[ \frac{\sqrt{2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{e x}}{\sqrt{2} \sqrt{e}}\right )\right |-1\right )}{\sqrt{b} \sqrt{e}} \]
Antiderivative was successfully verified.
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Rule 116
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{e x} \sqrt{2-b x} \sqrt{2+b x}} \, dx &=\frac{\sqrt{2} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{e x}}{\sqrt{2} \sqrt{e}}\right )\right |-1\right )}{\sqrt{b} \sqrt{e}}\\ \end{align*}
Mathematica [C] time = 0.0087796, size = 29, normalized size = 0.69 \[ \frac{x \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b^2 x^2}{4}\right )}{\sqrt{e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 34, normalized size = 0.8 \begin{align*}{\frac{1}{b}{\it EllipticF} \left ({\frac{\sqrt{2}}{2}\sqrt{bx+2}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-bx}{\frac{1}{\sqrt{ex}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x + 2} \sqrt{-b x + 2} \sqrt{e x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{b x + 2} \sqrt{-b x + 2} \sqrt{e x}}{b^{2} e x^{3} - 4 \, e x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 13.4268, size = 105, normalized size = 2.5 \begin{align*} \frac{\sqrt{2} 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{4}{b^{2} x^{2}}} \right )}}{8 \pi ^{\frac{3}{2}} \sqrt{b} \sqrt{e}} - \frac{\sqrt{2} 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{4 e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{8 \pi ^{\frac{3}{2}} \sqrt{b} \sqrt{e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x + 2} \sqrt{-b x + 2} \sqrt{e x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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