Optimal. Leaf size=87 \[ \frac{2 \sqrt{a} \sqrt{1-\frac{b x}{a}} \sqrt{\frac{b x}{a}+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{e x}}{\sqrt{a} \sqrt{e}}\right ),-1\right )}{\sqrt{b} \sqrt{e} \sqrt{a-b x} \sqrt{a+b x}} \]
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Rubi [A] time = 0.0307264, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {117, 116} \[ \frac{2 \sqrt{a} \sqrt{1-\frac{b x}{a}} \sqrt{\frac{b x}{a}+1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b} \sqrt{e x}}{\sqrt{a} \sqrt{e}}\right )\right |-1\right )}{\sqrt{b} \sqrt{e} \sqrt{a-b x} \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Rule 117
Rule 116
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{e x} \sqrt{a-b x} \sqrt{a+b x}} \, dx &=\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*}
Mathematica [C] time = 0.0189864, size = 66, normalized size = 0.76 \[ \frac{2 x \sqrt{1-\frac{b^2 x^2}{a^2}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{b^2 x^2}{a^2}\right )}{\sqrt{e x} \sqrt{a-b x} \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 93, normalized size = 1.1 \begin{align*} -{\frac{a}{b \left ({b}^{2}{x}^{2}-{a}^{2} \right ) }\sqrt{-bx+a}\sqrt{bx+a}\sqrt{{\frac{bx+a}{a}}}\sqrt{-2\,{\frac{bx-a}{a}}}\sqrt{-{\frac{bx}{a}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+a}{a}}},{\frac{\sqrt{2}}{2}} \right ){\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 + a} \sqrt{-b x + a} \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 + a} \sqrt{-b x + a} \sqrt{e x}}{b^{2} e x^{3} - a^{2} e x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.3597, size = 109, normalized size = 1.25 \begin{align*} \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}} \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 + a} \sqrt{-b x + a} \sqrt{e x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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