Optimal. Leaf size=37 \[ -\frac{2 E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{b x}}{\sqrt{-b}}\right )\right |-1\right )}{\sqrt{-b} \sqrt{c}} \]
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Rubi [A] time = 0.0217859, antiderivative size = 37, 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 = {110} \[ -\frac{2 E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{b x}}{\sqrt{-b}}\right )\right |-1\right )}{\sqrt{-b} \sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 110
Rubi steps
\begin{align*} \int \frac{\sqrt{1-c x}}{\sqrt{b x} \sqrt{1+c x}} \, dx &=-\frac{2 E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{b x}}{\sqrt{-b}}\right )\right |-1\right )}{\sqrt{-b} \sqrt{c}}\\ \end{align*}
Mathematica [C] time = 0.0316616, size = 52, normalized size = 1.41 \[ -\frac{2 x \left (c x \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};c^2 x^2\right )-3 \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};c^2 x^2\right )\right )}{3 \sqrt{b x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.012, size = 33, normalized size = 0.9 \begin{align*} 2\,{\frac{\sqrt{2}\sqrt{-cx}{\it EllipticE} \left ( \sqrt{cx+1},1/2\,\sqrt{2} \right ) }{c\sqrt{bx}}} \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{\sqrt{-c x + 1}}{\sqrt{b x} \sqrt{c x + 1}}\,{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} \sqrt{c x + 1} \sqrt{-c x + 1}}{b c x^{2} + b x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- c x + 1}}{\sqrt{b x} \sqrt{c x + 1}}\, dx \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{\sqrt{-c x + 1}}{\sqrt{b x} \sqrt{c x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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