Optimal. Leaf size=24 \[ -\frac{2 \sqrt{b-\frac{a}{x}}}{\sqrt{a-b x}} \]
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Rubi [A] time = 0.0343887, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {515, 23, 30} \[ -\frac{2 \sqrt{b-\frac{a}{x}}}{\sqrt{a-b x}} \]
Antiderivative was successfully verified.
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Rule 515
Rule 23
Rule 30
Rubi steps
\begin{align*} \int \frac{\sqrt{b-\frac{a}{x}}}{x \sqrt{a-b x}} \, dx &=\frac{\left (\sqrt{b-\frac{a}{x}} \sqrt{x}\right ) \int \frac{\sqrt{-a+b x}}{x^{3/2} \sqrt{a-b x}} \, dx}{\sqrt{-a+b x}}\\ &=\frac{\left (\sqrt{b-\frac{a}{x}} \sqrt{x}\right ) \int \frac{1}{x^{3/2}} \, dx}{\sqrt{a-b x}}\\ &=-\frac{2 \sqrt{b-\frac{a}{x}}}{\sqrt{a-b x}}\\ \end{align*}
Mathematica [A] time = 0.012466, size = 24, normalized size = 1. \[ -\frac{2 \sqrt{b-\frac{a}{x}}}{\sqrt{a-b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 24, normalized size = 1. \begin{align*} -2\,{\frac{1}{\sqrt{-bx+a}}\sqrt{-{\frac{-bx+a}{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 2.05264, size = 7, normalized size = 0.29 \begin{align*} \frac{2 i}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48782, size = 62, normalized size = 2.58 \begin{align*} \frac{2 \, \sqrt{-b x + a} \sqrt{\frac{b x - a}{x}}}{b x - a} \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{- \frac{a}{x} + b}}{x \sqrt{a - b x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12331, size = 57, normalized size = 2.38 \begin{align*} \frac{2 \,{\left (\frac{b^{3}}{\sqrt{-{\left (b x - a\right )} b - a b}} - \frac{b^{3}}{\sqrt{-a b}}\right )}{\left | b \right |} \mathrm{sgn}\left (x\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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