Optimal. Leaf size=29 \[ \frac{2 x^2 \sqrt{b-\frac{a}{x}}}{3 \sqrt{a-b x}} \]
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Rubi [A] time = 0.023956, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {515, 23, 30} \[ \frac{2 x^2 \sqrt{b-\frac{a}{x}}}{3 \sqrt{a-b x}} \]
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{x} \sqrt{-a+b x}}{\sqrt{a-b x}} \, dx}{\sqrt{-a+b x}}\\ &=\frac{\left (\sqrt{b-\frac{a}{x}} \sqrt{x}\right ) \int \sqrt{x} \, dx}{\sqrt{a-b x}}\\ &=\frac{2 \sqrt{b-\frac{a}{x}} x^2}{3 \sqrt{a-b x}}\\ \end{align*}
Mathematica [A] time = 0.0119389, size = 29, normalized size = 1. \[ \frac{2 x^2 \sqrt{b-\frac{a}{x}}}{3 \sqrt{a-b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 27, normalized size = 0.9 \begin{align*}{\frac{2\,{x}^{2}}{3}\sqrt{-{\frac{-bx+a}{x}}}{\frac{1}{\sqrt{-bx+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.18344, size = 7, normalized size = 0.24 \begin{align*} -\frac{2}{3} i \, x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47565, size = 72, normalized size = 2.48 \begin{align*} -\frac{2 \, \sqrt{-b x + a} x^{2} \sqrt{\frac{b x - a}{x}}}{3 \,{\left (b x - a\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{x \sqrt{- \frac{a}{x} + b}}{\sqrt{a - b x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13623, size = 76, normalized size = 2.62 \begin{align*} \frac{2 \, \sqrt{-a b} a{\left | b \right |} \mathrm{sgn}\left (x\right )}{3 \, b^{3}} - \frac{2 \,{\left (\sqrt{-a b} a + \frac{{\left (-{\left (b x - a\right )} b - a b\right )}^{\frac{3}{2}}}{b}\right )}{\left | b \right |} \mathrm{sgn}\left (x\right )}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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