Optimal. Leaf size=36 \[ \frac{2 x^{m+1} \sqrt{b-\frac{a}{x}}}{(2 m+1) \sqrt{a-b x}} \]
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Rubi [A] time = 0.036967, antiderivative size = 36, 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} \[ \frac{2 x^{m+1} \sqrt{b-\frac{a}{x}}}{(2 m+1) \sqrt{a-b x}} \]
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^m}{\sqrt{a-b x}} \, dx &=\frac{\left (\sqrt{b-\frac{a}{x}} \sqrt{x}\right ) \int \frac{x^{-\frac{1}{2}+m} \sqrt{-a+b x}}{\sqrt{a-b x}} \, dx}{\sqrt{-a+b x}}\\ &=\frac{\left (\sqrt{b-\frac{a}{x}} \sqrt{x}\right ) \int x^{-\frac{1}{2}+m} \, dx}{\sqrt{a-b x}}\\ &=\frac{2 \sqrt{b-\frac{a}{x}} x^{1+m}}{(1+2 m) \sqrt{a-b x}}\\ \end{align*}
Mathematica [A] time = 0.0212461, size = 35, normalized size = 0.97 \[ \frac{x^{m+1} \sqrt{b-\frac{a}{x}}}{\left (m+\frac{1}{2}\right ) \sqrt{a-b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 36, normalized size = 1. \begin{align*} 2\,{\frac{{x}^{1+m}}{ \left ( 1+2\,m \right ) \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 = 1.75715, size = 20, normalized size = 0.56 \begin{align*} \frac{2 \, \sqrt{x} x^{m}}{2 i \, m + i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56237, size = 95, normalized size = 2.64 \begin{align*} \frac{2 \, \sqrt{-b x + a} x x^{m} \sqrt{\frac{b x - a}{x}}}{2 \, a m -{\left (2 \, b m + b\right )} 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{x^{m} \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 = 1.2424, size = 212, normalized size = 5.89 \begin{align*} \frac{2 \, \sqrt{-a b} a{\left | b \right |} e^{\left (m \log \left (\frac{a}{b}\right ) - \log \left (\frac{a}{b}\right )\right )} \mathrm{sgn}\left (x\right )}{2 \, b^{3} m + b^{3}} - \frac{2 \,{\left (\frac{\sqrt{-a b} a e^{\left (m \log \left (\frac{a}{b}\right ) - \log \left (\frac{a}{b}\right )\right )}}{2 \, m + 1} + \frac{{\left (-{\left (b x - a\right )} b - a b\right )}^{\frac{3}{2}} e^{\left (m \log \left (\frac{{\left (b x - a\right )} b + a b}{b^{2}}\right ) - \log \left (\frac{{\left (b x - a\right )} b + a b}{b^{2}}\right )\right )}}{b{\left (2 \, m + 1\right )}}\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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