Optimal. Leaf size=34 \[ -\frac{(-x)^{m+1} \, _2F_1\left (\frac{1}{2},m+1;m+2;\frac{3 x}{2}\right )}{\sqrt{2} (m+1)} \]
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Rubi [A] time = 0.0048189, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {64} \[ -\frac{(-x)^{m+1} \, _2F_1\left (\frac{1}{2},m+1;m+2;\frac{3 x}{2}\right )}{\sqrt{2} (m+1)} \]
Antiderivative was successfully verified.
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Rule 64
Rubi steps
\begin{align*} \int \frac{(-x)^m}{\sqrt{2-3 x}} \, dx &=-\frac{(-x)^{1+m} \, _2F_1\left (\frac{1}{2},1+m;2+m;\frac{3 x}{2}\right )}{\sqrt{2} (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0050443, size = 32, normalized size = 0.94 \[ \frac{x (-x)^m \, _2F_1\left (\frac{1}{2},m+1;m+2;\frac{3 x}{2}\right )}{\sqrt{2} (m+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 30, normalized size = 0.9 \begin{align*}{\frac{\sqrt{2} \left ( -x \right ) ^{m}x}{2+2\,m}{\mbox{$_2$F$_1$}({\frac{1}{2}},1+m;\,2+m;\,{\frac{3\,x}{2}})}} \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{\left (-x\right )^{m}}{\sqrt{-3 \, x + 2}}\,{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{\left (-x\right )^{m} \sqrt{-3 \, x + 2}}{3 \, x - 2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.04734, size = 53, normalized size = 1.56 \begin{align*} - \frac{2 \cdot 2^{m} \sqrt{3} \cdot 3^{- m} i \sqrt{x - \frac{2}{3}} e^{i \pi m}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - m \\ \frac{3}{2} \end{matrix}\middle |{\frac{3 \left (x - \frac{2}{3}\right ) e^{i \pi }}{2}} \right )}}{3} \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{\left (-x\right )^{m}}{\sqrt{-3 \, x + 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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