Optimal. Leaf size=50 \[ -2^{m+1} 3^{-m-1} \sqrt{-3 x-2} (-x)^{-m} x^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{3 x}{2}+1\right ) \]
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Rubi [A] time = 0.0109464, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {67, 12, 65} \[ -2^{m+1} 3^{-m-1} \sqrt{-3 x-2} (-x)^{-m} x^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{3 x}{2}+1\right ) \]
Antiderivative was successfully verified.
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Rule 67
Rule 12
Rule 65
Rubi steps
\begin{align*} \int \frac{x^m}{\sqrt{-2-3 x}} \, dx &=\left (\left (\frac{2}{3}\right )^m (-x)^{-m} x^m\right ) \int \frac{\left (\frac{3}{2}\right )^m (-x)^m}{\sqrt{-2-3 x}} \, dx\\ &=\left ((-x)^{-m} x^m\right ) \int \frac{(-x)^m}{\sqrt{-2-3 x}} \, dx\\ &=-2^{1+m} 3^{-1-m} \sqrt{-2-3 x} (-x)^{-m} x^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};1+\frac{3 x}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.0228393, size = 48, normalized size = 0.96 \[ -\frac{2}{3} \sqrt{-3 x-2} \left (\frac{1}{2} (-3 x-2)+1\right )^{-m} x^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{3 x}{2}+1\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.016, size = 30, normalized size = 0.6 \begin{align*}{\frac{-{\frac{i}{2}}{x}^{1+m}\sqrt{2}}{1+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{x^{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{x^{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 = 1.01834, size = 41, normalized size = 0.82 \begin{align*} - \frac{\sqrt{2} i x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{3 x e^{i \pi }}{2}} \right )}}{2 \Gamma \left (m + 2\right )} \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{x^{m}}{\sqrt{-3 \, x - 2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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