Optimal. Leaf size=35 \[ -2^{n+1} \sqrt{1-x} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{1-x}{2}\right ) \]
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Rubi [A] time = 0.0055015, antiderivative size = 35, 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 = {69} \[ -2^{n+1} \sqrt{1-x} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{1-x}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 69
Rubi steps
\begin{align*} \int \frac{(1+x)^n}{\sqrt{1-x}} \, dx &=-2^{1+n} \sqrt{1-x} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{1-x}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.0063314, size = 35, normalized size = 1. \[ -2^{n+1} \sqrt{1-x} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{1-x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 1+x \right ) ^{n}{\frac{1}{\sqrt{1-x}}}}\, dx \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 + 1\right )}^{n}}{\sqrt{-x + 1}}\,{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 + 1\right )}^{n} \sqrt{-x + 1}}{x - 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.04889, size = 31, normalized size = 0.89 \begin{align*} - 2 \cdot 2^{n} i \sqrt{x - 1}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - n \\ \frac{3}{2} \end{matrix}\middle |{\frac{\left (x - 1\right ) e^{i \pi }}{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{{\left (x + 1\right )}^{n}}{\sqrt{-x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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