Optimal. Leaf size=30 \[ 2^{n+1} \sqrt{x+1} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{x+1}{2}\right ) \]
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Rubi [A] time = 0.0067607, antiderivative size = 30, 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} \[ 2^{n+1} \sqrt{x+1} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{x+1}{2}\right ) \]
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 = 0.0058675, size = 30, normalized size = 1. \[ 2^{n+1} \sqrt{x+1} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{x+1}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.031, 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\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.01301, size = 29, normalized size = 0.97 \begin{align*} 2 \cdot 2^{n} \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^{2 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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