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 = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 1.00 \[ -2^{n+1} \sqrt {1-x} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};\frac {1-x}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (x + 1\right )}^{n} \sqrt {-x + 1}}{x - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (x + 1\right )}^{n}}{\sqrt {-x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (x +1\right )^{n}}{\sqrt {-x +1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (x + 1\right )}^{n}}{\sqrt {-x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (x+1\right )}^n}{\sqrt {1-x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.12, size = 31, normalized size = 0.89 \[ - 2 \cdot 2^{n} i \sqrt {x - 1} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - n \\ \frac {3}{2} \end {matrix}\middle | {\frac {\left (x - 1\right ) e^{i \pi }}{2}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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