3.1886 \(\int \frac{(1+x)^n}{\sqrt{1-x}} \, dx\)

Optimal. Leaf size=35 \[ -2^{n+1} \sqrt{1-x} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{1-x}{2}\right ) \]

[Out]

-(2^(1 + n)*Sqrt[1 - x]*Hypergeometric2F1[1/2, -n, 3/2, (1 - x)/2])

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Rubi [A]  time = 0.0208974, 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 \[ -2^{n+1} \sqrt{1-x} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{1-x}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)^n/Sqrt[1 - x],x]

[Out]

-(2^(1 + n)*Sqrt[1 - x]*Hypergeometric2F1[1/2, -n, 3/2, (1 - x)/2])

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Rubi in Sympy [A]  time = 2.77172, size = 26, normalized size = 0.74 \[ - 2 \cdot 2^{n} \sqrt{- x + 1}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{x}{2} + \frac{1}{2}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)**n/(1-x)**(1/2),x)

[Out]

-2*2**n*sqrt(-x + 1)*hyper((-n, 1/2), (3/2,), -x/2 + 1/2)

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Mathematica [A]  time = 0.0170266, 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.

[In]  Integrate[(1 + x)^n/Sqrt[1 - x],x]

[Out]

-(2^(1 + n)*Sqrt[1 - x]*Hypergeometric2F1[1/2, -n, 3/2, (1 - x)/2])

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{ \left ( 1+x \right ) ^{n}{\frac{1}{\sqrt{1-x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^n/(1-x)^(1/2),x)

[Out]

int((1+x)^n/(1-x)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x + 1\right )}^{n}}{\sqrt{-x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^n/sqrt(-x + 1),x, algorithm="maxima")

[Out]

integrate((x + 1)^n/sqrt(-x + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (x + 1\right )}^{n}}{\sqrt{-x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^n/sqrt(-x + 1),x, algorithm="fricas")

[Out]

integral((x + 1)^n/sqrt(-x + 1), x)

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Sympy [A]  time = 3.62835, 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.

[In]  integrate((1+x)**n/(1-x)**(1/2),x)

[Out]

-2*2**n*I*sqrt(x - 1)*hyper((1/2, -n), (3/2,), (x - 1)*exp_polar(I*pi)/2)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x + 1\right )}^{n}}{\sqrt{-x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^n/sqrt(-x + 1),x, algorithm="giac")

[Out]

integrate((x + 1)^n/sqrt(-x + 1), x)