3.1885 \(\int \frac{(1-x)^n}{\sqrt{1+x}} \, dx\)

Optimal. Leaf size=30 \[ 2^{n+1} \sqrt{x+1} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{x+1}{2}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0220427, 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 \[ 2^{n+1} \sqrt{x+1} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{x+1}{2}\right ) \]

Antiderivative was successfully verified.

[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 = 2.832, size = 24, normalized size = 0.8 \[ 2 \cdot 2^{n} \sqrt{x + 1}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{x}{2} + \frac{1}{2}} \right )} \]

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 = 0.0199474, 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.

[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 = 0.039, 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 = 3.622, size = 29, normalized size = 0.97 \[ 2 \cdot 2^{n} \sqrt{x + 1}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - n \\ \frac{3}{2} \end{matrix}\middle |{\frac{\left (x + 1\right ) e^{2 i \pi }}{2}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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