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 = 0.0067607, 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.

[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]

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

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.

[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 = 0.031, 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.

[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. \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.

[In]

integrate((1-x)^n/(1+x)^(1/2),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. \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.

[In]

integrate((1-x)^n/(1+x)^(1/2),x, algorithm="fricas")

[Out]

integral((-x + 1)^n/sqrt(x + 1), x)

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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.

[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 [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.

[In]

integrate((1-x)^n/(1+x)^(1/2),x, algorithm="giac")

[Out]

integrate((-x + 1)^n/sqrt(x + 1), x)