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 = 0.0055015, 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, 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.

[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 = 0.0063314, 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 = 0.028, 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 - 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 - 1), x)

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Sympy [C]  time = 2.04889, size = 31, normalized size = 0.89 \begin{align*} - 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 )} \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*I*sqrt(x - 1)*hyper((1/2, -n), (3/2,), (x - 1)*exp_polar(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)