\(\int \frac {(1-x)^n}{\sqrt {1+x}} \, dx\) [1885]

Optimal result

Integrand size = 15, antiderivative size = 30 \[ \int \frac {(1-x)^n}{\sqrt {1+x}} \, dx=2^{1+n} \sqrt {1+x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},\frac {1+x}{2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {71} \[ \int \frac {(1-x)^n}{\sqrt {1+x}} \, dx=2^{n+1} \sqrt {x+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},\frac {x+1}{2}\right ) \]

[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 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], 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*} \text {integral}& = 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] (verified)

Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {(1-x)^n}{\sqrt {1+x}} \, dx=2^{1+n} \sqrt {1+x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},\frac {1+x}{2}\right ) \]

[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]

Maple [F]

\[\int \frac {\left (1-x \right )^{n}}{\sqrt {1+x}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(1-x)^n}{\sqrt {1+x}} \, dx=\int { \frac {{\left (-x + 1\right )}^{n}}{\sqrt {x + 1}} \,d x } \]

[In]

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

[Out]

integral((-x + 1)^n/sqrt(x + 1), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {(1-x)^n}{\sqrt {1+x}} \, dx=2 \cdot 2^{n} \sqrt {x + 1} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - n \\ \frac {3}{2} \end {matrix}\middle | {\frac {\left (x + 1\right ) e^{2 i \pi }}{2}} \right )} \]

[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)

Maxima [F]

\[ \int \frac {(1-x)^n}{\sqrt {1+x}} \, dx=\int { \frac {{\left (-x + 1\right )}^{n}}{\sqrt {x + 1}} \,d x } \]

[In]

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

[Out]

integrate((-x + 1)^n/sqrt(x + 1), x)

Giac [F]

\[ \int \frac {(1-x)^n}{\sqrt {1+x}} \, dx=\int { \frac {{\left (-x + 1\right )}^{n}}{\sqrt {x + 1}} \,d x } \]

[In]

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

[Out]

integrate((-x + 1)^n/sqrt(x + 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(1-x)^n}{\sqrt {1+x}} \, dx=\int \frac {{\left (1-x\right )}^n}{\sqrt {x+1}} \,d x \]

[In]

int((1 - x)^n/(x + 1)^(1/2),x)

[Out]

int((1 - x)^n/(x + 1)^(1/2), x)