\(\int \frac {x^n}{\sqrt {1-x}} \, dx\) [728]

Optimal result

Integrand size = 13, antiderivative size = 26 \[ \int \frac {x^n}{\sqrt {1-x}} \, dx=-2 \sqrt {1-x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-x\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {67} \[ \int \frac {x^n}{\sqrt {1-x}} \, dx=-2 \sqrt {1-x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-x\right ) \]

[In]

Int[x^n/Sqrt[1 - x],x]

[Out]

-2*Sqrt[1 - x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - x]

Rule 67

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

Rubi steps \begin{align*} \text {integral}& = -2 \sqrt {1-x} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-x\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {x^n}{\sqrt {1-x}} \, dx=-2 \sqrt {1-x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-x\right ) \]

[In]

Integrate[x^n/Sqrt[1 - x],x]

[Out]

-2*Sqrt[1 - x]*Hypergeometric2F1[1/2, -n, 3/2, 1 - x]

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88

method	result	size
meijerg	\(\frac {x^{1+n} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},1+n ;2+n ;x \right )}{1+n}\)	\(23\)

[In]

int(x^n/(1-x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(-x^n*sqrt(-x + 1)/(x - 1), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.72 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {x^n}{\sqrt {1-x}} \, dx=- 2 i \sqrt {x - 1} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - n \\ \frac {3}{2} \end {matrix}\middle | {\left (x - 1\right ) e^{i \pi }} \right )} \]

[In]

integrate(x**n/(1-x)**(1/2),x)

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(x^n/sqrt(-x + 1), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^n/sqrt(-x + 1), x)

Mupad [F(-1)]

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

[In]

int(x^n/(1 - x)^(1/2),x)

[Out]

int(x^n/(1 - x)^(1/2), x)