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

Optimal. Leaf size=26 \[ -2 \sqrt {1-x} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-x\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {67} \begin {gather*} -2 \sqrt {1-x} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-x\right ) \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {x^n}{\sqrt {1-x}} \, dx &=-2 \sqrt {1-x} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-x\right )\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 26, normalized size = 1.00 \begin {gather*} -2 \sqrt {1-x} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^n/Sqrt[1 - x],x]

[Out]

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

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Maple [A]
time = 0.11, size = 23, normalized size = 0.88

method	result	size
meijerg	\(\frac {x^{1+n} \hypergeom \left (\left [\frac {1}{2}, 1+n \right ], \left [2+n \right ], x\right )}{1+n}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^n/sqrt(-x + 1), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.48, size = 26, normalized size = 1.00 \begin {gather*} - 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^n/sqrt(-x + 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^n}{\sqrt {1-x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^n/(1 - x)^(1/2),x)

[Out]

int(x^n/(1 - x)^(1/2), x)

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