3.729 \(\int \frac {x^n}{\sqrt {a-a x}} \, dx\)

Optimal. Leaf size=30 \[ -\frac {2 \sqrt {a-a x} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-x\right )}{a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {65} \[ -\frac {2 \sqrt {a-a x} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-x\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[x^n/Sqrt[a - a*x],x]

[Out]

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

Rule 65

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 30, normalized size = 1.00 \[ -\frac {2 \sqrt {a-a x} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-x\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[x^n/Sqrt[a - a*x],x]

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a x + a} x^{n}}{a x - a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{n}}{\sqrt {-a x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^n/sqrt(-a*x + a), x)

________________________________________________________________________________________

maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {x^{n}}{\sqrt {-a x +a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{n}}{\sqrt {-a x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^n/sqrt(-a*x + a), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^n}{\sqrt {a-a\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^n/(a - a*x)^(1/2),x)

[Out]

int(x^n/(a - a*x)^(1/2), x)

________________________________________________________________________________________

sympy [C]  time = 1.16, size = 31, normalized size = 1.03 \[ - \frac {2 i \sqrt {x - 1} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - n \\ \frac {3}{2} \end {matrix}\middle | {\left (x - 1\right ) e^{i \pi }} \right )}}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________