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

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Rubi [A]  time = 0.0044462, antiderivative size = 30, normalized size of antiderivative = 1., 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 = 0.0073741, size = 30, normalized size = 1. \[ -\frac{2 \sqrt{a-a x} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-x\right )}{a} \]

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

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{{x}^{n}{\frac{1}{\sqrt{-ax+a}}}}\, dx \end{align*}

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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{n}}{\sqrt{-a x + a}}\,{d x} \end{align*}

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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a x + a} x^{n}}{a x - a}, x\right ) \end{align*}

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)

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Sympy [C]  time = 0.995525, size = 31, normalized size = 1.03 \begin{align*} - \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}} \end{align*}

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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{n}}{\sqrt{-a x + a}}\,{d x} \end{align*}

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)