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 = 0.0209464, 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 \[ -\frac{2 \sqrt{a-a x} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};1-x\right )}{a} \]

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

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Rubi in Sympy [A]  time = 3.06873, size = 22, normalized size = 0.73 \[ - \frac{2 \sqrt{- a x + a}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- x + 1} \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**n/(-a*x+a)**(1/2),x)

[Out]

-2*sqrt(-a*x + a)*hyper((-n, 1/2), (3/2,), -x + 1)/a

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Mathematica [A]  time = 0.0231201, 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 = 0.036, size = 0, normalized size = 0. \[ \int{{x}^{n}{\frac{1}{\sqrt{-ax+a}}}}\, dx \]

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^n/sqrt(-a*x + a),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. \[{\rm integral}\left (\frac{x^{n}}{\sqrt{-a x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^n/sqrt(-a*x + a),x, algorithm="fricas")

[Out]

integral(x^n/sqrt(-a*x + a), x)

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Sympy [A]  time = 3.33139, 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)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{n}}{\sqrt{-a x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^n/sqrt(-a*x + a),x, algorithm="giac")

[Out]

integrate(x^n/sqrt(-a*x + a), x)