\(\int \frac {x^n}{\sqrt {a-a x}} \, dx\) [729]

Optimal result

Integrand size = 14, antiderivative size = 30 \[ \int \frac {x^n}{\sqrt {a-a x}} \, dx=-\frac {2 \sqrt {a-a x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-x\right )}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {67} \[ \int \frac {x^n}{\sqrt {a-a x}} \, dx=-\frac {2 \sqrt {a-a x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-x\right )}{a} \]

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {x^n}{\sqrt {a-a x}} \, dx=-\frac {2 \sqrt {a-a x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},1-x\right )}{a} \]

[In]

Integrate[x^n/Sqrt[a - a*x],x]

[Out]

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

Maple [F]

\[\int \frac {x^{n}}{\sqrt {-a x +a}}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {x^n}{\sqrt {a-a x}} \, dx=- \frac {2 i \sqrt {x - 1} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - n \\ \frac {3}{2} \end {matrix}\middle | {\left (x - 1\right ) e^{i \pi }} \right )}}{\sqrt {a}} \]

[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)

Maxima [F]

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

[In]

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

[Out]

integrate(x^n/sqrt(-a*x + a), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^n/sqrt(-a*x + a), x)

Mupad [F(-1)]

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

[In]

int(x^n/(a - a*x)^(1/2),x)

[Out]

int(x^n/(a - a*x)^(1/2), x)