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

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

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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 = {67} \begin {gather*} -\frac {2 \sqrt {a-a x} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-x\right )}{a} \end {gather*}

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 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 {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*}

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Mathematica [A]
time = 0.10, size = 30, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {a-a x} \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-x\right )}{a} \end {gather*}

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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
time = 2.19, size = 26, normalized size = 0.87 \begin {gather*} \frac {-2 I \sqrt {-1+x} \text {hyper}\left [\left \{\frac {1}{2},-n\right \},\left \{\frac {3}{2}\right \},\left (-1+x\right ) \text {exp\_polar}\left [I \text {Pi}\right ]\right ]}{\sqrt {a}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^n/Sqrt[a - a*x],x]')

[Out]

-2 I Sqrt[-1 + x] hyper[{1 / 2, -n}, {3 / 2}, (-1 + x) exp_polar[I Pi]] / Sqrt[a]

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Maple [F]
time = 0.03, 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)

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

[Out]

integrate(x^n/sqrt(-a*x + a), x)

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Fricas [F]
time = 0.32, 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/(-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] Result contains complex when optimal does not.
time = 0.55, size = 31, normalized size = 1.03 \begin {gather*} - \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 {gather*}

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] N/A
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/(-a*x+a)^(1/2),x)

[Out]

Could not integrate

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x^n}{\sqrt {a-a\,x}} \,d x \end {gather*}

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)

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