3.727 \(\int \frac{(-x)^m}{\sqrt{-2-3 x}} \, dx\)

Optimal. Leaf size=37 \[ -\left (\frac{3}{2}\right )^{-m-1} \sqrt{-3 x-2} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{3 x}{2}+1\right ) \]

[Out]

-((3/2)^(-1 - m)*Sqrt[-2 - 3*x]*Hypergeometric2F1[1/2, -m, 3/2, 1 + (3*x)/2])

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Rubi [A]  time = 0.0046895, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {65} \[ -\left (\frac{3}{2}\right )^{-m-1} \sqrt{-3 x-2} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{3 x}{2}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-x)^m/Sqrt[-2 - 3*x],x]

[Out]

-((3/2)^(-1 - m)*Sqrt[-2 - 3*x]*Hypergeometric2F1[1/2, -m, 3/2, 1 + (3*x)/2])

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

Mathematica [A]  time = 0.0143071, size = 57, normalized size = 1.54 \[ -\frac{2}{3} \sqrt{-3 x-2} \left (\frac{1}{2} (-3 x-2)+1\right )^{-m} x^{-m} \left (-x^2\right )^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{3 x}{2}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-x)^m/Sqrt[-2 - 3*x],x]

[Out]

(-2*Sqrt[-2 - 3*x]*(-x^2)^m*Hypergeometric2F1[1/2, -m, 3/2, 1 + (3*x)/2])/(3*(1 + (-2 - 3*x)/2)^m*x^m)

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Maple [C]  time = 0.017, size = 31, normalized size = 0.8 \begin{align*}{\frac{-{\frac{i}{2}}\sqrt{2} \left ( -x \right ) ^{m}x}{1+m}{\mbox{$_2$F$_1$}({\frac{1}{2}},1+m;\,2+m;\,-{\frac{3\,x}{2}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x)^m/(-2-3*x)^(1/2),x)

[Out]

-1/2*I*2^(1/2)*(-x)^m/(1+m)*x*hypergeom([1/2,1+m],[2+m],-3/2*x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (-x\right )^{m}}{\sqrt{-3 \, x - 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x)^m/(-2-3*x)^(1/2),x, algorithm="maxima")

[Out]

integrate((-x)^m/sqrt(-3*x - 2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\left (-x\right )^{m} \sqrt{-3 \, x - 2}}{3 \, x + 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x)^m/(-2-3*x)^(1/2),x, algorithm="fricas")

[Out]

integral(-(-x)^m*sqrt(-3*x - 2)/(3*x + 2), x)

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Sympy [C]  time = 1.05786, size = 48, normalized size = 1.3 \begin{align*} - \frac{2 \cdot 2^{m} \sqrt{3} \cdot 3^{- m} i \sqrt{x + \frac{2}{3}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - m \\ \frac{3}{2} \end{matrix}\middle |{\frac{3 \left (x + \frac{2}{3}\right ) e^{2 i \pi }}{2}} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x)**m/(-2-3*x)**(1/2),x)

[Out]

-2*2**m*sqrt(3)*3**(-m)*I*sqrt(x + 2/3)*hyper((1/2, -m), (3/2,), 3*(x + 2/3)*exp_polar(2*I*pi)/2)/3

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (-x\right )^{m}}{\sqrt{-3 \, x - 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x)^m/(-2-3*x)^(1/2),x, algorithm="giac")

[Out]

integrate((-x)^m/sqrt(-3*x - 2), x)