3.8.27 \(\int \frac {(-x)^m}{\sqrt {-2-3 x}} \, dx\) [727]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 37, normalized size of antiderivative = 1.00, 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 = {67} \begin {gather*} -\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 ) \end {gather*}

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 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)^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*}

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Mathematica [A]
time = 0.01, size = 57, normalized size = 1.54 \begin {gather*} -\frac {2}{3} \left (1+\frac {1}{2} (-2-3 x)\right )^{-m} \sqrt {-2-3 x} x^{-m} \left (-x^2\right )^m \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1+\frac {3 x}{2}\right ) \end {gather*}

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] Result contains complex when optimal does not.
time = 0.12, size = 31, normalized size = 0.84

method	result	size
meijerg	\(-\frac {i \sqrt {2}\, \left (-x \right )^{m} x \hypergeom \left (\left [\frac {1}{2}, 1+m \right ], \left [2+m \right ], -\frac {3 x}{2}\right )}{2 \left (1+m \right )}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, 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)^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 = 0.00, 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)^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] Result contains complex when optimal does not.
time = 0.54, size = 48, normalized size = 1.30 \begin {gather*} - \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 {gather*}

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

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Giac [F]
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)^m/(-2-3*x)^(1/2),x, algorithm="giac")

[Out]

integrate((-x)^m/sqrt(-3*x - 2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (-x\right )}^m}{\sqrt {-3\,x-2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x)^m/(- 3*x - 2)^(1/2),x)

[Out]

int((-x)^m/(- 3*x - 2)^(1/2), x)

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