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

Optimal. Leaf size=50 \[ -2^{m+1} 3^{-m-1} \sqrt{-3 x-2} (-x)^{-m} x^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{3 x}{2}+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.0109464, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {67, 12, 65} \[ -2^{m+1} 3^{-m-1} \sqrt{-3 x-2} (-x)^{-m} x^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{3 x}{2}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^m/Sqrt[-2 - 3*x],x]

[Out]

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

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[((-((b*c)/d))^IntPart[m]*(b*x)^FracPart[m])/
(-((d*x)/c))^FracPart[m], Int[(-((d*x)/c))^m*(c + d*x)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-(d/(b*c)), 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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

Mathematica [A]  time = 0.0228393, size = 48, normalized size = 0.96 \[ -\frac{2}{3} \sqrt{-3 x-2} \left (\frac{1}{2} (-3 x-2)+1\right )^{-m} x^m \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{3 x}{2}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m/Sqrt[-2 - 3*x],x]

[Out]

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

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Maple [C]  time = 0.016, size = 30, normalized size = 0.6 \begin{align*}{\frac{-{\frac{i}{2}}{x}^{1+m}\sqrt{2}}{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*x^(1+m)*hypergeom([1/2,1+m],[2+m],-3/2*x)/(1+m)*2^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{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{x^{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 = 1.01834, size = 41, normalized size = 0.82 \begin{align*} - \frac{\sqrt{2} i x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{3 x e^{i \pi }}{2}} \right )}}{2 \Gamma \left (m + 2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{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)