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

Optimal. Leaf size=49 \[ 2^{m+1} 3^{-m-1} \sqrt{3 x-2} (-x)^m x^{-m} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};1-\frac{3 x}{2}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0110271, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {67, 12, 65} \[ 2^{m+1} 3^{-m-1} \sqrt{3 x-2} (-x)^m x^{-m} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};1-\frac{3 x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-x)^m/Sqrt[-2 + 3*x],x]

[Out]

(2^(1 + m)*3^(-1 - m)*(-x)^m*Sqrt[-2 + 3*x]*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} (-x)^m x^{-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 = 0.0074754, size = 49, normalized size = 1. \[ 2^{m+1} 3^{-m-1} \sqrt{3 x-2} (-x)^m x^{-m} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};1-\frac{3 x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-x)^m/Sqrt[-2 + 3*x],x]

[Out]

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

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Maple [C]  time = 0.033, size = 44, normalized size = 0.9 \begin{align*}{\frac{\sqrt{2} \left ( -x \right ) ^{m}x}{2+2\,m}\sqrt{-{\it signum} \left ( x-{\frac{2}{3}} \right ) }{\mbox{$_2$F$_1$}({\frac{1}{2}},1+m;\,2+m;\,{\frac{3\,x}{2}})}{\frac{1}{\sqrt{{\it signum} \left ( x-{\frac{2}{3}} \right ) }}}} \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*2^(1/2)*(-x)^m/signum(x-2/3)^(1/2)*(-signum(x-2/3))^(1/2)/(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}}, 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), x)

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Sympy [C]  time = 1.03686, size = 42, normalized size = 0.86 \begin{align*} - \frac{\sqrt{2} i x x^{m} e^{i \pi 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}{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*exp(I*pi*m)*gamma(m + 1)*hyper((1/2, m + 1), (m + 2,), 3*x/2)/(2*gamma(m + 2))

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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)