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

Optimal. Leaf size=36 \[ \left (\frac{3}{2}\right )^{-m-1} \sqrt{3 x-2} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};1-\frac{3 x}{2}\right ) \]

[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 = 0.0083152, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {65} \[ \left (\frac{3}{2}\right )^{-m-1} \sqrt{3 x-2} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};1-\frac{3 x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^m/Sqrt[-2 + 3*x],x]

[Out]

(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 = 0.006302, size = 36, normalized size = 1. \[ \left (\frac{3}{2}\right )^{-m-1} \sqrt{3 x-2} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};1-\frac{3 x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m/Sqrt[-2 + 3*x],x]

[Out]

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

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Maple [C]  time = 0.033, size = 43, normalized size = 1.2 \begin{align*}{\frac{\sqrt{2}{x}^{1+m}}{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)/signum(x-2/3)^(1/2)*(-signum(x-2/3))^(1/2)/(1+m)*x^(1+m)*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{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}}, 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 = 1.01632, size = 36, normalized size = 1. \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}{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/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)