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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 = 0.01, size = 36, normalized size = 1.00 \[ \left (\frac {3}{2}\right )^{-m-1} \sqrt {3 x-2} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1-\frac {3 x}{2}\right ) \]

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]

________________________________________________________________________________________

fricas [F]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{\sqrt {3 \, x - 2}}, x\right ) \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\sqrt {3 \, x - 2}}\,{d x} \]

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)

________________________________________________________________________________________

maple [C]  time = 0.06, size = 43, normalized size = 1.19 \[ \frac {\sqrt {2}\, \sqrt {-\mathrm {signum}\left (x -\frac {2}{3}\right )}\, x^{m +1} \hypergeom \left (\left [\frac {1}{2}, m +1\right ], \left [m +2\right ], \frac {3 x}{2}\right )}{2 \left (m +1\right ) \sqrt {\mathrm {signum}\left (x -\frac {2}{3}\right )}} \]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\sqrt {3 \, x - 2}}\,{d x} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^m}{\sqrt {3\,x-2}} \,d x \]

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)

________________________________________________________________________________________

sympy [C]  time = 1.22, size = 36, normalized size = 1.00 \[ - \frac {\sqrt {2} i x x^{m} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {3 x}{2}} \right )}}{2 \Gamma \left (m + 2\right )} \]

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))

________________________________________________________________________________________