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]

_______________________________________________________________________________________

Rubi [A]  time = 0.0236413, 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 \[ \left (\frac{3}{2}\right )^{-m-1} \sqrt{3 x-2} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};1-\frac{3 x}{2}\right ) \]

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]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.61238, size = 29, normalized size = 0.81 \[ \frac{2 \left (\frac{2}{3}\right )^{m} \sqrt{3 x - 2}{{}_{2}F_{1}\left (\begin{matrix} - m, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{3 x}{2} + 1} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m/(-2+3*x)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0160436, 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]

_______________________________________________________________________________________

Maple [C]  time = 0.075, size = 43, normalized size = 1.2 \[{\frac{\sqrt{2}{x}^{1+m}}{2+2\,m}\sqrt{-{\it signum} \left ( x-{\frac{2}{3}} \right ) }{\mbox{$_2$F$_1$}({\frac{1}{2}},1+m;\,2+m;\,{\frac{3\,x}{2}})}{\frac{1}{\sqrt{{\it signum} \left ( x-{\frac{2}{3}} \right ) }}}} \]

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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/sqrt(3*x - 2),x, algorithm="maxima")

[Out]

integrate(x^m/sqrt(3*x - 2), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/sqrt(3*x - 2),x, algorithm="fricas")

[Out]

integral(x^m/sqrt(3*x - 2), x)

_______________________________________________________________________________________

Sympy [A]  time = 3.33074, size = 36, normalized size = 1. \[ - \frac{\sqrt{2} i x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{3 x}{2}} \right )}}{2 \Gamma \left (m + 2\right )} \]

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))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{\sqrt{3 \, x - 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/sqrt(3*x - 2),x, algorithm="giac")

[Out]

integrate(x^m/sqrt(3*x - 2), x)