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 = 0.0332155, 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 \[ 2^{m+1} 3^{-m-1} \sqrt{3 x-2} (-x)^m x^{-m} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};1-\frac{3 x}{2}\right ) \]

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

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Rubi in Sympy [A]  time = 3.72673, size = 37, normalized size = 0.76 \[ \frac{2 \left (- x\right )^{m} \left (\frac{3 x}{2}\right )^{- m} \sqrt{3 x - 2}{{}_{2}F_{1}\left (\begin{matrix} - m, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{3 x}{2} + 1} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0165527, 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 = 0.047, size = 44, normalized size = 0.9 \[{\frac{\sqrt{2} \left ( -x \right ) ^{m}x}{2+2\,m}\sqrt{-{\it signum} \left ( x-{\frac{2}{3}} \right ) }{\mbox{$_2$F$_1$}({\frac{1}{2}},1+m;\,2+m;\,{\frac{3\,x}{2}})}{\frac{1}{\sqrt{{\it signum} \left ( x-{\frac{2}{3}} \right ) }}}} \]

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. \[ \int \frac{\left (-x\right )^{m}}{\sqrt{3 \, x - 2}}\,{d x} \]

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)

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

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)

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Sympy [A]  time = 3.45889, size = 42, normalized size = 0.86 \[ - \frac{\sqrt{2} i x x^{m} e^{i \pi m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{3 x}{2}} \right )}}{2 \Gamma \left (m + 2\right )} \]

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

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)