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

Optimal. Leaf size=31 \[ \frac{x^{m+1} \, _2F_1\left (\frac{1}{2},m+1;m+2;\frac{3 x}{2}\right )}{\sqrt{2} (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.017481, antiderivative size = 31, 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 \[ \frac{x^{m+1} \, _2F_1\left (\frac{1}{2},m+1;m+2;\frac{3 x}{2}\right )}{\sqrt{2} (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^m/Sqrt[2 - 3*x],x]

[Out]

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

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Rubi in Sympy [A]  time = 2.60922, size = 26, normalized size = 0.84 \[ \frac{\sqrt{2} x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{3 x}{2}} \right )}}{2 \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0214933, size = 37, normalized size = 1.19 \[ -\left (\frac{3}{2}\right )^{-m-1} \sqrt{2-3 x} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};1-\frac{3 x}{2}\right ) \]

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 [A]  time = 0.04, size = 29, normalized size = 0.9 \[{\frac{{x}^{1+m}\sqrt{2}}{2+2\,m}{\mbox{$_2$F$_1$}({\frac{1}{2}},1+m;\,2+m;\,{\frac{3\,x}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(2-3*x)^(1/2),x)

[Out]

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

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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)

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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)

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Sympy [A]  time = 3.33253, size = 46, normalized size = 1.48 \[ - \frac{2 \cdot 2^{m} \sqrt{3} \cdot 3^{- m} i \sqrt{x - \frac{2}{3}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - m \\ \frac{3}{2} \end{matrix}\middle |{\frac{3 \left (x - \frac{2}{3}\right ) e^{i \pi }}{2}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*2**m*sqrt(3)*3**(-m)*I*sqrt(x - 2/3)*hyper((1/2, -m), (3/2,), 3*(x - 2/3)*exp
_polar(I*pi)/2)/3

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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)