3.4 \(\int \frac{x}{\sqrt{2-3 x}} \, dx\)

Optimal. Leaf size=27 \[ \frac{2}{27} (2-3 x)^{3/2}-\frac{4}{9} \sqrt{2-3 x} \]

[Out]

(-4*Sqrt[2 - 3*x])/9 + (2*(2 - 3*x)^(3/2))/27

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Rubi [A]  time = 0.0154312, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2}{27} (2-3 x)^{3/2}-\frac{4}{9} \sqrt{2-3 x} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[2 - 3*x],x]

[Out]

(-4*Sqrt[2 - 3*x])/9 + (2*(2 - 3*x)^(3/2))/27

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Rubi in Sympy [A]  time = 1.55981, size = 22, normalized size = 0.81 \[ \frac{2 \left (- 3 x + 2\right )^{\frac{3}{2}}}{27} - \frac{4 \sqrt{- 3 x + 2}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-3*x + 2)**(3/2)/27 - 4*sqrt(-3*x + 2)/9

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Mathematica [A]  time = 0.00579681, size = 18, normalized size = 0.67 \[ -\frac{2}{27} \sqrt{2-3 x} (3 x+4) \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[2 - 3*x],x]

[Out]

(-2*Sqrt[2 - 3*x]*(4 + 3*x))/27

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Maple [A]  time = 0.004, size = 15, normalized size = 0.6 \[ -{\frac{6\,x+8}{27}\sqrt{2-3\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(2-3*x)^(1/2),x)

[Out]

-2/27*(3*x+4)*(2-3*x)^(1/2)

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Maxima [A]  time = 1.34264, size = 26, normalized size = 0.96 \[ \frac{2}{27} \,{\left (-3 \, x + 2\right )}^{\frac{3}{2}} - \frac{4}{9} \, \sqrt{-3 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/27*(-3*x + 2)^(3/2) - 4/9*sqrt(-3*x + 2)

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Fricas [A]  time = 0.2114, size = 19, normalized size = 0.7 \[ -\frac{2}{27} \,{\left (3 \, x + 4\right )} \sqrt{-3 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/27*(3*x + 4)*sqrt(-3*x + 2)

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Sympy [A]  time = 1.5892, size = 61, normalized size = 2.26 \[ \begin{cases} - \frac{2 i x \sqrt{3 x - 2}}{9} - \frac{8 i \sqrt{3 x - 2}}{27} & \text{for}\: \frac{3 \left |{x}\right |}{2} > 1 \\- \frac{2 x \sqrt{- 3 x + 2}}{9} - \frac{8 \sqrt{- 3 x + 2}}{27} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(2-3*x)**(1/2),x)

[Out]

Piecewise((-2*I*x*sqrt(3*x - 2)/9 - 8*I*sqrt(3*x - 2)/27, 3*Abs(x)/2 > 1), (-2*x
*sqrt(-3*x + 2)/9 - 8*sqrt(-3*x + 2)/27, True))

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GIAC/XCAS [A]  time = 0.221197, size = 26, normalized size = 0.96 \[ \frac{2}{27} \,{\left (-3 \, x + 2\right )}^{\frac{3}{2}} - \frac{4}{9} \, \sqrt{-3 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/27*(-3*x + 2)^(3/2) - 4/9*sqrt(-3*x + 2)