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 = 0.0046574, 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, Rules used = {43} \[ \frac{2}{27} (2-3 x)^{3/2}-\frac{4}{9} \sqrt{2-3 x} \]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x}{\sqrt{2-3 x}} \, dx &=\int \left (\frac{2}{3 \sqrt{2-3 x}}-\frac{1}{3} \sqrt{2-3 x}\right ) \, dx\\ &=-\frac{4}{9} \sqrt{2-3 x}+\frac{2}{27} (2-3 x)^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0049493, 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 \begin{align*} -{\frac{6\,x+8}{27}\sqrt{2-3\,x}} \end{align*}

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 = 0.947613, size = 26, normalized size = 0.96 \begin{align*} \frac{2}{27} \,{\left (-3 \, x + 2\right )}^{\frac{3}{2}} - \frac{4}{9} \, \sqrt{-3 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(2-3*x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.416598, size = 43, normalized size = 1.59 \begin{align*} -\frac{2}{27} \,{\left (3 \, x + 4\right )} \sqrt{-3 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(2-3*x)^(1/2),x, algorithm="fricas")

[Out]

-2/27*(3*x + 4)*sqrt(-3*x + 2)

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Sympy [A]  time = 0.82036, size = 61, normalized size = 2.26 \begin{align*} \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{2 - 3 x}}{9} - \frac{8 \sqrt{2 - 3 x}}{27} & \text{otherwise} \end{cases} \end{align*}

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(2 - 3*x)/9 - 8*sqrt(2 -
3*x)/27, True))

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Giac [A]  time = 1.06409, size = 26, normalized size = 0.96 \begin{align*} \frac{2}{27} \,{\left (-3 \, x + 2\right )}^{\frac{3}{2}} - \frac{4}{9} \, \sqrt{-3 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(2-3*x)^(1/2),x, algorithm="giac")

[Out]

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