3.1.4 \(\int \frac {x}{\sqrt {2-3 x}} \, dx\) [4]

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

[Out]

2/27*(2-3*x)^(3/2)-4/9*(2-3*x)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 = {45} \begin {gather*} \frac {2}{27} (2-3 x)^{3/2}-\frac {4}{9} \sqrt {2-3 x} \end {gather*}

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 45

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*}

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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.67 \begin {gather*} -\frac {2}{27} \sqrt {2-3 x} (4+3 x) \end {gather*}

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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 2.09, size = 41, normalized size = 1.52 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 I \left (-4-3 x\right ) \sqrt {-2+3 x}}{27},\text {Abs}\left [x\right ]>\frac {2}{3}\right \}\right \},\frac {-2 x \sqrt {2-3 x}}{9}-\frac {8 \sqrt {2-3 x}}{27}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x/Sqrt[2 - 3*x],x]')

[Out]

Piecewise[{{2 I / 27 (-4 - 3 x) Sqrt[-2 + 3 x], Abs[x] > 2 / 3}}, -2 x Sqrt[2 - 3 x] / 9 - 8 Sqrt[2 - 3 x] / 2
7]

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Maple [A]
time = 0.07, size = 20, normalized size = 0.74

method	result	size
trager	\(\left (-\frac {2 x}{9}-\frac {8}{27}\right ) \sqrt {2-3 x}\)	\(14\)
gosper	\(-\frac {2 \left (3 x +4\right ) \sqrt {2-3 x}}{27}\)	\(15\)
derivativedivides	\(\frac {2 \left (2-3 x \right )^{\frac {3}{2}}}{27}-\frac {4 \sqrt {2-3 x}}{9}\)	\(20\)
default	\(\frac {2 \left (2-3 x \right )^{\frac {3}{2}}}{27}-\frac {4 \sqrt {2-3 x}}{9}\)	\(20\)
risch	\(\frac {2 \left (-2+3 x \right ) \left (3 x +4\right )}{27 \sqrt {2-3 x}}\)	\(20\)
meijerg	\(\frac {2 \sqrt {2}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (6 x +8\right ) \sqrt {1-\frac {3 x}{2}}}{6}\right )}{9 \sqrt {\pi }}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(2-3*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/27*(2-3*x)^(3/2)-4/9*(2-3*x)^(1/2)

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Maxima [A]
time = 0.24, size = 19, normalized size = 0.70 \begin {gather*} \frac {2}{27} \, {\left (-3 \, x + 2\right )}^{\frac {3}{2}} - \frac {4}{9} \, \sqrt {-3 \, x + 2} \end {gather*}

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 = 0.33, size = 14, normalized size = 0.52 \begin {gather*} -\frac {2}{27} \, {\left (3 \, x + 4\right )} \sqrt {-3 \, x + 2} \end {gather*}

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 = 0.49, size = 60, normalized size = 2.22 \begin {gather*} \begin {cases} - \frac {2 i x \sqrt {3 x - 2}}{9} - \frac {8 i \sqrt {3 x - 2}}{27} & \text {for}\: \left |{x}\right | > \frac {2}{3} \\- \frac {2 x \sqrt {2 - 3 x}}{9} - \frac {8 \sqrt {2 - 3 x}}{27} & \text {otherwise} \end {cases} \end {gather*}

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

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Giac [A]
time = 0.00, size = 37, normalized size = 1.37 \begin {gather*} \frac {2}{9} \left (\frac {1}{3} \sqrt {-3 x+2} \left (-3 x+2\right )-2 \sqrt {-3 x+2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(2-3*x)^(1/2),x)

[Out]

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

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Mupad [B]
time = 0.06, size = 14, normalized size = 0.52 \begin {gather*} -\frac {2\,\sqrt {2-3\,x}\,\left (3\,x+4\right )}{27} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(2 - 3*x)^(1/2),x)

[Out]

-(2*(2 - 3*x)^(1/2)*(3*x + 4))/27

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