### 3.217 $$\int \frac{x}{\sqrt{-4+12 x-9 x^2}} \, dx$$

Optimal. Leaf size=48 $-\frac{1}{9} \sqrt{-9 x^2+12 x-4}-\frac{2 (2-3 x) \log (2-3 x)}{9 \sqrt{-9 x^2+12 x-4}}$

[Out]

-Sqrt[-4 + 12*x - 9*x^2]/9 - (2*(2 - 3*x)*Log[2 - 3*x])/(9*Sqrt[-4 + 12*x - 9*x^2])

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Rubi [A]  time = 0.0105578, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.188, Rules used = {640, 608, 31} $-\frac{1}{9} \sqrt{-9 x^2+12 x-4}-\frac{2 (2-3 x) \log (2-3 x)}{9 \sqrt{-9 x^2+12 x-4}}$

Antiderivative was successfully veriﬁed.

[In]

Int[x/Sqrt[-4 + 12*x - 9*x^2],x]

[Out]

-Sqrt[-4 + 12*x - 9*x^2]/9 - (2*(2 - 3*x)*Log[2 - 3*x])/(9*Sqrt[-4 + 12*x - 9*x^2])

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 608

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(b/2 + c*x)/Sqrt[a + b*x + c*x^2], Int[1/(b/2
+ c*x), x], x] /; FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{x}{\sqrt{-4+12 x-9 x^2}} \, dx &=-\frac{1}{9} \sqrt{-4+12 x-9 x^2}+\frac{2}{3} \int \frac{1}{\sqrt{-4+12 x-9 x^2}} \, dx\\ &=-\frac{1}{9} \sqrt{-4+12 x-9 x^2}+\frac{(2 (6-9 x)) \int \frac{1}{6-9 x} \, dx}{3 \sqrt{-4+12 x-9 x^2}}\\ &=-\frac{1}{9} \sqrt{-4+12 x-9 x^2}-\frac{2 (2-3 x) \log (2-3 x)}{9 \sqrt{-4+12 x-9 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0111983, size = 35, normalized size = 0.73 $\frac{(3 x-2) (3 x+2 \log (2-3 x)-2)}{9 \sqrt{-(2-3 x)^2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x/Sqrt[-4 + 12*x - 9*x^2],x]

[Out]

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

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Maple [A]  time = 0.096, size = 31, normalized size = 0.7 \begin{align*}{\frac{ \left ( -2+3\,x \right ) \left ( 3\,x+2\,\ln \left ( -2+3\,x \right ) \right ) }{9}{\frac{1}{\sqrt{- \left ( -2+3\,x \right ) ^{2}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-(-2+3*x)^2)^(1/2),x)

[Out]

1/9*(-2+3*x)*(3*x+2*ln(-2+3*x))/(-(-2+3*x)^2)^(1/2)

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Maxima [C]  time = 1.70781, size = 28, normalized size = 0.58 \begin{align*} -\frac{1}{9} \, \sqrt{-9 \, x^{2} + 12 \, x - 4} + \frac{2}{9} i \, \log \left (x - \frac{2}{3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*sqrt(-9*x^2 + 12*x - 4) + 2/9*I*log(x - 2/3)

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Fricas [C]  time = 1.59583, size = 42, normalized size = 0.88 \begin{align*} -\frac{1}{3} i \, x - \frac{2}{9} i \, \log \left (x - \frac{2}{3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*I*x - 2/9*I*log(x - 2/3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{- \left (3 x - 2\right )^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/sqrt(-(3*x - 2)**2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef