### 3.2412 $$\int \frac{x}{\sqrt{2+4 x-3 x^2}} \, dx$$

Optimal. Leaf size=40 $-\frac{1}{3} \sqrt{-3 x^2+4 x+2}-\frac{2 \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{3 \sqrt{3}}$

[Out]

-Sqrt[2 + 4*x - 3*x^2]/3 - (2*ArcSin[(2 - 3*x)/Sqrt[10]])/(3*Sqrt[3])

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Rubi [A]  time = 0.0225357, antiderivative size = 40, 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, 619, 216} $-\frac{1}{3} \sqrt{-3 x^2+4 x+2}-\frac{2 \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{3 \sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

Int[x/Sqrt[2 + 4*x - 3*x^2],x]

[Out]

-Sqrt[2 + 4*x - 3*x^2]/3 - (2*ArcSin[(2 - 3*x)/Sqrt[10]])/(3*Sqrt[3])

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 619

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

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{x}{\sqrt{2+4 x-3 x^2}} \, dx &=-\frac{1}{3} \sqrt{2+4 x-3 x^2}+\frac{2}{3} \int \frac{1}{\sqrt{2+4 x-3 x^2}} \, dx\\ &=-\frac{1}{3} \sqrt{2+4 x-3 x^2}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{40}}} \, dx,x,4-6 x\right )}{3 \sqrt{30}}\\ &=-\frac{1}{3} \sqrt{2+4 x-3 x^2}-\frac{2 \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{3 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0162402, size = 40, normalized size = 1. $\frac{1}{9} \left (-3 \sqrt{-3 x^2+4 x+2}-2 \sqrt{3} \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x/Sqrt[2 + 4*x - 3*x^2],x]

[Out]

(-3*Sqrt[2 + 4*x - 3*x^2] - 2*Sqrt[3]*ArcSin[(2 - 3*x)/Sqrt[10]])/9

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Maple [A]  time = 0.043, size = 30, normalized size = 0.8 \begin{align*} -{\frac{1}{3}\sqrt{-3\,{x}^{2}+4\,x+2}}+{\frac{2\,\sqrt{3}}{9}\arcsin \left ({\frac{3\,\sqrt{10}}{10} \left ( x-{\frac{2}{3}} \right ) } \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(-3*x^2+4*x+2)^(1/2)+2/9*3^(1/2)*arcsin(3/10*10^(1/2)*(x-2/3))

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Maxima [A]  time = 1.51413, size = 42, normalized size = 1.05 \begin{align*} -\frac{2}{9} \, \sqrt{3} \arcsin \left (-\frac{1}{10} \, \sqrt{10}{\left (3 \, x - 2\right )}\right ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 4 \, x + 2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-3*x^2+4*x+2)^(1/2),x, algorithm="maxima")

[Out]

-2/9*sqrt(3)*arcsin(-1/10*sqrt(10)*(3*x - 2)) - 1/3*sqrt(-3*x^2 + 4*x + 2)

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Fricas [A]  time = 2.03475, size = 154, normalized size = 3.85 \begin{align*} -\frac{2}{9} \, \sqrt{3} \arctan \left (\frac{\sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x + 2}{\left (3 \, x - 2\right )}}{3 \,{\left (3 \, x^{2} - 4 \, x - 2\right )}}\right ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 4 \, x + 2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-3*x^2+4*x+2)^(1/2),x, algorithm="fricas")

[Out]

-2/9*sqrt(3)*arctan(1/3*sqrt(3)*sqrt(-3*x^2 + 4*x + 2)*(3*x - 2)/(3*x^2 - 4*x - 2)) - 1/3*sqrt(-3*x^2 + 4*x +
2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{- 3 x^{2} + 4 x + 2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-3*x**2+4*x+2)**(1/2),x)

[Out]

Integral(x/sqrt(-3*x**2 + 4*x + 2), x)

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Giac [A]  time = 1.15499, size = 42, normalized size = 1.05 \begin{align*} \frac{2}{9} \, \sqrt{3} \arcsin \left (\frac{1}{10} \, \sqrt{10}{\left (3 \, x - 2\right )}\right ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 4 \, x + 2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-3*x^2+4*x+2)^(1/2),x, algorithm="giac")

[Out]

2/9*sqrt(3)*arcsin(1/10*sqrt(10)*(3*x - 2)) - 1/3*sqrt(-3*x^2 + 4*x + 2)