3.967 \(\int \frac{x \sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx\)

Optimal. Leaf size=39 \[ -\frac{1}{6} \sqrt{2-3 x^2} \sqrt{3 x^2-1}-\frac{1}{12} \sin ^{-1}\left (3-6 x^2\right ) \]

[Out]

-(Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x^2])/6 - ArcSin[3 - 6*x^2]/12

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Rubi [A]  time = 0.0310672, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {444, 50, 53, 619, 216} \[ -\frac{1}{6} \sqrt{2-3 x^2} \sqrt{3 x^2-1}-\frac{1}{12} \sin ^{-1}\left (3-6 x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[(x*Sqrt[-1 + 3*x^2])/Sqrt[2 - 3*x^2],x]

[Out]

-(Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x^2])/6 - ArcSin[3 - 6*x^2]/12

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 53

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

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{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{-1+3 x}}{\sqrt{2-3 x}} \, dx,x,x^2\right )\\ &=-\frac{1}{6} \sqrt{2-3 x^2} \sqrt{-1+3 x^2}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-3 x} \sqrt{-1+3 x}} \, dx,x,x^2\right )\\ &=-\frac{1}{6} \sqrt{2-3 x^2} \sqrt{-1+3 x^2}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-2+9 x-9 x^2}} \, dx,x,x^2\right )\\ &=-\frac{1}{6} \sqrt{2-3 x^2} \sqrt{-1+3 x^2}-\frac{1}{36} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{9}}} \, dx,x,9 \left (1-2 x^2\right )\right )\\ &=-\frac{1}{6} \sqrt{2-3 x^2} \sqrt{-1+3 x^2}-\frac{1}{12} \sin ^{-1}\left (3-6 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0123984, size = 37, normalized size = 0.95 \[ \frac{1}{6} \left (-\sqrt{-9 x^4+9 x^2-2}-\sin ^{-1}\left (\sqrt{2-3 x^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x*Sqrt[-1 + 3*x^2])/Sqrt[2 - 3*x^2],x]

[Out]

(-Sqrt[-2 + 9*x^2 - 9*x^4] - ArcSin[Sqrt[2 - 3*x^2]])/6

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Maple [A]  time = 0.009, size = 60, normalized size = 1.5 \begin{align*}{\frac{1}{12}\sqrt{-3\,{x}^{2}+2}\sqrt{3\,{x}^{2}-1} \left ( -2\,\sqrt{-9\,{x}^{4}+9\,{x}^{2}-2}+\arcsin \left ( 6\,{x}^{2}-3 \right ) \right ){\frac{1}{\sqrt{-9\,{x}^{4}+9\,{x}^{2}-2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*x^2-1)^(1/2)*(-3*x^2+2)^(1/2)*(-2*(-9*x^4+9*x^2-2)^(1/2)+arcsin(6*x^2-3))/(-9*x^4+9*x^2-2)^(1/2)

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Maxima [A]  time = 1.47783, size = 36, normalized size = 0.92 \begin{align*} -\frac{1}{6} \, \sqrt{-9 \, x^{4} + 9 \, x^{2} - 2} + \frac{1}{12} \, \arcsin \left (6 \, x^{2} - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(-9*x^4 + 9*x^2 - 2) + 1/12*arcsin(6*x^2 - 3)

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Fricas [B]  time = 1.80314, size = 166, normalized size = 4.26 \begin{align*} -\frac{1}{6} \, \sqrt{3 \, x^{2} - 1} \sqrt{-3 \, x^{2} + 2} - \frac{1}{12} \, \arctan \left (\frac{3 \, \sqrt{3 \, x^{2} - 1}{\left (2 \, x^{2} - 1\right )} \sqrt{-3 \, x^{2} + 2}}{2 \,{\left (9 \, x^{4} - 9 \, x^{2} + 2\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(3*x^2 - 1)*sqrt(-3*x^2 + 2) - 1/12*arctan(3/2*sqrt(3*x^2 - 1)*(2*x^2 - 1)*sqrt(-3*x^2 + 2)/(9*x^4 -
9*x^2 + 2))

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Sympy [A]  time = 6.1469, size = 66, normalized size = 1.69 \begin{align*} \frac{\begin{cases} - \frac{\sqrt{2 - 3 x^{2}} \sqrt{3 x^{2} - 1}}{2} + \frac{\operatorname{asin}{\left (\sqrt{3 x^{2} - 1} \right )}}{2} & \text{for}\: \left (x \geq \frac{\sqrt{3}}{3} \wedge x < \frac{\sqrt{6}}{3}\right ) \vee \left (x \leq - \frac{\sqrt{3}}{3} \wedge x > - \frac{\sqrt{6}}{3}\right ) \end{cases}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-sqrt(2 - 3*x**2)*sqrt(3*x**2 - 1)/2 + asin(sqrt(3*x**2 - 1))/2, ((x >= sqrt(3)/3) & (x < sqrt(6)/3
)) | ((x <= -sqrt(3)/3) & (x > -sqrt(6)/3))))/3

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Giac [A]  time = 1.16384, size = 45, normalized size = 1.15 \begin{align*} -\frac{1}{6} \, \sqrt{3 \, x^{2} - 1} \sqrt{-3 \, x^{2} + 2} + \frac{1}{6} \, \arcsin \left (\sqrt{3 \, x^{2} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(3*x^2 - 1)*sqrt(-3*x^2 + 2) + 1/6*arcsin(sqrt(3*x^2 - 1))