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

Optimal. Leaf size=65 \[ -\frac{1}{36} \sqrt{2-3 x^2} \left (3 x^2-1\right )^{3/2}-\frac{7}{72} \sqrt{2-3 x^2} \sqrt{3 x^2-1}-\frac{7}{144} \sin ^{-1}\left (3-6 x^2\right ) \]

[Out]

(-7*Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x^2])/72 - (Sqrt[2 - 3*x^2]*(-1 + 3*x^2)^(3/2))/36 - (7*ArcSin[3 - 6*x^2])/144

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Rubi [A]  time = 0.0492191, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {446, 80, 50, 53, 619, 216} \[ -\frac{1}{36} \sqrt{2-3 x^2} \left (3 x^2-1\right )^{3/2}-\frac{7}{72} \sqrt{2-3 x^2} \sqrt{3 x^2-1}-\frac{7}{144} \sin ^{-1}\left (3-6 x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x^2])/72 - (Sqrt[2 - 3*x^2]*(-1 + 3*x^2)^(3/2))/36 - (7*ArcSin[3 - 6*x^2])/144

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
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^3 \sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x \sqrt{-1+3 x}}{\sqrt{2-3 x}} \, dx,x,x^2\right )\\ &=-\frac{1}{36} \sqrt{2-3 x^2} \left (-1+3 x^2\right )^{3/2}+\frac{7}{24} \operatorname{Subst}\left (\int \frac{\sqrt{-1+3 x}}{\sqrt{2-3 x}} \, dx,x,x^2\right )\\ &=-\frac{7}{72} \sqrt{2-3 x^2} \sqrt{-1+3 x^2}-\frac{1}{36} \sqrt{2-3 x^2} \left (-1+3 x^2\right )^{3/2}+\frac{7}{48} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-3 x} \sqrt{-1+3 x}} \, dx,x,x^2\right )\\ &=-\frac{7}{72} \sqrt{2-3 x^2} \sqrt{-1+3 x^2}-\frac{1}{36} \sqrt{2-3 x^2} \left (-1+3 x^2\right )^{3/2}+\frac{7}{48} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-2+9 x-9 x^2}} \, dx,x,x^2\right )\\ &=-\frac{7}{72} \sqrt{2-3 x^2} \sqrt{-1+3 x^2}-\frac{1}{36} \sqrt{2-3 x^2} \left (-1+3 x^2\right )^{3/2}-\frac{7}{432} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{9}}} \, dx,x,9 \left (1-2 x^2\right )\right )\\ &=-\frac{7}{72} \sqrt{2-3 x^2} \sqrt{-1+3 x^2}-\frac{1}{36} \sqrt{2-3 x^2} \left (-1+3 x^2\right )^{3/2}-\frac{7}{144} \sin ^{-1}\left (3-6 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0270139, size = 44, normalized size = 0.68 \[ \frac{1}{72} \left (-\sqrt{-9 x^4+9 x^2-2} \left (6 x^2+5\right )-7 \sin ^{-1}\left (\sqrt{2-3 x^2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((5 + 6*x^2)*Sqrt[-2 + 9*x^2 - 9*x^4]) - 7*ArcSin[Sqrt[2 - 3*x^2]])/72

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Maple [A]  time = 0.023, size = 81, normalized size = 1.3 \begin{align*}{\frac{1}{144}\sqrt{-3\,{x}^{2}+2}\sqrt{3\,{x}^{2}-1} \left ( -12\,{x}^{2}\sqrt{-9\,{x}^{4}+9\,{x}^{2}-2}+7\,\arcsin \left ( 6\,{x}^{2}-3 \right ) -10\,\sqrt{-9\,{x}^{4}+9\,{x}^{2}-2} \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*(3*x^2-1)^(1/2)/(-3*x^2+2)^(1/2),x)

[Out]

1/144*(3*x^2-1)^(1/2)*(-3*x^2+2)^(1/2)*(-12*x^2*(-9*x^4+9*x^2-2)^(1/2)+7*arcsin(6*x^2-3)-10*(-9*x^4+9*x^2-2)^(
1/2))/(-9*x^4+9*x^2-2)^(1/2)

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Maxima [A]  time = 1.45833, size = 62, normalized size = 0.95 \begin{align*} -\frac{1}{12} \, \sqrt{-9 \, x^{4} + 9 \, x^{2} - 2} x^{2} - \frac{5}{72} \, \sqrt{-9 \, x^{4} + 9 \, x^{2} - 2} + \frac{7}{144} \, \arcsin \left (6 \, x^{2} - 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(-9*x^4 + 9*x^2 - 2)*x^2 - 5/72*sqrt(-9*x^4 + 9*x^2 - 2) + 7/144*arcsin(6*x^2 - 3)

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Fricas [A]  time = 1.78888, size = 185, normalized size = 2.85 \begin{align*} -\frac{1}{72} \,{\left (6 \, x^{2} + 5\right )} \sqrt{3 \, x^{2} - 1} \sqrt{-3 \, x^{2} + 2} - \frac{7}{144} \, \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*(3*x^2-1)^(1/2)/(-3*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

-1/72*(6*x^2 + 5)*sqrt(3*x^2 - 1)*sqrt(-3*x^2 + 2) - 7/144*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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \sqrt{3 x^{2} - 1}}{\sqrt{2 - 3 x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*sqrt(3*x**2 - 1)/sqrt(2 - 3*x**2), x)

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Giac [A]  time = 1.1517, size = 54, normalized size = 0.83 \begin{align*} -\frac{1}{72} \,{\left (6 \, x^{2} + 5\right )} \sqrt{3 \, x^{2} - 1} \sqrt{-3 \, x^{2} + 2} + \frac{7}{72} \, \arcsin \left (\sqrt{3 \, x^{2} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72*(6*x^2 + 5)*sqrt(3*x^2 - 1)*sqrt(-3*x^2 + 2) + 7/72*arcsin(sqrt(3*x^2 - 1))