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

Optimal. Leaf size=70 \[ -\frac {1}{9} \sqrt {2-3 x^2} \sqrt {3 x^2-1} x-\frac {F\left (\left .\cos ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{9 \sqrt {3}}-\frac {E\left (\left .\cos ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{3 \sqrt {3}} \]

[Out]

-1/9*(x^2)^(1/2)/x*EllipticE(1/2*(-6*x^2+4)^(1/2),2^(1/2))*3^(1/2)-1/27*(x^2)^(1/2)/x*EllipticF(1/2*(-6*x^2+4)
^(1/2),2^(1/2))*3^(1/2)-1/9*x*(-3*x^2+2)^(1/2)*(3*x^2-1)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {478, 524, 425, 420} \[ -\frac {1}{9} \sqrt {2-3 x^2} \sqrt {3 x^2-1} x-\frac {F\left (\left .\cos ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{9 \sqrt {3}}-\frac {E\left (\left .\cos ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{3 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x*Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x^2])/9 - EllipticE[ArcCos[Sqrt[3/2]*x], 2]/(3*Sqrt[3]) - EllipticF[ArcCos[Sqr
t[3/2]*x], 2]/(9*Sqrt[3])

Rule 420

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

Rule 425

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

Rule 478

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

Rule 524

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-(b/a), -(d/c)]))))))

Rubi steps

\begin {align*} \int \frac {x^2 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx &=-\frac {1}{9} x \sqrt {2-3 x^2} \sqrt {-1+3 x^2}+\frac {1}{9} \int \frac {-2+9 x^2}{\sqrt {2-3 x^2} \sqrt {-1+3 x^2}} \, dx\\ &=-\frac {1}{9} x \sqrt {2-3 x^2} \sqrt {-1+3 x^2}+\frac {1}{9} \int \frac {1}{\sqrt {2-3 x^2} \sqrt {-1+3 x^2}} \, dx+\frac {1}{3} \int \frac {\sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx\\ &=-\frac {1}{9} x \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {E\left (\left .\cos ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{3 \sqrt {3}}-\frac {F\left (\left .\cos ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{9 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 86, normalized size = 1.23 \[ \frac {3 x \sqrt {2-3 x^2} \left (1-3 x^2\right )+\sqrt {3-9 x^2} F\left (\left .\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )-3 \sqrt {3-9 x^2} E\left (\left .\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{27 \sqrt {3 x^2-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x*(1 - 3*x^2)*Sqrt[2 - 3*x^2] - 3*Sqrt[3 - 9*x^2]*EllipticE[ArcSin[Sqrt[3/2]*x], 2] + Sqrt[3 - 9*x^2]*Ellip
ticF[ArcSin[Sqrt[3/2]*x], 2])/(27*Sqrt[-1 + 3*x^2])

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fricas [F]  time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {3 \, x^{2} - 1} \sqrt {-3 \, x^{2} + 2} x^{2}}{3 \, x^{2} - 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {3 \, x^{2} - 1} x^{2}}{\sqrt {-3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.02, size = 129, normalized size = 1.84 \[ -\frac {\sqrt {3 x^{2}-1}\, \sqrt {2}\, \sqrt {-6 x^{2}+4}\, \left (54 x^{5}-54 x^{3}+12 x -3 \sqrt {3}\, \sqrt {2}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, \EllipticE \left (\frac {\sqrt {3}\, \sqrt {2}\, x}{2}, \sqrt {2}\right )+\sqrt {3}\, \sqrt {2}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, \EllipticF \left (\frac {\sqrt {3}\, \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{108 \left (9 x^{4}-9 x^{2}+2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/108*(3*x^2-1)^(1/2)*2^(1/2)*(-6*x^2+4)^(1/2)*(54*x^5+3^(1/2)*2^(1/2)*(-6*x^2+4)^(1/2)*(-3*x^2+1)^(1/2)*Elli
pticF(1/2*3^(1/2)*2^(1/2)*x,2^(1/2))-3*3^(1/2)*2^(1/2)*(-6*x^2+4)^(1/2)*(-3*x^2+1)^(1/2)*EllipticE(1/2*3^(1/2)
*2^(1/2)*x,2^(1/2))-54*x^3+12*x)/(9*x^4-9*x^2+2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {3 \, x^{2} - 1} x^{2}}{\sqrt {-3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\sqrt {3\,x^2-1}}{\sqrt {2-3\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \sqrt {3 x^{2} - 1}}{\sqrt {2 - 3 x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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