3.179 \(\int \sqrt{3-6 x^2} \sqrt{2+4 x^2} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{2} x\right ),-1\right )}{\sqrt{3}}+\sqrt{\frac{2}{3}} \sqrt{1-4 x^4} x \]

[Out]

Sqrt[2/3]*x*Sqrt[1 - 4*x^4] + (2*EllipticF[ArcSin[Sqrt[2]*x], -1])/Sqrt[3]

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Rubi [A]  time = 0.009185, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {248, 195, 221} \[ \sqrt{\frac{2}{3}} \sqrt{1-4 x^4} x+\frac{2 F\left (\left .\sin ^{-1}\left (\sqrt{2} x\right )\right |-1\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[3 - 6*x^2]*Sqrt[2 + 4*x^2],x]

[Out]

Sqrt[2/3]*x*Sqrt[1 - 4*x^4] + (2*EllipticF[ArcSin[Sqrt[2]*x], -1])/Sqrt[3]

Rule 248

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_.)*((a2_.) + (b2_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[(a1*a2 + b1*b2*x^(2*
n))^p, x] /; FreeQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a
2, 0]))

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \sqrt{3-6 x^2} \sqrt{2+4 x^2} \, dx &=\int \sqrt{6-24 x^4} \, dx\\ &=\sqrt{\frac{2}{3}} x \sqrt{1-4 x^4}+4 \int \frac{1}{\sqrt{6-24 x^4}} \, dx\\ &=\sqrt{\frac{2}{3}} x \sqrt{1-4 x^4}+\frac{2 F\left (\left .\sin ^{-1}\left (\sqrt{2} x\right )\right |-1\right )}{\sqrt{3}}\\ \end{align*}

Mathematica [C]  time = 0.0116507, size = 22, normalized size = 0.58 \[ \sqrt{6} x \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};4 x^4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[3 - 6*x^2]*Sqrt[2 + 4*x^2],x]

[Out]

Sqrt[6]*x*Hypergeometric2F1[-1/2, 1/4, 5/4, 4*x^4]

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Maple [B]  time = 0.026, size = 75, normalized size = 2. \begin{align*} -{\frac{\sqrt{2}}{36\,{x}^{4}-9}\sqrt{-6\,{x}^{2}+3}\sqrt{2\,{x}^{2}+1} \left ( \sqrt{2}\sqrt{3}\sqrt{-6\,{x}^{2}+3}\sqrt{2\,{x}^{2}+1}{\it EllipticF} \left ( x\sqrt{2},i \right ) -12\,{x}^{5}+3\,x \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*x^2+3)^(1/2)*(4*x^2+2)^(1/2),x)

[Out]

-1/9*(-6*x^2+3)^(1/2)*2^(1/2)*(2*x^2+1)^(1/2)*(2^(1/2)*3^(1/2)*(-6*x^2+3)^(1/2)*(2*x^2+1)^(1/2)*EllipticF(x*2^
(1/2),I)-12*x^5+3*x)/(4*x^4-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{4 \, x^{2} + 2} \sqrt{-6 \, x^{2} + 3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2+3)^(1/2)*(4*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(4*x^2 + 2)*sqrt(-6*x^2 + 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{4 \, x^{2} + 2} \sqrt{-6 \, x^{2} + 3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2+3)^(1/2)*(4*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(4*x^2 + 2)*sqrt(-6*x^2 + 3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \sqrt{6} \int \sqrt{1 - 2 x^{2}} \sqrt{2 x^{2} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x**2+3)**(1/2)*(4*x**2+2)**(1/2),x)

[Out]

sqrt(6)*Integral(sqrt(1 - 2*x**2)*sqrt(2*x**2 + 1), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{4 \, x^{2} + 2} \sqrt{-6 \, x^{2} + 3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2+3)^(1/2)*(4*x^2+2)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(4*x^2 + 2)*sqrt(-6*x^2 + 3), x)