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

Optimal. Leaf size=20 \[ 2 \sqrt{\frac{2}{3}} x^3+\sqrt{6} x \]

[Out]

Sqrt[6]*x + 2*Sqrt[2/3]*x^3

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Rubi [A]  time = 0.0038428, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {22} \[ 2 \sqrt{\frac{2}{3}} x^3+\sqrt{6} x \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[6]*x + 2*Sqrt[2/3]*x^3

Rule 22

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m + n
), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] && GtQ[b/d, 0] &&  !(IntegerQ[m] || IntegerQ[n]
)

Rubi steps

\begin{align*} \int \sqrt{2+4 x^2} \sqrt{3+6 x^2} \, dx &=\sqrt{\frac{2}{3}} \int \left (3+6 x^2\right ) \, dx\\ &=\sqrt{6} x+2 \sqrt{\frac{2}{3}} x^3\\ \end{align*}

Mathematica [A]  time = 0.0019961, size = 15, normalized size = 0.75 \[ \sqrt{6} \left (\frac{2 x^3}{3}+x\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[6]*(x + (2*x^3)/3)

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Maple [C]  time = 0.001, size = 38, normalized size = 1.9 \begin{align*}{\frac{x \left ( 2\,{x}^{2}+3 \right ) }{6\,{x}^{2}+3}\sqrt{4\,{x}^{2}+2}\sqrt{6\,{x}^{2}+3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.76583, size = 85, normalized size = 4.25 \begin{align*} \frac{{\left (2 \, x^{3} + 3 \, x\right )} \sqrt{6 \, x^{2} + 3} \sqrt{4 \, x^{2} + 2}}{3 \,{\left (2 \, x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(2*x^3 + 3*x)*sqrt(6*x^2 + 3)*sqrt(4*x^2 + 2)/(2*x^2 + 1)

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Sympy [A]  time = 6.09365, size = 17, normalized size = 0.85 \begin{align*} \frac{2 \sqrt{6} x^{3}}{3} + \sqrt{6} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(6)*x**3/3 + sqrt(6)*x

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Giac [A]  time = 1.08836, size = 23, normalized size = 1.15 \begin{align*} \frac{1}{3} \, \sqrt{3} \sqrt{2}{\left (2 \, x^{3} + 3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(3)*sqrt(2)*(2*x^3 + 3*x)