\(\int \sqrt {2+4 x^2} \sqrt {3+6 x^2} \, dx\) [180]

Optimal result

Integrand size = 23, antiderivative size = 20 \[ \int \sqrt {2+4 x^2} \sqrt {3+6 x^2} \, dx=\sqrt {6} x+2 \sqrt {\frac {2}{3}} x^3 \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {22} \[ \int \sqrt {2+4 x^2} \sqrt {3+6 x^2} \, dx=2 \sqrt {\frac {2}{3}} x^3+\sqrt {6} x \]

[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*} \text {integral}& = \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] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \sqrt {2+4 x^2} \sqrt {3+6 x^2} \, dx=\sqrt {6} \left (x+\frac {2 x^3}{3}\right ) \]

[In]

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

[Out]

Sqrt[6]*(x + (2*x^3)/3)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 2 vs. order 1.

Time = 2.36 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.90

method	result	size
gosper	\(\frac {x \left (2 x^{2}+3\right ) \sqrt {4 x^{2}+2}}{\sqrt {6 x^{2}+3}}\)	\(38\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (14) = 28\).

Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.90 \[ \int \sqrt {2+4 x^2} \sqrt {3+6 x^2} \, dx=\frac {{\left (2 \, x^{3} + 3 \, x\right )} \sqrt {6 \, x^{2} + 3} \sqrt {4 \, x^{2} + 2}}{3 \, {\left (2 \, x^{2} + 1\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 1.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \sqrt {2+4 x^2} \sqrt {3+6 x^2} \, dx=\frac {2 \sqrt {6} x^{3}}{3} + \sqrt {6} x \]

[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

Maxima [F]

\[ \int \sqrt {2+4 x^2} \sqrt {3+6 x^2} \, dx=\int { \sqrt {6 \, x^{2} + 3} \sqrt {4 \, x^{2} + 2} \,d x } \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \sqrt {2+4 x^2} \sqrt {3+6 x^2} \, dx=\frac {1}{3} \, \sqrt {3} \sqrt {2} {\left (2 \, x^{3} + 3 \, x\right )} \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {2+4 x^2} \sqrt {3+6 x^2} \, dx=\int \sqrt {4\,x^2+2}\,\sqrt {6\,x^2+3} \,d x \]

[In]

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

[Out]

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