3.1155 \(\int \sqrt{3-6 x} \sqrt{2+4 x} \, dx\)

Optimal. Leaf size=43 \[ \sqrt{\frac{3}{2}} \sqrt{1-2 x} \sqrt{2 x+1} x+\frac{1}{2} \sqrt{\frac{3}{2}} \sin ^{-1}(2 x) \]

[Out]

Sqrt[3/2]*Sqrt[1 - 2*x]*x*Sqrt[1 + 2*x] + (Sqrt[3/2]*ArcSin[2*x])/2

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Rubi [A]  time = 0.0057703, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {38, 41, 216} \[ \sqrt{\frac{3}{2}} \sqrt{1-2 x} \sqrt{2 x+1} x+\frac{1}{2} \sqrt{\frac{3}{2}} \sin ^{-1}(2 x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[3 - 6*x]*Sqrt[2 + 4*x],x]

[Out]

Sqrt[3/2]*Sqrt[1 - 2*x]*x*Sqrt[1 + 2*x] + (Sqrt[3/2]*ArcSin[2*x])/2

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(x*(a + b*x)^m*(c + d*x)^m)/(2*m + 1)
, x] + Dist[(2*a*c*m)/(2*m + 1), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[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 \sqrt{3-6 x} \sqrt{2+4 x} \, dx &=\sqrt{\frac{3}{2}} \sqrt{1-2 x} x \sqrt{1+2 x}+3 \int \frac{1}{\sqrt{3-6 x} \sqrt{2+4 x}} \, dx\\ &=\sqrt{\frac{3}{2}} \sqrt{1-2 x} x \sqrt{1+2 x}+3 \int \frac{1}{\sqrt{6-24 x^2}} \, dx\\ &=\sqrt{\frac{3}{2}} \sqrt{1-2 x} x \sqrt{1+2 x}+\frac{1}{2} \sqrt{\frac{3}{2}} \sin ^{-1}(2 x)\\ \end{align*}

Mathematica [A]  time = 0.009989, size = 30, normalized size = 0.7 \[ \frac{1}{2} \sqrt{\frac{3}{2}} \left (2 \sqrt{1-4 x^2} x+\sin ^{-1}(2 x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.003, size = 70, normalized size = 1.6 \begin{align*} -{\frac{1}{12}\sqrt{2+4\,x} \left ( 3-6\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1}{4}\sqrt{3-6\,x}\sqrt{2+4\,x}}+{\frac{\arcsin \left ( 2\,x \right ) \sqrt{6}}{4}\sqrt{ \left ( 2+4\,x \right ) \left ( 3-6\,x \right ) }{\frac{1}{\sqrt{3-6\,x}}}{\frac{1}{\sqrt{2+4\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(2+4*x)^(1/2)*(3-6*x)^(3/2)+1/4*(3-6*x)^(1/2)*(2+4*x)^(1/2)+1/4*((2+4*x)*(3-6*x))^(1/2)/(2+4*x)^(1/2)/(3
-6*x)^(1/2)*6^(1/2)*arcsin(2*x)

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Maxima [A]  time = 1.48093, size = 30, normalized size = 0.7 \begin{align*} \frac{1}{2} \, \sqrt{-24 \, x^{2} + 6} x + \frac{1}{4} \, \sqrt{6} \arcsin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-24*x^2 + 6)*x + 1/4*sqrt(6)*arcsin(2*x)

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Fricas [A]  time = 1.46261, size = 159, normalized size = 3.7 \begin{align*} \frac{1}{2} \, \sqrt{4 \, x + 2} x \sqrt{-6 \, x + 3} - \frac{1}{4} \, \sqrt{3} \sqrt{2} \arctan \left (\frac{\sqrt{3} \sqrt{2} \sqrt{4 \, x + 2} \sqrt{-6 \, x + 3}}{12 \, x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(4*x + 2)*x*sqrt(-6*x + 3) - 1/4*sqrt(3)*sqrt(2)*arctan(1/12*sqrt(3)*sqrt(2)*sqrt(4*x + 2)*sqrt(-6*x +
3)/x)

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Sympy [B]  time = 5.52426, size = 187, normalized size = 4.35 \begin{align*} \begin{cases} - \frac{\sqrt{6} i \operatorname{acosh}{\left (\sqrt{x + \frac{1}{2}} \right )}}{2} + \frac{\sqrt{6} i \left (x + \frac{1}{2}\right )^{\frac{5}{2}}}{\sqrt{x - \frac{1}{2}}} - \frac{3 \sqrt{6} i \left (x + \frac{1}{2}\right )^{\frac{3}{2}}}{2 \sqrt{x - \frac{1}{2}}} + \frac{\sqrt{6} i \sqrt{x + \frac{1}{2}}}{2 \sqrt{x - \frac{1}{2}}} & \text{for}\: \left |{x + \frac{1}{2}}\right | > 1 \\\frac{\sqrt{6} \operatorname{asin}{\left (\sqrt{x + \frac{1}{2}} \right )}}{2} - \frac{\sqrt{6} \left (x + \frac{1}{2}\right )^{\frac{5}{2}}}{\sqrt{\frac{1}{2} - x}} + \frac{3 \sqrt{6} \left (x + \frac{1}{2}\right )^{\frac{3}{2}}}{2 \sqrt{\frac{1}{2} - x}} - \frac{\sqrt{6} \sqrt{x + \frac{1}{2}}}{2 \sqrt{\frac{1}{2} - x}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-sqrt(6)*I*acosh(sqrt(x + 1/2))/2 + sqrt(6)*I*(x + 1/2)**(5/2)/sqrt(x - 1/2) - 3*sqrt(6)*I*(x + 1/2
)**(3/2)/(2*sqrt(x - 1/2)) + sqrt(6)*I*sqrt(x + 1/2)/(2*sqrt(x - 1/2)), Abs(x + 1/2) > 1), (sqrt(6)*asin(sqrt(
x + 1/2))/2 - sqrt(6)*(x + 1/2)**(5/2)/sqrt(1/2 - x) + 3*sqrt(6)*(x + 1/2)**(3/2)/(2*sqrt(1/2 - x)) - sqrt(6)*
sqrt(x + 1/2)/(2*sqrt(1/2 - x)), True))

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Giac [A]  time = 1.06266, size = 51, normalized size = 1.19 \begin{align*} \frac{1}{2} \, \sqrt{3} \sqrt{2}{\left (\sqrt{2 \, x + 1} x \sqrt{-2 \, x + 1} + \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{2 \, x + 1}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(3)*sqrt(2)*(sqrt(2*x + 1)*x*sqrt(-2*x + 1) + arcsin(1/2*sqrt(2)*sqrt(2*x + 1)))