3.12.55 \(\int \sqrt {3-6 x} \sqrt {2+4 x} \, dx\) [1155]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 43, normalized size of antiderivative = 1.00, 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, 222} \begin {gather*} \frac {1}{2} \sqrt {\frac {3}{2}} \text {ArcSin}(2 x)+\sqrt {\frac {3}{2}} \sqrt {1-2 x} \sqrt {2 x+1} x \end {gather*}

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 222

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 [B] Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(43)=86\).
time = 0.66, size = 108, normalized size = 2.51 \begin {gather*} \frac {\sqrt {3-6 x} x \left (7+4 x^2-5 \sqrt {2+4 x}+x \left (16-6 \sqrt {2+4 x}\right )\right )}{-10+7 \sqrt {2+4 x}+2 x \left (-6+\sqrt {2+4 x}\right )}+\sqrt {6} \tan ^{-1}\left (\frac {-\sqrt {2}+\sqrt {1+2 x}}{\sqrt {1-2 x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[3 - 6*x]*x*(7 + 4*x^2 - 5*Sqrt[2 + 4*x] + x*(16 - 6*Sqrt[2 + 4*x])))/(-10 + 7*Sqrt[2 + 4*x] + 2*x*(-6 +
Sqrt[2 + 4*x])) + Sqrt[6]*ArcTan[(-Sqrt[2] + Sqrt[1 + 2*x])/Sqrt[1 - 2*x]]

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(30)=60\).
time = 0.18, size = 70, normalized size = 1.63

method	result	size
default	\(\frac {\left (2+4 x \right )^{\frac {3}{2}} \sqrt {3-6 x}}{8}-\frac {\sqrt {3-6 x}\, \sqrt {2+4 x}}{4}+\frac {\sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \arcsin \left (2 x \right ) \sqrt {6}}{4 \sqrt {2+4 x}\, \sqrt {3-6 x}}\)	\(70\)
risch	\(-\frac {x \left (2 x -1\right ) \left (1+2 x \right ) \sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \sqrt {6}}{2 \sqrt {-\left (2 x -1\right ) \left (1+2 x \right )}\, \sqrt {3-6 x}\, \sqrt {2+4 x}}+\frac {\sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \arcsin \left (2 x \right ) \sqrt {6}}{4 \sqrt {2+4 x}\, \sqrt {3-6 x}}\)	\(95\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2+4*x)^(3/2)*(3-6*x)^(1/2)-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)*arcsin(2*x)*6^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.48, size = 22, normalized size = 0.51 \begin {gather*} \frac {1}{2} \, \sqrt {-24 \, x^{2} + 6} x + \frac {1}{4} \, \sqrt {6} \arcsin \left (2 \, x\right ) \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 1.17, size = 52, normalized size = 1.21 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 2.10, size = 187, normalized size = 4.35 \begin {gather*} \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 {gather*}

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))

________________________________________________________________________________________

Giac [A]
time = 1.30, size = 55, normalized size = 1.28 \begin {gather*} \frac {1}{2} \, \sqrt {3} \sqrt {2} {\left (\sqrt {2 \, x + 1} {\left (x - 1\right )} \sqrt {-2 \, x + 1} + \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} + \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {2 \, x + 1}\right )\right )} \end {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 0.26, size = 44, normalized size = 1.02 \begin {gather*} \frac {x\,\sqrt {4\,x+2}\,\sqrt {3-6\,x}}{2}-\frac {\sqrt {6}\,\ln \left (x-\frac {\sqrt {1-2\,x}\,\sqrt {2\,x+1}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________