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

Optimal. Leaf size=74 \[ 3 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (2 x+1)^{3/2}+\frac {9}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {2 x+1}+\frac {9}{4} \sqrt {\frac {3}{2}} \sin ^{-1}(2 x) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {38, 41, 216} \begin {gather*} 3 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (2 x+1)^{3/2}+\frac {9}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {2 x+1}+\frac {9}{4} \sqrt {\frac {3}{2}} \sin ^{-1}(2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 - 6*x)^(3/2)*(2 + 4*x)^(3/2),x]

[Out]

(9*Sqrt[3/2]*Sqrt[1 - 2*x]*x*Sqrt[1 + 2*x])/2 + 3*Sqrt[3/2]*(1 - 2*x)^(3/2)*x*(1 + 2*x)^(3/2) + (9*Sqrt[3/2]*A
rcSin[2*x])/4

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 (3-6 x)^{3/2} (2+4 x)^{3/2} \, dx &=3 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+\frac {9}{2} \int \sqrt {3-6 x} \sqrt {2+4 x} \, dx\\ &=\frac {9}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+3 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+\frac {27}{2} \int \frac {1}{\sqrt {3-6 x} \sqrt {2+4 x}} \, dx\\ &=\frac {9}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+3 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+\frac {27}{2} \int \frac {1}{\sqrt {6-24 x^2}} \, dx\\ &=\frac {9}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+3 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+\frac {9}{4} \sqrt {\frac {3}{2}} \sin ^{-1}(2 x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 39, normalized size = 0.53 \begin {gather*} \frac {3}{4} \sqrt {\frac {3}{2}} \left (2 x \sqrt {1-4 x^2} \left (5-8 x^2\right )+3 \sin ^{-1}(2 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 - 6*x)^(3/2)*(2 + 4*x)^(3/2),x]

[Out]

(3*Sqrt[3/2]*(2*x*(5 - 8*x^2)*Sqrt[1 - 4*x^2] + 3*ArcSin[2*x]))/4

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 0.84, size = 179, normalized size = 2.42 \begin {gather*} \frac {-12 \sqrt {6} \sqrt {1-2 x} x \sqrt {2 x+1} \left (8 x^2-5\right ) \left (-56 x^3-364 x^2-490 x-169\right )-12 \sqrt {3} \sqrt {1-2 x} x \left (8 x^2-5\right ) \left (16 x^4+368 x^3+1088 x^2+932 x+239\right )}{-896 x^3-5824 x^2+\sqrt {2} \sqrt {2 x+1} \left (64 x^3+1440 x^2+3632 x+1912\right )-7840 x-2704}+9 \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {2 x+1}-\sqrt {2}}{\sqrt {1-2 x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(3 - 6*x)^(3/2)*(2 + 4*x)^(3/2),x]

[Out]

(-12*Sqrt[6]*Sqrt[1 - 2*x]*x*Sqrt[1 + 2*x]*(-5 + 8*x^2)*(-169 - 490*x - 364*x^2 - 56*x^3) - 12*Sqrt[3]*Sqrt[1
- 2*x]*x*(-5 + 8*x^2)*(239 + 932*x + 1088*x^2 + 368*x^3 + 16*x^4))/(-2704 - 7840*x - 5824*x^2 - 896*x^3 + Sqrt
[2]*Sqrt[1 + 2*x]*(1912 + 3632*x + 1440*x^2 + 64*x^3)) + 9*Sqrt[3/2]*ArcTan[(-Sqrt[2] + Sqrt[1 + 2*x])/Sqrt[1
- 2*x]]

________________________________________________________________________________________

fricas [A]  time = 0.87, size = 60, normalized size = 0.81 \begin {gather*} -\frac {3}{4} \, {\left (8 \, x^{3} - 5 \, x\right )} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3} - \frac {9}{8} \, \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)^(3/2)*(4*x+2)^(3/2),x, algorithm="fricas")

[Out]

-3/4*(8*x^3 - 5*x)*sqrt(4*x + 2)*sqrt(-6*x + 3) - 9/8*sqrt(3)*sqrt(2)*arctan(1/12*sqrt(3)*sqrt(2)*sqrt(4*x + 2
)*sqrt(-6*x + 3)/x)

________________________________________________________________________________________

giac [B]  time = 0.98, size = 125, normalized size = 1.69 \begin {gather*} -\frac {1}{8} \, \sqrt {3} \sqrt {2} {\left ({\left ({\left (4 \, {\left (3 \, x - 5\right )} {\left (2 \, x + 1\right )} + 43\right )} {\left (2 \, x + 1\right )} - 39\right )} \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} + 4 \, {\left ({\left (4 \, x - 5\right )} {\left (2 \, x + 1\right )} + 9\right )} \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} - 24 \, \sqrt {2 \, x + 1} {\left (x - 1\right )} \sqrt {-2 \, x + 1} - 24 \, \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} - 18 \, \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)^(3/2)*(4*x+2)^(3/2),x, algorithm="giac")

[Out]

-1/8*sqrt(3)*sqrt(2)*(((4*(3*x - 5)*(2*x + 1) + 43)*(2*x + 1) - 39)*sqrt(2*x + 1)*sqrt(-2*x + 1) + 4*((4*x - 5
)*(2*x + 1) + 9)*sqrt(2*x + 1)*sqrt(-2*x + 1) - 24*sqrt(2*x + 1)*(x - 1)*sqrt(-2*x + 1) - 24*sqrt(2*x + 1)*sqr
t(-2*x + 1) - 18*arcsin(1/2*sqrt(2)*sqrt(2*x + 1)))

________________________________________________________________________________________

maple [B]  time = 0.00, size = 102, normalized size = 1.38 \begin {gather*} \frac {9 \sqrt {\left (4 x +2\right ) \left (-6 x +3\right )}\, \sqrt {6}\, \arcsin \left (2 x \right )}{8 \sqrt {4 x +2}\, \sqrt {-6 x +3}}+\frac {\left (-6 x +3\right )^{\frac {3}{2}} \left (4 x +2\right )^{\frac {5}{2}}}{16}+\frac {3 \left (4 x +2\right )^{\frac {5}{2}} \sqrt {-6 x +3}}{16}-\frac {3 \left (4 x +2\right )^{\frac {3}{2}} \sqrt {-6 x +3}}{16}-\frac {9 \sqrt {-6 x +3}\, \sqrt {4 x +2}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(-6*x+3)^(3/2)*(4*x+2)^(5/2)+3/16*(4*x+2)^(5/2)*(-6*x+3)^(1/2)-3/16*(4*x+2)^(3/2)*(-6*x+3)^(1/2)-9/8*(-6*
x+3)^(1/2)*(4*x+2)^(1/2)+9/8*((4*x+2)*(-6*x+3))^(1/2)/(4*x+2)^(1/2)/(-6*x+3)^(1/2)*6^(1/2)*arcsin(2*x)

________________________________________________________________________________________

maxima [A]  time = 2.86, size = 34, normalized size = 0.46 \begin {gather*} \frac {1}{4} \, {\left (-24 \, x^{2} + 6\right )}^{\frac {3}{2}} x + \frac {9}{4} \, \sqrt {-24 \, x^{2} + 6} x + \frac {9}{8} \, \sqrt {6} \arcsin \left (2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(-24*x^2 + 6)^(3/2)*x + 9/4*sqrt(-24*x^2 + 6)*x + 9/8*sqrt(6)*arcsin(2*x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (4\,x+2\right )}^{3/2}\,{\left (3-6\,x\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________