\(\int (3-6 x)^{3/2} (2+4 x)^{3/2} \, dx\) [1154]

Optimal result

Integrand size = 19, antiderivative size = 74 \[ \int (3-6 x)^{3/2} (2+4 x)^{3/2} \, dx=\frac {9}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+3 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+\frac {9}{4} \sqrt {\frac {3}{2}} \arcsin (2 x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {38, 41, 222} \[ \int (3-6 x)^{3/2} (2+4 x)^{3/2} \, dx=\frac {9}{4} \sqrt {\frac {3}{2}} \arcsin (2 x)+3 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (2 x+1)^{3/2}+\frac {9}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {2 x+1} \]

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

Time = 0.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.73 \[ \int (3-6 x)^{3/2} (2+4 x)^{3/2} \, dx=-\frac {3}{2} \sqrt {\frac {3}{2}} \left (x \sqrt {1-4 x^2} \left (-5+8 x^2\right )+3 \arctan \left (\frac {\sqrt {1-4 x^2}}{-1+2 x}\right )\right ) \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(101\) vs. \(2(50)=100\).

Time = 0.19 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.38

method	result	size
default	\(\frac {\left (3-6 x \right )^{\frac {3}{2}} \left (2+4 x \right )^{\frac {5}{2}}}{16}+\frac {3 \left (2+4 x \right )^{\frac {5}{2}} \sqrt {3-6 x}}{16}-\frac {3 \left (2+4 x \right )^{\frac {3}{2}} \sqrt {3-6 x}}{16}-\frac {9 \sqrt {3-6 x}\, \sqrt {2+4 x}}{8}+\frac {9 \sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \arcsin \left (2 x \right ) \sqrt {6}}{8 \sqrt {2+4 x}\, \sqrt {3-6 x}}\)	\(102\)
risch	\(\frac {3 x \left (8 x^{2}-5\right ) \left (-1+2 x \right ) \left (1+2 x \right ) \sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \sqrt {6}}{4 \sqrt {-\left (-1+2 x \right ) \left (1+2 x \right )}\, \sqrt {3-6 x}\, \sqrt {2+4 x}}+\frac {9 \sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \arcsin \left (2 x \right ) \sqrt {6}}{8 \sqrt {2+4 x}\, \sqrt {3-6 x}}\)	\(102\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.81 \[ \int (3-6 x)^{3/2} (2+4 x)^{3/2} \, dx=-\frac {3}{4} \, {\left (8 \, x^{3} - 5 \, x\right )} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3} - \frac {9}{8} \, \sqrt {3} \sqrt {2} \arctan \left (\frac {\sqrt {3} \sqrt {2} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3}}{12 \, x}\right ) \]

[In]

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

Sympy [F(-1)]

Timed out. \[ \int (3-6 x)^{3/2} (2+4 x)^{3/2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.46 \[ \int (3-6 x)^{3/2} (2+4 x)^{3/2} \, dx=\frac {1}{4} \, {\left (-24 \, x^{2} + 6\right )}^{\frac {3}{2}} x + \frac {9}{4} \, \sqrt {-24 \, x^{2} + 6} x + \frac {9}{8} \, \sqrt {6} \arcsin \left (2 \, x\right ) \]

[In]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (50) = 100\).

Time = 0.32 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.69 \[ \int (3-6 x)^{3/2} (2+4 x)^{3/2} \, dx=-\frac {1}{8} \, \sqrt {3} \sqrt {2} {\left ({\left ({\left (4 \, {\left (3 \, x - 5\right )} {\left (2 \, x + 1\right )} + 43\right )} {\left (2 \, x + 1\right )} - 39\right )} \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} + 4 \, {\left ({\left (4 \, x - 5\right )} {\left (2 \, x + 1\right )} + 9\right )} \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} - 24 \, \sqrt {2 \, x + 1} {\left (x - 1\right )} \sqrt {-2 \, x + 1} - 24 \, \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} - 18 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {2 \, x + 1}\right )\right )} \]

[In]

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

Mupad [F(-1)]

Timed out. \[ \int (3-6 x)^{3/2} (2+4 x)^{3/2} \, dx=\int {\left (4\,x+2\right )}^{3/2}\,{\left (3-6\,x\right )}^{3/2} \,d x \]

[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)