\(\int \frac {1}{(6-2 b x)^{3/2} (3+b x)^{3/2}} \, dx\) [1168]

Optimal result

Integrand size = 20, antiderivative size = 29 \[ \int \frac {1}{(6-2 b x)^{3/2} (3+b x)^{3/2}} \, dx=\frac {x}{18 \sqrt {2} \sqrt {3-b x} \sqrt {3+b x}} \]

[Out]

1/36*x*2^(1/2)/(-b*x+3)^(1/2)/(b*x+3)^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {39} \[ \int \frac {1}{(6-2 b x)^{3/2} (3+b x)^{3/2}} \, dx=\frac {x}{18 \sqrt {2} \sqrt {3-b x} \sqrt {b x+3}} \]

[In]

Int[1/((6 - 2*b*x)^(3/2)*(3 + b*x)^(3/2)),x]

[Out]

x/(18*Sqrt[2]*Sqrt[3 - b*x]*Sqrt[3 + b*x])

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {x}{18 \sqrt {2} \sqrt {3-b x} \sqrt {3+b x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {1}{(6-2 b x)^{3/2} (3+b x)^{3/2}} \, dx=\frac {x}{18 \sqrt {18-2 b^2 x^2}} \]

[In]

Integrate[1/((6 - 2*b*x)^(3/2)*(3 + b*x)^(3/2)),x]

[Out]

x/(18*Sqrt[18 - 2*b^2*x^2])

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83

method	result	size
gosper	\(-\frac {\left (b x -3\right ) x}{9 \sqrt {b x +3}\, \left (-2 b x +6\right )^{\frac {3}{2}}}\)	\(24\)
default	\(\frac {1}{6 b \sqrt {-2 b x +6}\, \sqrt {b x +3}}-\frac {\sqrt {-2 b x +6}}{36 b \sqrt {b x +3}}\)	\(42\)

[In]

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

[Out]

-1/9*(b*x-3)/(b*x+3)^(1/2)*x/(-2*b*x+6)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(6-2 b x)^{3/2} (3+b x)^{3/2}} \, dx=-\frac {\sqrt {b x + 3} \sqrt {-2 \, b x + 6} x}{36 \, {\left (b^{2} x^{2} - 9\right )}} \]

[In]

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

[Out]

-1/36*sqrt(b*x + 3)*sqrt(-2*b*x + 6)*x/(b^2*x^2 - 9)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 13.19 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.86 \[ \int \frac {1}{(6-2 b x)^{3/2} (3+b x)^{3/2}} \, dx=- \frac {\sqrt {2} i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, 1 & \frac {1}{2}, \frac {3}{2}, 2 \\\frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 2 & 0 \end {matrix} \middle | {\frac {9}{b^{2} x^{2}}} \right )}}{72 \pi ^{\frac {3}{2}} b} + \frac {\sqrt {2} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1 & \\\frac {1}{4}, \frac {3}{4} & - \frac {1}{2}, 0, 1, 0 \end {matrix} \middle | {\frac {9 e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{72 \pi ^{\frac {3}{2}} b} \]

[In]

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

[Out]

-sqrt(2)*I*meijerg(((3/4, 5/4, 1), (1/2, 3/2, 2)), ((3/4, 1, 5/4, 3/2, 2), (0,)), 9/(b**2*x**2))/(72*pi**(3/2)
*b) + sqrt(2)*meijerg(((-1/2, 0, 1/4, 1/2, 3/4, 1), ()), ((1/4, 3/4), (-1/2, 0, 1, 0)), 9*exp_polar(-2*I*pi)/(
b**2*x**2))/(72*pi**(3/2)*b)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.52 \[ \int \frac {1}{(6-2 b x)^{3/2} (3+b x)^{3/2}} \, dx=\frac {x}{18 \, \sqrt {-2 \, b^{2} x^{2} + 18}} \]

[In]

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

[Out]

1/18*x/sqrt(-2*b^2*x^2 + 18)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (21) = 42\).

Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.72 \[ \int \frac {1}{(6-2 b x)^{3/2} (3+b x)^{3/2}} \, dx=-\frac {\frac {\sqrt {2} {\left (\sqrt {6} - \sqrt {b x + 3}\right )}}{\sqrt {-b x + 3}} + \frac {2 \, \sqrt {2} \sqrt {-b x + 3}}{\sqrt {b x + 3}} - \frac {\sqrt {2} \sqrt {-b x + 3}}{\sqrt {6} - \sqrt {b x + 3}}}{144 \, b} \]

[In]

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

[Out]

-1/144*(sqrt(2)*(sqrt(6) - sqrt(b*x + 3))/sqrt(-b*x + 3) + 2*sqrt(2)*sqrt(-b*x + 3)/sqrt(b*x + 3) - sqrt(2)*sq
rt(-b*x + 3)/(sqrt(6) - sqrt(b*x + 3)))/b

Mupad [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(6-2 b x)^{3/2} (3+b x)^{3/2}} \, dx=-\frac {x\,\sqrt {6-2\,b\,x}}{\sqrt {b\,x+3}\,\left (36\,b\,x-108\right )} \]

[In]

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

[Out]

-(x*(6 - 2*b*x)^(1/2))/((b*x + 3)^(1/2)*(36*b*x - 108))