3.12.66 \(\int \frac {1}{(3-b x)^{3/2} (3+b x)^{3/2}} \, dx\) [1166]

Optimal. Leaf size=24 \[ \frac {x}{9 \sqrt {3-b x} \sqrt {3+b x}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {39} \begin {gather*} \frac {x}{9 \sqrt {3-b x} \sqrt {b x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.04, size = 19, normalized size = 0.79 \begin {gather*} \frac {x}{9 \sqrt {9-b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/(9*Sqrt[9 - b^2*x^2])

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 4.27, size = 69, normalized size = 2.88 \begin {gather*} \frac {-I \text {meijerg}\left [\left \{\left \{\frac {3}{4},\frac {5}{4},1\right \},\left \{\frac {1}{2},\frac {3}{2},2\right \}\right \},\left \{\left \{\frac {3}{4},1,\frac {5}{4},\frac {3}{2},2\right \},\left \{0\right \}\right \},\frac {9}{b^2 x^2}\right ]+\text {meijerg}\left [\left \{\left \{-\frac {1}{2},0,\frac {1}{4},\frac {1}{2},\frac {3}{4},1\right \},\left \{\right \}\right \},\left \{\left \{\frac {1}{4},\frac {3}{4}\right \},\left \{-\frac {1}{2},0,1,0\right \}\right \},\frac {9 \text {exp\_polar}\left [-2 I \text {Pi}\right ]}{b^2 x^2}\right ]}{18 \text {Pi}^{\frac {3}{2}} b} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-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)] + m
eijerg[{{-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)]) / (18 Pi ^ (3 / 2) b)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(41\) vs. \(2(18)=36\).
time = 0.15, size = 42, normalized size = 1.75

method	result	size
gosper	\(\frac {x}{9 \sqrt {-b x +3}\, \sqrt {b x +3}}\)	\(19\)
default	\(\frac {1}{3 b \sqrt {-b x +3}\, \sqrt {b x +3}}-\frac {\sqrt {-b x +3}}{9 b \sqrt {b x +3}}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 15, normalized size = 0.62 \begin {gather*} \frac {x}{9 \, \sqrt {-b^{2} x^{2} + 9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x/sqrt(-b^2*x^2 + 9)

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Fricas [A]
time = 0.29, size = 29, normalized size = 1.21 \begin {gather*} -\frac {\sqrt {b x + 3} \sqrt {-b x + 3} x}{9 \, {\left (b^{2} x^{2} - 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 2.40, size = 73, normalized size = 3.04 \begin {gather*} - \frac {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 )}}{18 \pi ^{\frac {3}{2}} b} + \frac {{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 )}}{18 \pi ^{\frac {3}{2}} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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))/(18*pi**(3/2)*b) + me
ijerg(((-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))/(18*
pi**(3/2)*b)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (18) = 36\).
time = 0.00, size = 103, normalized size = 4.29 \begin {gather*} \frac {2 \left (\frac {\sqrt {-b x+3}}{36 \left (2 \sqrt {6}-2 \sqrt {b x+3}\right )}-\frac {2 \sqrt {6}-2 \sqrt {b x+3}}{144 \sqrt {-b x+3}}-\frac {\sqrt {-b x+3} \sqrt {b x+3}}{36 \left (b x+3\right )}\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36*((sqrt(6) - sqrt(b*x + 3))/sqrt(-b*x + 3) + 2*sqrt(-b*x + 3)/sqrt(b*x + 3) - sqrt(-b*x + 3)/(sqrt(6) - s
qrt(b*x + 3)))/b

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Mupad [B]
time = 0.46, size = 26, normalized size = 1.08 \begin {gather*} -\frac {x\,\sqrt {3-b\,x}}{\sqrt {b\,x+3}\,\left (9\,b\,x-27\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*(3 - b*x)^(1/2))/((b*x + 3)^(1/2)*(9*b*x - 27))

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