3.1167 \(\int \frac{1}{(6-2 x)^{3/2} (3+x)^{3/2}} \, dx\)

Optimal. Leaf size=26 \[ \frac{x}{18 \sqrt{2} \sqrt{3-x} \sqrt{x+3}} \]

[Out]

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

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Rubi [A]  time = 0.0016731, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {39} \[ \frac{x}{18 \sqrt{2} \sqrt{3-x} \sqrt{x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(18*Sqrt[2]*Sqrt[3 - x]*Sqrt[3 + 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}{(6-2 x)^{3/2} (3+x)^{3/2}} \, dx &=\frac{x}{18 \sqrt{2} \sqrt{3-x} \sqrt{3+x}}\\ \end{align*}

Mathematica [A]  time = 0.0123053, size = 21, normalized size = 0.81 \[ \frac{x}{18 \sqrt{6-2 x} \sqrt{x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(18*Sqrt[6 - 2*x]*Sqrt[3 + x])

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Maple [A]  time = 0.001, size = 19, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -3+x \right ) x}{9}{\frac{1}{\sqrt{3+x}}} \left ( 6-2\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00331, size = 16, normalized size = 0.62 \begin{align*} \frac{x}{18 \, \sqrt{-2 \, x^{2} + 18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*x/sqrt(-2*x^2 + 18)

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Fricas [A]  time = 1.48798, size = 62, normalized size = 2.38 \begin{align*} -\frac{\sqrt{x + 3} x \sqrt{-2 \, x + 6}}{36 \,{\left (x^{2} - 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 123.053, size = 90, normalized size = 3.46 \begin{align*} \begin{cases} \frac{\sqrt{2}}{36 \sqrt{-1 + \frac{6}{x + 3}}} - \frac{\sqrt{2}}{12 \sqrt{-1 + \frac{6}{x + 3}} \left (x + 3\right )} & \text{for}\: \frac{6}{\left |{x + 3}\right |} > 1 \\- \frac{\sqrt{2} i \sqrt{1 - \frac{6}{x + 3}} \left (x + 3\right )}{36 x - 108} + \frac{3 \sqrt{2} i \sqrt{1 - \frac{6}{x + 3}}}{36 x - 108} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((sqrt(2)/(36*sqrt(-1 + 6/(x + 3))) - sqrt(2)/(12*sqrt(-1 + 6/(x + 3))*(x + 3)), 6/Abs(x + 3) > 1), (
-sqrt(2)*I*sqrt(1 - 6/(x + 3))*(x + 3)/(36*x - 108) + 3*sqrt(2)*I*sqrt(1 - 6/(x + 3))/(36*x - 108), True))

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Giac [B]  time = 1.07278, size = 96, normalized size = 3.69 \begin{align*} \frac{\sqrt{2}{\left (\sqrt{6} - \sqrt{-x + 3}\right )}}{144 \, \sqrt{x + 3}} - \frac{\sqrt{2} \sqrt{x + 3} \sqrt{-x + 3}}{72 \,{\left (x - 3\right )}} - \frac{\sqrt{2} \sqrt{x + 3}}{144 \,{\left (\sqrt{6} - \sqrt{-x + 3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*sqrt(2)*(sqrt(6) - sqrt(-x + 3))/sqrt(x + 3) - 1/72*sqrt(2)*sqrt(x + 3)*sqrt(-x + 3)/(x - 3) - 1/144*sqr
t(2)*sqrt(x + 3)/(sqrt(6) - sqrt(-x + 3))