3.10.22 \(\int \frac {(2+e x)^{3/2}}{(12-3 e^2 x^2)^{3/2}} \, dx\) [922]

Optimal. Leaf size=22 \[ \frac {2}{3 \sqrt {3} e \sqrt {2-e x}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {641, 32} \begin {gather*} \frac {2}{3 \sqrt {3} e \sqrt {2-e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + e*x)^(3/2)/(12 - 3*e^2*x^2)^(3/2),x]

[Out]

2/(3*Sqrt[3]*e*Sqrt[2 - e*x])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 641

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rubi steps

\begin {align*} \int \frac {(2+e x)^{3/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx &=\int \frac {1}{(6-3 e x)^{3/2}} \, dx\\ &=\frac {2}{3 \sqrt {3} e \sqrt {2-e x}}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 38, normalized size = 1.73 \begin {gather*} -\frac {2 \sqrt {4-e^2 x^2}}{3 e (-2+e x) \sqrt {6+3 e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + e*x)^(3/2)/(12 - 3*e^2*x^2)^(3/2),x]

[Out]

(-2*Sqrt[4 - e^2*x^2])/(3*e*(-2 + e*x)*Sqrt[6 + 3*e*x])

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Maple [A]
time = 0.48, size = 32, normalized size = 1.45

method	result	size
gosper	\(-\frac {2 \left (e x -2\right ) \left (e x +2\right )^{\frac {3}{2}}}{e \left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}}}\)	\(30\)
default	\(-\frac {2 \sqrt {-3 e^{2} x^{2}+12}}{9 \sqrt {e x +2}\, \left (e x -2\right ) e}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+2)^(3/2)/(-3*e^2*x^2+12)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/9/(e*x+2)^(1/2)*(-3*e^2*x^2+12)^(1/2)/(e*x-2)/e

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Maxima [C] Result contains complex when optimal does not.
time = 0.49, size = 15, normalized size = 0.68 \begin {gather*} -\frac {2 i \, \sqrt {3} e^{\left (-1\right )}}{9 \, \sqrt {x e - 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(3/2)/(-3*e^2*x^2+12)^(3/2),x, algorithm="maxima")

[Out]

-2/9*I*sqrt(3)*e^(-1)/sqrt(x*e - 2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (16) = 32\).
time = 3.36, size = 34, normalized size = 1.55 \begin {gather*} -\frac {2 \, \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2}}{9 \, {\left (x^{2} e^{3} - 4 \, e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(3/2)/(-3*e^2*x^2+12)^(3/2),x, algorithm="fricas")

[Out]

-2/9*sqrt(-3*x^2*e^2 + 12)*sqrt(x*e + 2)/(x^2*e^3 - 4*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\sqrt {3} \left (\int \frac {2 \sqrt {e x + 2}}{- e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {- e^{2} x^{2} + 4}}\, dx + \int \frac {e x \sqrt {e x + 2}}{- e^{2} x^{2} \sqrt {- e^{2} x^{2} + 4} + 4 \sqrt {- e^{2} x^{2} + 4}}\, dx\right )}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)**(3/2)/(-3*e**2*x**2+12)**(3/2),x)

[Out]

sqrt(3)*(Integral(2*sqrt(e*x + 2)/(-e**2*x**2*sqrt(-e**2*x**2 + 4) + 4*sqrt(-e**2*x**2 + 4)), x) + Integral(e*
x*sqrt(e*x + 2)/(-e**2*x**2*sqrt(-e**2*x**2 + 4) + 4*sqrt(-e**2*x**2 + 4)), x))/9

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Giac [A]
time = 0.97, size = 24, normalized size = 1.09 \begin {gather*} -\frac {1}{9} \, \sqrt {3} e^{\left (-1\right )} + \frac {2 \, \sqrt {3} e^{\left (-1\right )}}{9 \, \sqrt {-x e + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(3/2)/(-3*e^2*x^2+12)^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.21, size = 24, normalized size = 1.09 \begin {gather*} \frac {2\,\sqrt {e\,x+2}}{3\,e\,\sqrt {12-3\,e^2\,x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x + 2)^(3/2)/(12 - 3*e^2*x^2)^(3/2),x)

[Out]

(2*(e*x + 2)^(1/2))/(3*e*(12 - 3*e^2*x^2)^(1/2))

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