3.922 \(\int \frac {(2+e x)^{3/2}}{(12-3 e^2 x^2)^{3/2}} \, dx\)

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 = {627, 32} \[ \frac {2}{3 \sqrt {3} e \sqrt {2-e x}} \]

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 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
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 = 0.04, size = 30, normalized size = 1.36 \[ \frac {2 \sqrt {e x+2}}{3 e \sqrt {12-3 e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.96, size = 34, normalized size = 1.55 \[ -\frac {2 \, \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{9 \, {\left (e^{3} x^{2} - 4 \, e\right )}} \]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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]

sage0*x

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maple [A]  time = 0.05, size = 30, normalized size = 1.36 \[ -\frac {2 \left (e x -2\right ) \left (e x +2\right )^{\frac {3}{2}}}{\left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}} e} \]

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)

[Out]

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

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maxima [C]  time = 3.02, size = 15, normalized size = 0.68 \[ -\frac {2 i \, \sqrt {3}}{9 \, \sqrt {e x - 2} e} \]

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)/(sqrt(e*x - 2)*e)

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

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

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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