### 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/(3*Sqrt[3]*e*Sqrt[2 - e*x])

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Rubi [A]  time = 0.0096814, antiderivative size = 22, normalized size of antiderivative = 1., 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 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]))

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]

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*}

Mathematica [A]  time = 0.0455999, 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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Maple [A]  time = 0.04, size = 30, normalized size = 1.4 \begin{align*} -2\,{\frac{ \left ( ex-2 \right ) \left ( ex+2 \right ) ^{3/2}}{e \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{3/2}}} \end{align*}

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 = 1.71926, size = 20, normalized size = 0.91 \begin{align*} -\frac{2 i \, \sqrt{3}}{9 \, \sqrt{e x - 2} e} \end{align*}

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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Fricas [B]  time = 1.67795, size = 78, normalized size = 3.55 \begin{align*} -\frac{2 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{9 \,{\left (e^{3} x^{2} - 4 \, e\right )}} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

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