### 3.905 $$\int \frac{(12-3 e^2 x^2)^{3/2}}{(2+e x)^{3/2}} \, dx$$

Optimal. Leaf size=22 $-\frac{6 \sqrt{3} (2-e x)^{5/2}}{5 e}$

[Out]

(-6*Sqrt[3]*(2 - e*x)^(5/2))/(5*e)

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Rubi [A]  time = 0.0094595, 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} $-\frac{6 \sqrt{3} (2-e x)^{5/2}}{5 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-6*Sqrt[3]*(2 - e*x)^(5/2))/(5*e)

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{\left (12-3 e^2 x^2\right )^{3/2}}{(2+e x)^{3/2}} \, dx &=\int (6-3 e x)^{3/2} \, dx\\ &=-\frac{6 \sqrt{3} (2-e x)^{5/2}}{5 e}\\ \end{align*}

Mathematica [A]  time = 0.0447272, size = 37, normalized size = 1.68 $-\frac{6 (e x-2)^2 \sqrt{12-3 e^2 x^2}}{5 e \sqrt{e x+2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-6*(-2 + e*x)^2*Sqrt[12 - 3*e^2*x^2])/(5*e*Sqrt[2 + e*x])

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Maple [A]  time = 0.042, size = 30, normalized size = 1.4 \begin{align*}{\frac{2\,ex-4}{5\,e} \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{{\frac{3}{2}}} \left ( ex+2 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(e*x-2)*(-3*e^2*x^2+12)^(3/2)/e/(e*x+2)^(3/2)

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Maxima [C]  time = 1.78105, size = 49, normalized size = 2.23 \begin{align*} -\frac{{\left (6 i \, \sqrt{3} e^{2} x^{2} - 24 i \, \sqrt{3} e x + 24 i \, \sqrt{3}\right )} \sqrt{e x - 2}}{5 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(6*I*sqrt(3)*e^2*x^2 - 24*I*sqrt(3)*e*x + 24*I*sqrt(3))*sqrt(e*x - 2)/e

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Fricas [B]  time = 1.7918, size = 105, normalized size = 4.77 \begin{align*} -\frac{6 \,{\left (e^{2} x^{2} - 4 \, e x + 4\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{5 \,{\left (e^{2} x + 2 \, e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/5*(e^2*x^2 - 4*e*x + 4)*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)/(e^2*x + 2*e)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{3}{2}}}{{\left (e x + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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