### 3.897 $$\int \frac{\sqrt{12-3 e^2 x^2}}{\sqrt{2+e x}} \, dx$$

Optimal. Leaf size=20 $-\frac{2 (2-e x)^{3/2}}{\sqrt{3} e}$

[Out]

(-2*(2 - e*x)^(3/2))/(Sqrt*e)

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Rubi [A]  time = 0.0090526, antiderivative size = 20, 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} $-\frac{2 (2-e x)^{3/2}}{\sqrt{3} e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0380533, size = 34, normalized size = 1.7 $\frac{2 (e x-2) \sqrt{4-e^2 x^2}}{e \sqrt{3 e x+6}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.04, size = 30, normalized size = 1.5 \begin{align*}{\frac{2\,ex-4}{3\,e}\sqrt{-3\,{e}^{2}{x}^{2}+12}{\frac{1}{\sqrt{ex+2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C]  time = 1.87714, size = 34, normalized size = 1.7 \begin{align*} \frac{{\left (2 i \, \sqrt{3} e x - 4 i \, \sqrt{3}\right )} \sqrt{e x - 2}}{3 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*e^2*x^2+12)^(1/2)/(e*x+2)^(1/2),x, algorithm="maxima")

[Out]

1/3*(2*I*sqrt(3)*e*x - 4*I*sqrt(3))*sqrt(e*x - 2)/e

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Fricas [B]  time = 2.04526, size = 88, normalized size = 4.4 \begin{align*} \frac{2 \, \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}{\left (e x - 2\right )}}{3 \,{\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)^(1/2)/(e*x+2)^(1/2),x, algorithm="fricas")

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*e^2*x^2+12)^(1/2)/(e*x+2)^(1/2),x, algorithm="giac")

[Out]

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