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]

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

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Rubi [A]  time = 0.01, antiderivative size = 20, 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} \[ -\frac {2 (2-e x)^{3/2}}{\sqrt {3} e} \]

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[3]*e)

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 {\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*}

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Mathematica [A]  time = 0.04, size = 34, normalized size = 1.70 \[ \frac {2 (e x-2) \sqrt {4-e^2 x^2}}{e \sqrt {3 e x+6}} \]

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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fricas [B]  time = 1.02, size = 37, normalized size = 1.85 \[ \frac {2 \, \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} {\left (e x - 2\right )}}{3 \, {\left (e^{2} x + 2 \, e\right )}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-3 \, e^{2} x^{2} + 12}}{\sqrt {e x + 2}}\,{d x} \]

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)

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maple [A]  time = 0.04, size = 30, normalized size = 1.50 \[ \frac {2 \left (e x -2\right ) \sqrt {-3 e^{2} x^{2}+12}}{3 \sqrt {e x +2}\, e} \]

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 = 3.08, size = 25, normalized size = 1.25 \[ \frac {{\left (2 i \, \sqrt {3} e x - 4 i \, \sqrt {3}\right )} \sqrt {e x - 2}}{3 \, e} \]

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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mupad [B]  time = 0.11, size = 29, normalized size = 1.45 \[ \frac {\left (\frac {2\,x}{3}-\frac {4}{3\,e}\right )\,\sqrt {12-3\,e^2\,x^2}}{\sqrt {e\,x+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((2*x)/3 - 4/(3*e))*(12 - 3*e^2*x^2)^(1/2))/(e*x + 2)^(1/2)

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

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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