3.10.14 \(\int \frac {\sqrt {2+e x}}{\sqrt {12-3 e^2 x^2}} \, dx\) [914]

Optimal. Leaf size=20 \[ -\frac {2 \sqrt {2-e x}}{\sqrt {3} e} \]

[Out]

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

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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 = {641, 32} \begin {gather*} -\frac {2 \sqrt {2-e x}}{\sqrt {3} e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2 - e*x])/(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 641

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^
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 {2+e x}}{\sqrt {12-3 e^2 x^2}} \, dx &=\int \frac {1}{\sqrt {6-3 e x}} \, dx\\ &=-\frac {2 \sqrt {2-e x}}{\sqrt {3} e}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 29, normalized size = 1.45 \begin {gather*} -\frac {2 \sqrt {4-e^2 x^2}}{e \sqrt {6+3 e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.48, size = 25, normalized size = 1.25

method	result	size
default	\(-\frac {2 \sqrt {-3 e^{2} x^{2}+12}}{3 \sqrt {e x +2}\, e}\)	\(25\)
gosper	\(\frac {2 \left (e x -2\right ) \sqrt {e x +2}}{e \sqrt {-3 e^{2} x^{2}+12}}\)	\(30\)
risch	\(\frac {2 \left (e x -2\right ) \sqrt {\frac {-3 e^{2} x^{2}+12}{e x +2}}\, \sqrt {e x +2}}{e \sqrt {-3 e x +6}\, \sqrt {-3 e^{2} x^{2}+12}}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.50, size = 26, normalized size = 1.30 \begin {gather*} \frac {2 \, {\left (-i \, \sqrt {3} x e + 2 i \, \sqrt {3}\right )} e^{\left (-1\right )}}{3 \, \sqrt {x e - 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.65, size = 32, normalized size = 1.60 \begin {gather*} -\frac {2 \, \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2}}{3 \, {\left (x e^{2} + 2 \, e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.39, size = 18, normalized size = 0.90 \begin {gather*} -\frac {2}{3} \, \sqrt {3} {\left (\sqrt {-x e + 2} - 2\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.15, size = 24, normalized size = 1.20 \begin {gather*} -\frac {2\,\sqrt {12-3\,e^2\,x^2}}{3\,e\,\sqrt {e\,x+2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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