### 3.941 $$\int \frac{1}{(2+e x)^{3/2} \sqrt {12-3 e^2 x^2}} \, dx$$

Optimal. Leaf size=35 $-\frac{\left (4-e^2 x^2\right )^{3/4}}{3 \sqrt {3} e (e x+2)^{3/2}}$

[Out]

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

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Rubi [A]  time = 0.0141082, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.042, Rules used = {651} $-\frac{\left (4-e^2 x^2\right )^{3/4}}{3 \sqrt {3} e (e x+2)^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((2 + e*x)^(3/2)*(12 - 3*e^2*x^2)^(1/4)),x]

[Out]

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

Rule 651

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))
/(2*c*d*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rubi steps

\begin{align*} \int \frac{1}{(2+e x)^{3/2} \sqrt {12-3 e^2 x^2}} \, dx &=-\frac{\left (4-e^2 x^2\right )^{3/4}}{3 \sqrt {3} e (2+e x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0460585, size = 35, normalized size = 1. $\frac{e x-2}{3 e \sqrt{e x+2} \sqrt {12-3 e^2 x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((2 + e*x)^(3/2)*(12 - 3*e^2*x^2)^(1/4)),x]

[Out]

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

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Maple [A]  time = 0.043, size = 30, normalized size = 0.9 \begin{align*}{\frac{ex-2}{3\,e}{\frac{1}{\sqrt{ex+2}}}{\frac{1}{\sqrt {-3\,{e}^{2}{x}^{2}+12}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}{\left (e x + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+2)^(3/2)/(-3*e^2*x^2+12)^(1/4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.85474, size = 95, normalized size = 2.71 \begin{align*} -\frac{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{3}{4}} \sqrt{e x + 2}}{9 \,{\left (e^{3} x^{2} + 4 \, e^{2} x + 4 \, e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+2)^(3/2)/(-3*e^2*x^2+12)^(1/4),x, algorithm="fricas")

[Out]

-1/9*(-3*e^2*x^2 + 12)^(3/4)*sqrt(e*x + 2)/(e^3*x^2 + 4*e^2*x + 4*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3^{\frac{3}{4}} \int \frac{1}{e x \sqrt{e x + 2} \sqrt {- e^{2} x^{2} + 4} + 2 \sqrt{e x + 2} \sqrt {- e^{2} x^{2} + 4}}\, dx}{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3**(3/4)*Integral(1/(e*x*sqrt(e*x + 2)*(-e**2*x**2 + 4)**(1/4) + 2*sqrt(e*x + 2)*(-e**2*x**2 + 4)**(1/4)), x)/
3

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}{\left (e x + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+2)^(3/2)/(-3*e^2*x^2+12)^(1/4),x, algorithm="giac")

[Out]

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