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]

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

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Rubi [A]  time = 0.01, antiderivative size = 22, 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} \[ -\frac {6 \sqrt {3} (2-e x)^{5/2}}{5 e} \]

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 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 {\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*}

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Mathematica [A]  time = 0.05, 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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fricas [B]  time = 0.98, size = 45, normalized size = 2.05 \[ -\frac {6 \, {\left (e^{2} x^{2} - 4 \, e x + 4\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{5 \, {\left (e^{2} x + 2 \, e\right )}} \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Erro
r index.cc index_gcd Error: Bad Argument ValueError index.cc index_gcd Error: Bad Argument Value

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maple [A]  time = 0.04, size = 30, normalized size = 1.36 \[ \frac {2 \left (e x -2\right ) \left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}}}{5 \left (e x +2\right )^{\frac {3}{2}} e} \]

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 = 3.09, size = 36, normalized size = 1.64 \[ -\frac {{\left (6 i \, \sqrt {3} e^{2} x^{2} - 24 i \, \sqrt {3} e x + 24 i \, \sqrt {3}\right )} \sqrt {e x - 2}}{5 \, e} \]

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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mupad [B]  time = 0.50, size = 36, normalized size = 1.64 \[ -\frac {\sqrt {12-3\,e^2\,x^2}\,\left (\frac {6\,e\,x^2}{5}-\frac {24\,x}{5}+\frac {24}{5\,e}\right )}{\sqrt {e\,x+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((12 - 3*e^2*x^2)^(1/2)*((6*e*x^2)/5 - (24*x)/5 + 24/(5*e)))/(e*x + 2)^(1/2)

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

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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