3.10.5 \(\int \frac {(12-3 e^2 x^2)^{3/2}}{(2+e x)^{3/2}} \, dx\) [905]

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 = {641, 32} \begin {gather*} -\frac {6 \sqrt {3} (2-e x)^{5/2}}{5 e} \end {gather*}

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 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 {\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 = 0.18, size = 35, normalized size = 1.59 \begin {gather*} -\frac {6 \sqrt {3} \left (4-e^2 x^2\right )^{5/2}}{5 e (2+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[3]*(4 - e^2*x^2)^(5/2))/(5*e*(2 + e*x)^(5/2))

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Maple [A]
time = 0.47, size = 32, normalized size = 1.45

method	result	size
gosper	\(\frac {2 \left (e x -2\right ) \left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}}}{5 e \left (e x +2\right )^{\frac {3}{2}}}\)	\(30\)
default	\(-\frac {6 \sqrt {-3 e^{2} x^{2}+12}\, \left (e x -2\right )^{2}}{5 \sqrt {e x +2}\, e}\)	\(32\)
risch	\(\frac {18 \sqrt {\frac {-3 e^{2} x^{2}+12}{e x +2}}\, \sqrt {e x +2}\, \left (e^{2} x^{2}-4 e x +4\right ) \left (e x -2\right )}{5 \sqrt {-3 e^{2} x^{2}+12}\, e \sqrt {-3 e x +6}}\)	\(71\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.50, size = 36, normalized size = 1.64 \begin {gather*} \frac {6}{5} \, {\left (-i \, \sqrt {3} x^{2} e^{2} + 4 i \, \sqrt {3} x e - 4 i \, \sqrt {3}\right )} \sqrt {x e - 2} e^{\left (-1\right )} \end {gather*}

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]

6/5*(-I*sqrt(3)*x^2*e^2 + 4*I*sqrt(3)*x*e - 4*I*sqrt(3))*sqrt(x*e - 2)*e^(-1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (16) = 32\).
time = 2.30, size = 45, normalized size = 2.05 \begin {gather*} -\frac {6 \, {\left (x^{2} e^{2} - 4 \, x e + 4\right )} \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2}}{5 \, {\left (x e^{2} + 2 \, e\right )}} \end {gather*}

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*(x^2*e^2 - 4*x*e + 4)*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*} 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 ) \end {gather*}

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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Giac [A]
time = 0.84, size = 27, normalized size = 1.23 \begin {gather*} -\frac {6}{5} \, \sqrt {3} {\left ({\left (x e - 2\right )}^{2} \sqrt {-x e + 2} - 32\right )} e^{\left (-1\right )} \end {gather*}

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]

-6/5*sqrt(3)*((x*e - 2)^2*sqrt(-x*e + 2) - 32)*e^(-1)

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Mupad [B]
time = 0.50, size = 36, normalized size = 1.64 \begin {gather*} -\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}} \end {gather*}

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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