\(\int \frac {\sqrt {12-3 e^2 x^2}}{\sqrt {2+e x}} \, dx\) [897]

Optimal result

Integrand size = 24, antiderivative size = 20 \[ \int \frac {\sqrt {12-3 e^2 x^2}}{\sqrt {2+e x}} \, dx=-\frac {2 (2-e x)^{3/2}}{\sqrt {3} e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {641, 32} \[ \int \frac {\sqrt {12-3 e^2 x^2}}{\sqrt {2+e x}} \, dx=-\frac {2 (2-e x)^{3/2}}{\sqrt {3} e} \]

[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 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*} \text {integral}& = \int \sqrt {6-3 e x} \, dx \\ & = -\frac {2 (2-e x)^{3/2}}{\sqrt {3} e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65 \[ \int \frac {\sqrt {12-3 e^2 x^2}}{\sqrt {2+e x}} \, dx=-\frac {2 \left (4-e^2 x^2\right )^{3/2}}{\sqrt {3} e (2+e x)^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.17 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.50

method	result	size
gosper	\(\frac {2 \left (e x -2\right ) \sqrt {-3 x^{2} e^{2}+12}}{3 e \sqrt {e x +2}}\)	\(30\)
default	\(\frac {2 \left (e x -2\right ) \sqrt {-3 x^{2} e^{2}+12}}{3 e \sqrt {e x +2}}\)	\(30\)
risch	\(-\frac {2 \sqrt {\frac {-3 x^{2} e^{2}+12}{e x +2}}\, \sqrt {e x +2}\, \left (e x -2\right )^{2}}{\sqrt {-3 x^{2} e^{2}+12}\, e \sqrt {-3 e x +6}}\)	\(60\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (16) = 32\).

Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.85 \[ \int \frac {\sqrt {12-3 e^2 x^2}}{\sqrt {2+e x}} \, dx=\frac {2 \, \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} {\left (e x - 2\right )}}{3 \, {\left (e^{2} x + 2 \, e\right )}} \]

[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)

Sympy [F]

\[ \int \frac {\sqrt {12-3 e^2 x^2}}{\sqrt {2+e x}} \, dx=\sqrt {3} \int \frac {\sqrt {- e^{2} x^{2} + 4}}{\sqrt {e x + 2}}\, dx \]

[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)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt {12-3 e^2 x^2}}{\sqrt {2+e x}} \, dx=-\frac {2 \, {\left (-i \, \sqrt {3} e x + 2 i \, \sqrt {3}\right )} \sqrt {e x - 2}}{3 \, e} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {12-3 e^2 x^2}}{\sqrt {2+e x}} \, dx=-\frac {2 \, \sqrt {3} {\left ({\left (-e x + 2\right )}^{\frac {3}{2}} - 8\right )}}{3 \, e} \]

[In]

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

[Out]

-2/3*sqrt(3)*((-e*x + 2)^(3/2) - 8)/e

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {\sqrt {12-3 e^2 x^2}}{\sqrt {2+e x}} \, dx=\frac {\left (\frac {2\,x}{3}-\frac {4}{3\,e}\right )\,\sqrt {12-3\,e^2\,x^2}}{\sqrt {e\,x+2}} \]

[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)