\(\int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx\) [899]

Optimal result

Integrand size = 24, antiderivative size = 55 \[ \int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx=-\frac {\sqrt {3} \sqrt {2-e x}}{e (2+e x)}+\frac {\sqrt {3} \text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{2 e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {641, 43, 65, 212} \[ \int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx=\frac {\sqrt {3} \text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{2 e}-\frac {\sqrt {3} \sqrt {2-e x}}{e (e x+2)} \]

[In]

Int[Sqrt[12 - 3*e^2*x^2]/(2 + e*x)^(5/2),x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 \frac {\sqrt {6-3 e x}}{(2+e x)^2} \, dx \\ & = -\frac {\sqrt {3} \sqrt {2-e x}}{e (2+e x)}-\frac {3}{2} \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx \\ & = -\frac {\sqrt {3} \sqrt {2-e x}}{e (2+e x)}+\frac {\text {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{e} \\ & = -\frac {\sqrt {3} \sqrt {2-e x}}{e (2+e x)}+\frac {\sqrt {3} \tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{2 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx=\frac {\sqrt {3} \left (-\frac {2 \sqrt {4-e^2 x^2}}{(2+e x)^{3/2}}+\text {arctanh}\left (\frac {2 \sqrt {2+e x}}{\sqrt {4-e^2 x^2}}\right )\right )}{2 e} \]

[In]

Integrate[Sqrt[12 - 3*e^2*x^2]/(2 + e*x)^(5/2),x]

[Out]

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

Maple [A] (verified)

Time = 2.18 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.56

method	result	size
default	\(\frac {\sqrt {-x^{2} e^{2}+4}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) e x +2 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right )-2 \sqrt {-3 e x +6}\right ) \sqrt {3}}{2 \left (e x +2\right )^{\frac {3}{2}} \sqrt {-3 e x +6}\, e}\)	\(86\)

[In]

int((-3*e^2*x^2+12)^(1/2)/(e*x+2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/2*(-e^2*x^2+4)^(1/2)*(3^(1/2)*arctanh(1/6*(-3*e*x+6)^(1/2)*3^(1/2))*e*x+2*3^(1/2)*arctanh(1/6*(-3*e*x+6)^(1/
2)*3^(1/2))-2*(-3*e*x+6)^(1/2))/(e*x+2)^(3/2)/(-3*e*x+6)^(1/2)*3^(1/2)/e

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (43) = 86\).

Time = 0.38 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.11 \[ \int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx=\frac {\sqrt {3} {\left (e^{2} x^{2} + 4 \, e x + 4\right )} \log \left (-\frac {3 \, e^{2} x^{2} - 12 \, e x - 4 \, \sqrt {3} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \, \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{4 \, {\left (e^{3} x^{2} + 4 \, e^{2} x + 4 \, e\right )}} \]

[In]

integrate((-3*e^2*x^2+12)^(1/2)/(e*x+2)^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx=\sqrt {3} \int \frac {\sqrt {- e^{2} x^{2} + 4}}{e^{2} x^{2} \sqrt {e x + 2} + 4 e x \sqrt {e x + 2} + 4 \sqrt {e x + 2}}\, dx \]

[In]

integrate((-3*e**2*x**2+12)**(1/2)/(e*x+2)**(5/2),x)

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx=\int { \frac {\sqrt {-3 \, e^{2} x^{2} + 12}}{{\left (e x + 2\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((-3*e^2*x^2+12)^(1/2)/(e*x+2)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-3*e^2*x^2 + 12)/(e*x + 2)^(5/2), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx=-\frac {\sqrt {3} {\left (\frac {4 \, \sqrt {-e x + 2}}{e x + 2} - \log \left (\sqrt {-e x + 2} + 2\right ) + \log \left (-\sqrt {-e x + 2} + 2\right )\right )}}{4 \, e} \]

[In]

integrate((-3*e^2*x^2+12)^(1/2)/(e*x+2)^(5/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx=\int \frac {\sqrt {12-3\,e^2\,x^2}}{{\left (e\,x+2\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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