\(\int \frac {1}{(2+e x)^{3/2} \sqrt {12-3 e^2 x^2}} \, dx\) [916]

Optimal result

Integrand size = 24, antiderivative size = 57 \[ \int \frac {1}{(2+e x)^{3/2} \sqrt {12-3 e^2 x^2}} \, dx=-\frac {\sqrt {2-e x}}{4 \sqrt {3} e (2+e x)}-\frac {\text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{8 \sqrt {3} e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 57, 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, 44, 65, 212} \[ \int \frac {1}{(2+e x)^{3/2} \sqrt {12-3 e^2 x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{8 \sqrt {3} e}-\frac {\sqrt {2-e x}}{4 \sqrt {3} e (e x+2)} \]

[In]

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

[Out]

-1/4*Sqrt[2 - e*x]/(Sqrt[3]*e*(2 + e*x)) - ArcTanh[Sqrt[2 - e*x]/2]/(8*Sqrt[3]*e)

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[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 {1}{\sqrt {6-3 e x} (2+e x)^2} \, dx \\ & = -\frac {\sqrt {2-e x}}{4 \sqrt {3} e (2+e x)}+\frac {1}{8} \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx \\ & = -\frac {\sqrt {2-e x}}{4 \sqrt {3} e (2+e x)}-\frac {\text {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{12 e} \\ & = -\frac {\sqrt {2-e x}}{4 \sqrt {3} e (2+e x)}-\frac {\tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{8 \sqrt {3} e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.30 \[ \int \frac {1}{(2+e x)^{3/2} \sqrt {12-3 e^2 x^2}} \, dx=-\frac {2 \sqrt {4-e^2 x^2}+(2+e x)^{3/2} \text {arctanh}\left (\frac {2 \sqrt {2+e x}}{\sqrt {4-e^2 x^2}}\right )}{8 \sqrt {3} e (2+e x)^{3/2}} \]

[In]

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

[Out]

-1/8*(2*Sqrt[4 - e^2*x^2] + (2 + e*x)^(3/2)*ArcTanh[(2*Sqrt[2 + e*x])/Sqrt[4 - e^2*x^2]])/(Sqrt[3]*e*(2 + e*x)
^(3/2))

Maple [A] (verified)

Time = 2.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.51

method	result	size
default	\(-\frac {\sqrt {-x^{2} e^{2}+4}\, \left (\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) e x +2 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right )+2 \sqrt {-3 e x +6}\right ) \sqrt {3}}{24 \left (e x +2\right )^{\frac {3}{2}} \sqrt {-3 e x +6}\, e}\)	\(86\)

[In]

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

[Out]

-1/24*(-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 = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.04 \[ \int \frac {1}{(2+e x)^{3/2} \sqrt {12-3 e^2 x^2}} \, dx=\frac {\sqrt {3} {\left (e^{2} x^{2} + 4 \, e x + 4\right )} \log \left (-\frac {3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt {3} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right ) - 4 \, \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{48 \, {\left (e^{3} x^{2} + 4 \, e^{2} x + 4 \, e\right )}} \]

[In]

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

[Out]

1/48*(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]

\[ \int \frac {1}{(2+e x)^{3/2} \sqrt {12-3 e^2 x^2}} \, dx=\frac {\sqrt {3} \int \frac {1}{e x \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4} + 2 \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}}\, dx}{3} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{(2+e x)^{3/2} \sqrt {12-3 e^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-3 \, e^{2} x^{2} + 12} {\left (e x + 2\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{(2+e x)^{3/2} \sqrt {12-3 e^2 x^2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(2+e x)^{3/2} \sqrt {12-3 e^2 x^2}} \, dx=\int \frac {1}{\sqrt {12-3\,e^2\,x^2}\,{\left (e\,x+2\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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