∫ 1__________ (2+ex)3∕2√ _____

Optimal. Leaf size=57 \[ -\frac {\sqrt {2-e x}}{4 \sqrt {3} e (2+e x)}-\frac {\tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{8 \sqrt {3} e} \]

-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]
time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {641, 44, 65, 212} \begin {gather*} -\frac {\sqrt {2-e x}}{4 \sqrt {3} e (e x+2)}-\frac {\tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{8 \sqrt {3} e} \end {gather*}

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

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]

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,

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]

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

\begin {align*} \int \frac {1}{(2+e x)^{3/2} \sqrt {12-3 e^2 x^2}} \, dx &=\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]
time = 0.26, size = 74, normalized size = 1.30 \begin {gather*} -\frac {2 \sqrt {4-e^2 x^2}+(2+e x)^{3/2} \tanh ^{-1}\left (\frac {2 \sqrt {2+e x}}{\sqrt {4-e^2 x^2}}\right )}{8 \sqrt {3} e (2+e x)^{3/2}} \end {gather*}

-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)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(87\) vs. \(2(43)=86\).
time = 0.55, size = 88, normalized size = 1.54


method	result	size

default	\(-\frac {\sqrt {-e^{2} x^{2}+4}\, \left (\sqrt {3}\, \arctanh \left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right ) e x +2 \sqrt {3}\, \arctanh \left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right )+2 \sqrt {-3 e x +6}\right ) \sqrt {3}}{24 \sqrt {\left (e x +2\right )^{3}}\, \sqrt {-3 e x +6}\, e}\)	\(88\)

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (44) = 88\).
time = 3.33, size = 115, normalized size = 2.02 \begin {gather*} \frac {\sqrt {3} {\left (x^{2} e^{2} + 4 \, x e + 4\right )} \log \left (-\frac {3 \, x^{2} e^{2} - 12 \, x e + 4 \, \sqrt {3} \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2} - 36}{x^{2} e^{2} + 4 \, x e + 4}\right ) - 4 \, \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2}}{48 \, {\left (x^{2} e^{3} + 4 \, x e^{2} + 4 \, e\right )}} \end {gather*}

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

1/48*(sqrt(3)*(x^2*e^2 + 4*x*e + 4)*log(-(3*x^2*e^2 - 12*x*e + 4*sqrt(3)*sqrt(-3*x^2*e^2 + 12)*sqrt(x*e + 2) -

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

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

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP

UT:Error index.cc index_gcd Error: Bad Argument ValueError index.cc index_gcd Error: Bad Argument Value

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {12-3\,e^2\,x^2}\,{\left (e\,x+2\right )}^{3/2}} \,d x \end {gather*}

________________________________________________________________________________________

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