\(\int \frac {1}{\sqrt {2+e x} \sqrt [4]{12-3 e^2 x^2}} \, dx\) [314]

Optimal result

Integrand size = 24, antiderivative size = 151 \[ \int \frac {1}{\sqrt {2+e x} \sqrt [4]{12-3 e^2 x^2}} \, dx=-\frac {\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2+e x}}{\sqrt [4]{2-e x}}\right )}{\sqrt [4]{3} e}+\frac {\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{2+e x}}{\sqrt [4]{2-e x}}\right )}{\sqrt [4]{3} e}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{2+e x}}{\sqrt [4]{2-e x} \left (1+\frac {\sqrt {2+e x}}{\sqrt {2-e x}}\right )}\right )}{\sqrt [4]{3} e} \] Output:

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\sqrt {2+e x} \sqrt [4]{12-3 e^2 x^2}} \, dx=\frac {\sqrt {2} \left (-\arctan \left (\frac {\sqrt {4+2 e x} \sqrt [4]{4-e^2 x^2}}{2+e x-\sqrt {4-e^2 x^2}}\right )+\text {arctanh}\left (\frac {2+e x+\sqrt {4-e^2 x^2}}{\sqrt {4+2 e x} \sqrt [4]{4-e^2 x^2}}\right )\right )}{\sqrt [4]{3} e} \] Input:

Integrate[1/(Sqrt[2 + e*x]*(12 - 3*e^2*x^2)^(1/4)),x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.57 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.26, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {456, 73, 27, 854, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 456

\(\displaystyle \int \frac {1}{\sqrt [4]{6-3 e x} (e x+2)^{3/4}}dx\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {4 \int \frac {3^{3/4} \sqrt {6-3 e x}}{(3 e x+6)^{3/4}}d\sqrt [4]{6-3 e x}}{3 e}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {4 \int \frac {\sqrt {6-3 e x}}{(3 e x+6)^{3/4}}d\sqrt [4]{6-3 e x}}{\sqrt [4]{3} e}\)

\(\Big \downarrow \) 854

\(\displaystyle -\frac {4 \int \frac {\sqrt {6-3 e x}}{7-3 e x}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}}{\sqrt [4]{3} e}\)

\(\Big \downarrow \) 826

\(\displaystyle -\frac {4 \left (\frac {1}{2} \int \frac {\sqrt {6-3 e x}+1}{7-3 e x}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}-\frac {1}{2} \int \frac {1-\sqrt {6-3 e x}}{7-3 e x}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )}{\sqrt [4]{3} e}\)

\(\Big \downarrow \) 1476

\(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {6-3 e x}-\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+\frac {1}{2} \int \frac {1}{\sqrt {6-3 e x}+\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )-\frac {1}{2} \int \frac {1-\sqrt {6-3 e x}}{7-3 e x}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )}{\sqrt [4]{3} e}\)

\(\Big \downarrow \) 1082

\(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {6-3 e x}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {6-3 e x}-1}d\left (\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {6-3 e x}}{7-3 e x}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )}{\sqrt [4]{3} e}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\sqrt {6-3 e x}}{7-3 e x}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )}{\sqrt [4]{3} e}\)

\(\Big \downarrow \) 1479

\(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}}{\sqrt {6-3 e x}-\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1\right )}{\sqrt {6-3 e x}+\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )}{\sqrt {2}}\right )\right )}{\sqrt [4]{3} e}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {4 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}}{\sqrt {6-3 e x}-\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1\right )}{\sqrt {6-3 e x}+\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )}{\sqrt {2}}\right )\right )}{\sqrt [4]{3} e}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {4 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}}{\sqrt {6-3 e x}-\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1}{\sqrt {6-3 e x}+\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1}d\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )}{\sqrt {2}}\right )\right )}{\sqrt [4]{3} e}\)

\(\Big \downarrow \) 1103

\(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {6-3 e x}-\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {6-3 e x}+\frac {\sqrt {2} \sqrt [4]{6-3 e x}}{\sqrt [4]{3 e x+6}}+1\right )}{2 \sqrt {2}}\right )\right )}{\sqrt [4]{3} e}\)

Input:

Int[1/(Sqrt[2 + e*x]*(12 - 3*e^2*x^2)^(1/4)),x]
 

Output:

(-4*((-(ArcTan[1 - (Sqrt[2]*(6 - 3*e*x)^(1/4))/(6 + 3*e*x)^(1/4)]/Sqrt[2]) 
 + ArcTan[1 + (Sqrt[2]*(6 - 3*e*x)^(1/4))/(6 + 3*e*x)^(1/4)]/Sqrt[2])/2 + 
(Log[1 + Sqrt[6 - 3*e*x] - (Sqrt[2]*(6 - 3*e*x)^(1/4))/(6 + 3*e*x)^(1/4)]/ 
(2*Sqrt[2]) - Log[1 + Sqrt[6 - 3*e*x] + (Sqrt[2]*(6 - 3*e*x)^(1/4))/(6 + 3 
*e*x)^(1/4)]/(2*Sqrt[2]))/2))/(3^(1/4)*e)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 217

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

rule 456

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ 
(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p}, x] && 
EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] &&  !Integ 
erQ[n]))
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 854

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 
 1)/n)   Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n 
)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 
2^(-1)] && IntegersQ[m, p + (m + 1)/n]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [F]

\[\int \frac {1}{\sqrt {e x +2}\, \left (-3 e^{2} x^{2}+12\right )^{\frac {1}{4}}}d x\]

Input:

int(1/(e*x+2)^(1/2)/(-3*e^2*x^2+12)^(1/4),x)
 

Output:

int(1/(e*x+2)^(1/2)/(-3*e^2*x^2+12)^(1/4),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 261 vs. \(2 (119) = 238\).

Time = 0.10 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.73 \[ \int \frac {1}{\sqrt {2+e x} \sqrt [4]{12-3 e^2 x^2}} \, dx=-\frac {2 \cdot 12^{\frac {3}{4}} \arctan \left (\frac {3 \, e^{2} x^{2} + 12^{\frac {1}{4}} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2} - 12}{3 \, {\left (e^{2} x^{2} - 4\right )}}\right ) + 2 \cdot 12^{\frac {3}{4}} \arctan \left (-\frac {3 \, e^{2} x^{2} - 12^{\frac {1}{4}} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2} - 12}{3 \, {\left (e^{2} x^{2} - 4\right )}}\right ) + 12^{\frac {3}{4}} \log \left (\frac {12^{\frac {3}{4}} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2} + 6 \, \sqrt {3} {\left (e^{2} x^{2} - 4\right )} - 6 \, \sqrt {-3 \, e^{2} x^{2} + 12} {\left (e x + 2\right )}}{e^{2} x^{2} - 4}\right ) - 12^{\frac {3}{4}} \log \left (-\frac {12^{\frac {3}{4}} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2} - 6 \, \sqrt {3} {\left (e^{2} x^{2} - 4\right )} + 6 \, \sqrt {-3 \, e^{2} x^{2} + 12} {\left (e x + 2\right )}}{e^{2} x^{2} - 4}\right )}{12 \, e} \] Input:

integrate(1/(e*x+2)^(1/2)/(-3*e^2*x^2+12)^(1/4),x, algorithm="fricas")
 

Output:

-1/12*(2*12^(3/4)*arctan(1/3*(3*e^2*x^2 + 12^(1/4)*(-3*e^2*x^2 + 12)^(3/4) 
*sqrt(e*x + 2) - 12)/(e^2*x^2 - 4)) + 2*12^(3/4)*arctan(-1/3*(3*e^2*x^2 - 
12^(1/4)*(-3*e^2*x^2 + 12)^(3/4)*sqrt(e*x + 2) - 12)/(e^2*x^2 - 4)) + 12^( 
3/4)*log((12^(3/4)*(-3*e^2*x^2 + 12)^(3/4)*sqrt(e*x + 2) + 6*sqrt(3)*(e^2* 
x^2 - 4) - 6*sqrt(-3*e^2*x^2 + 12)*(e*x + 2))/(e^2*x^2 - 4)) - 12^(3/4)*lo 
g(-(12^(3/4)*(-3*e^2*x^2 + 12)^(3/4)*sqrt(e*x + 2) - 6*sqrt(3)*(e^2*x^2 - 
4) + 6*sqrt(-3*e^2*x^2 + 12)*(e*x + 2))/(e^2*x^2 - 4)))/e
 

Sympy [F]

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

integrate(1/(e*x+2)**(1/2)/(-3*e**2*x**2+12)**(1/4),x)
 

Output:

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

Maxima [F]

\[ \int \frac {1}{\sqrt {2+e x} \sqrt [4]{12-3 e^2 x^2}} \, dx=\int { \frac {1}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}} \,d x } \] Input:

integrate(1/(e*x+2)^(1/2)/(-3*e^2*x^2+12)^(1/4),x, algorithm="maxima")
 

Output:

integrate(1/((-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2)), x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt {2+e x} \sqrt [4]{12-3 e^2 x^2}} \, dx=-\frac {3^{\frac {3}{4}} {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (\frac {4}{e x + 2} - 1\right )}^{\frac {1}{4}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (\frac {4}{e x + 2} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} {\left (\frac {4}{e x + 2} - 1\right )}^{\frac {1}{4}} + \sqrt {\frac {4}{e x + 2} - 1} + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} {\left (\frac {4}{e x + 2} - 1\right )}^{\frac {1}{4}} + \sqrt {\frac {4}{e x + 2} - 1} + 1\right )\right )}}{6 \, e} \] Input:

integrate(1/(e*x+2)^(1/2)/(-3*e^2*x^2+12)^(1/4),x, algorithm="giac")
 

Output:

-1/6*3^(3/4)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(4/(e*x + 2) - 1)^ 
(1/4))) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(4/(e*x + 2) - 1)^(1/ 
4))) - sqrt(2)*log(sqrt(2)*(4/(e*x + 2) - 1)^(1/4) + sqrt(4/(e*x + 2) - 1) 
 + 1) + sqrt(2)*log(-sqrt(2)*(4/(e*x + 2) - 1)^(1/4) + sqrt(4/(e*x + 2) - 
1) + 1))/e
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {2+e x} \sqrt [4]{12-3 e^2 x^2}} \, dx=\int \frac {1}{{\left (12-3\,e^2\,x^2\right )}^{1/4}\,\sqrt {e\,x+2}} \,d x \] Input:

int(1/((12 - 3*e^2*x^2)^(1/4)*(e*x + 2)^(1/2)),x)
 

Output:

int(1/((12 - 3*e^2*x^2)^(1/4)*(e*x + 2)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\sqrt {2+e x} \sqrt [4]{12-3 e^2 x^2}} \, dx=\frac {3^{\frac {1}{4}} \left (3 \sqrt {e x +2}\, \left (-e^{2} x^{2}+4\right )^{\frac {1}{4}} e x -6 \sqrt {e x +2}\, \left (-e^{2} x^{2}+4\right )^{\frac {1}{4}}-\left (e x +2\right )^{\frac {1}{4}} \left (-e x +2\right )^{\frac {3}{4}} \sqrt {-e^{2} x^{2}+4}\, \sqrt {9}\right )}{3 \sqrt {-e^{2} x^{2}+4}\, \sqrt {3}\, e \left (e x -2\right )} \] Input:

int(1/(e*x+2)^(1/2)/(-3*e^2*x^2+12)^(1/4),x)
 

Output:

(3**(1/4)*(3*sqrt(e*x + 2)*( - e**2*x**2 + 4)**(1/4)*e*x - 6*sqrt(e*x + 2) 
*( - e**2*x**2 + 4)**(1/4) - (e*x + 2)**(1/4)*( - e*x + 2)**(3/4)*sqrt( - 
e**2*x**2 + 4)*sqrt(9)))/(3*sqrt( - e**2*x**2 + 4)*sqrt(3)*e*(e*x - 2))