\(\int \frac {\sqrt {2+e x}}{\sqrt [4]{12-3 e^2 x^2}} \, dx\) [313]

Optimal result

Integrand size = 24, antiderivative size = 184 \[ \int \frac {\sqrt {2+e x}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=-\frac {\left (4-e^2 x^2\right )^{3/4}}{\sqrt [4]{3} e \sqrt {2+e x}}-\frac {\sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2+e x}}{\sqrt [4]{2-e x}}\right )}{\sqrt [4]{3} e}+\frac {\sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{2+e x}}{\sqrt [4]{2-e x}}\right )}{\sqrt [4]{3} e}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{2+e x}}{\sqrt [4]{2-e x} \left (1+\frac {\sqrt {2+e x}}{\sqrt {2-e x}}\right )}\right )}{\sqrt [4]{3} e} \] Output:

-1/3*3^(3/4)*(-e^2*x^2+4)^(3/4)/e/(e*x+2)^(1/2)-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 = 0.58 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {2+e x}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=\frac {-\left (4-e^2 x^2\right )^{3/4}-\sqrt {4+2 e x} \arctan \left (\frac {\sqrt {4+2 e x} \sqrt [4]{4-e^2 x^2}}{2+e x-\sqrt {4-e^2 x^2}}\right )+\sqrt {4+2 e x} \text {arctanh}\left (\frac {2+e x+\sqrt {4-e^2 x^2}}{\sqrt {4+2 e x} \sqrt [4]{4-e^2 x^2}}\right )}{\sqrt [4]{3} e \sqrt {2+e x}} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.64 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.17, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {456, 60, 27, 73, 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.

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

\(\Big \downarrow \) 456

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 854

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

\(\Big \downarrow \) 826

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

-(((2 - e*x)^(3/4)*(2 + e*x)^(1/4))/(3^(1/4)*e)) - (4*((-(ArcTan[1 - (Sqrt 
[2]*(2 - e*x)^(1/4))/(2 + e*x)^(1/4)]/Sqrt[2]) + ArcTan[1 + (Sqrt[2]*(2 - 
e*x)^(1/4))/(2 + e*x)^(1/4)]/Sqrt[2])/2 + (Log[1 + Sqrt[2 - e*x] - (Sqrt[2 
]*(2 - e*x)^(1/4))/(2 + e*x)^(1/4)]/(2*Sqrt[2]) - Log[1 + Sqrt[2 - e*x] + 
(Sqrt[2]*(2 - e*x)^(1/4))/(2 + 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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, 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]

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

Input:

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

Output:

int((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. 310 vs. \(2 (146) = 292\).

Time = 0.10 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.68 \[ \int \frac {\sqrt {2+e x}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=-\frac {2 \cdot 12^{\frac {3}{4}} {\left (e x + 2\right )} \arctan \left (\frac {3 \, e^{2} x^{2} + 12^{\frac {1}{4}} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2} - 12}{3 \, {\left (e^{2} x^{2} - 4\right )}}\right ) + 2 \cdot 12^{\frac {3}{4}} {\left (e x + 2\right )} \arctan \left (-\frac {3 \, e^{2} x^{2} - 12^{\frac {1}{4}} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2} - 12}{3 \, {\left (e^{2} x^{2} - 4\right )}}\right ) + 12^{\frac {3}{4}} {\left (e x + 2\right )} \log \left (\frac {12^{\frac {3}{4}} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2} + 6 \, \sqrt {3} {\left (e^{2} x^{2} - 4\right )} - 6 \, \sqrt {-3 \, e^{2} x^{2} + 12} {\left (e x + 2\right )}}{e^{2} x^{2} - 4}\right ) - 12^{\frac {3}{4}} {\left (e x + 2\right )} \log \left (-\frac {12^{\frac {3}{4}} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2} - 6 \, \sqrt {3} {\left (e^{2} x^{2} - 4\right )} + 6 \, \sqrt {-3 \, e^{2} x^{2} + 12} {\left (e x + 2\right )}}{e^{2} x^{2} - 4}\right ) + 4 \, {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2}}{12 \, {\left (e^{2} x + 2 \, e\right )}} \] Input:

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

Output:

-1/12*(2*12^(3/4)*(e*x + 2)*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)*(e*x + 2)*arcta 
n(-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)*(e*x + 2)*log((12^(3/4)*(-3*e^2*x^2 + 12)^(3/4)*s 
qrt(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)*(e*x + 2)*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)) + 4*(-3*e^2*x^2 + 12)^(3/4)*sqrt(e*x + 2))/(e^2*x + 2 
*e)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {2+e x}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=-\frac {3^{\frac {3}{4}} {\left (2 \, {\left (e x + 2\right )} {\left (\frac {4}{e x + 2} - 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (\frac {4}{e x + 2} - 1\right )}^{\frac {1}{4}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (\frac {4}{e x + 2} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} {\left (\frac {4}{e x + 2} - 1\right )}^{\frac {1}{4}} + \sqrt {\frac {4}{e x + 2} - 1} + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} {\left (\frac {4}{e x + 2} - 1\right )}^{\frac {1}{4}} + \sqrt {\frac {4}{e x + 2} - 1} + 1\right )\right )}}{6 \, e} \] Input:

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

Output:

-1/6*3^(3/4)*(2*(e*x + 2)*(4/(e*x + 2) - 1)^(3/4) + 2*sqrt(2)*arctan(1/2*s 
qrt(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. \[ \int \frac {\sqrt {2+e x}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=\int \frac {\sqrt {e\,x+2}}{{\left (12-3\,e^2\,x^2\right )}^{1/4}} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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