\(\int \frac {1}{\sqrt [4]{1-x} \sqrt {e x} \sqrt [4]{1+x}} \, dx\) [519]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\sqrt [4]{1-x} \sqrt {e x} \sqrt [4]{1+x}} \, dx=\frac {\sqrt {x} \left (\arctan \left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1-x^2}}{-x+\sqrt {1-x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1-x^2}}{x+\sqrt {1-x^2}}\right )\right )}{\sqrt {2} \sqrt {e x}} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.37 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.36, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {135, 266, 770, 755, 27, 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.

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

\(\Big \downarrow \) 135

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

\(\Big \downarrow \) 266

\(\displaystyle \frac {2 \int \frac {1}{\sqrt [4]{1-x^2}}d\sqrt {e x}}{e}\)

\(\Big \downarrow \) 770

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

\(\Big \downarrow \) 755

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

(2*((e*(-(ArcTan[1 - (Sqrt[2]*Sqrt[e*x])/(Sqrt[e]*(1 - x^2)^(1/4))]/(Sqrt[ 
2]*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e*x])/(Sqrt[e]*(1 - x^2)^(1/4))]/( 
Sqrt[2]*Sqrt[e])))/2 + (e*(-1/2*Log[e + e*x - (Sqrt[2]*Sqrt[e]*Sqrt[e*x])/ 
(1 - x^2)^(1/4)]/(Sqrt[2]*Sqrt[e]) + Log[e + e*x + (Sqrt[2]*Sqrt[e]*Sqrt[e 
*x])/(1 - x^2)^(1/4)]/(2*Sqrt[2]*Sqrt[e])))/2))/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 135

Int[((f_.)*(x_))^(p_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), 
x_] :> Int[(a*c + b*d*x^2)^m*(f*x)^p, x] /; FreeQ[{a, b, c, d, f, m, n, p}, 
 x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && GtQ[a, 0] && GtQ[c, 0]
 

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 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   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 770

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n)   Subst[In 
t[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p + 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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\sqrt [4]{1-x} \sqrt {e x} \sqrt [4]{1+x}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {x^{2} + \frac {\sqrt {2} \sqrt {e x} {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{\sqrt {e}} - 1}{x^{2} - 1}\right )}{2 \, \sqrt {e}} - \frac {\sqrt {2} \arctan \left (-\frac {x^{2} - \frac {\sqrt {2} \sqrt {e x} {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{\sqrt {e}} - 1}{x^{2} - 1}\right )}{2 \, \sqrt {e}} - \frac {\sqrt {2} \log \left (\frac {x^{2} - \sqrt {x + 1} x \sqrt {-x + 1} + \frac {\sqrt {2} \sqrt {e x} {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{\sqrt {e}} - 1}{x^{2} - 1}\right )}{4 \, \sqrt {e}} + \frac {\sqrt {2} \log \left (\frac {x^{2} - \sqrt {x + 1} x \sqrt {-x + 1} - \frac {\sqrt {2} \sqrt {e x} {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{\sqrt {e}} - 1}{x^{2} - 1}\right )}{4 \, \sqrt {e}} \] Input:

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.39 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.60 \[ \int \frac {1}{\sqrt [4]{1-x} \sqrt {e x} \sqrt [4]{1+x}} \, dx=- \frac {i {G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {3}{8}, \frac {7}{8} & \frac {1}{2}, \frac {3}{4}, 1, 1 \\0, \frac {3}{8}, \frac {1}{2}, \frac {7}{8}, 1, 0 & \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )} e^{\frac {i \pi }{4}}}{4 \pi \sqrt {e} \Gamma \left (\frac {1}{4}\right )} - \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{8}, \frac {1}{4}, \frac {3}{8}, \frac {3}{4}, 1 & \\- \frac {1}{8}, \frac {3}{8} & - \frac {1}{4}, 0, \frac {1}{4}, 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi \sqrt {e} \Gamma \left (\frac {1}{4}\right )} \] Input:

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

Output:

-I*meijerg(((3/8, 7/8), (1/2, 3/4, 1, 1)), ((0, 3/8, 1/2, 7/8, 1, 0), ()), 
 exp_polar(-2*I*pi)/x**2)*exp(I*pi/4)/(4*pi*sqrt(e)*gamma(1/4)) - meijerg( 
((-1/4, -1/8, 1/4, 3/8, 3/4, 1), ()), ((-1/8, 3/8), (-1/4, 0, 1/4, 0)), x* 
*(-2))/(4*pi*sqrt(e)*gamma(1/4))
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.21 \[ \int \frac {1}{\sqrt [4]{1-x} \sqrt {e x} \sqrt [4]{1+x}} \, dx=\frac {\sqrt {x}\, \sqrt {e}\, \left (2 x +2\right )}{\left (x +1\right )^{\frac {3}{4}} \sqrt {x +1}\, \left (1-x \right )^{\frac {1}{4}} e} \] Input:

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

Output:

(sqrt(x)*sqrt(e)*(x + 1)**(1/4)*(x + 1 + x + 1))/(sqrt(x + 1)*( - x + 1)** 
(1/4)*e*(x + 1))