\(\int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx\) [518]

Optimal result

Integrand size = 24, antiderivative size = 182 \[ \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=-\frac {1}{2} e \sqrt {e x} \left (1-x^2\right )^{3/4}-\frac {e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt {2}}+\frac {e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e x}}{\sqrt {e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt {2}}+\frac {e^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e x}}{\sqrt [4]{1-x^2} \left (\sqrt {e}+\frac {\sqrt {e} x}{\sqrt {1-x^2}}\right )}\right )}{4 \sqrt {2}} \] Output:

-1/2*e*(e*x)^(1/2)*(-x^2+1)^(3/4)-1/8*e^(3/2)*arctan(1-2^(1/2)*(e*x)^(1/2) 
/e^(1/2)/(-x^2+1)^(1/4))*2^(1/2)+1/8*e^(3/2)*arctan(1+2^(1/2)*(e*x)^(1/2)/ 
e^(1/2)/(-x^2+1)^(1/4))*2^(1/2)+1/8*e^(3/2)*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)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.69 \[ \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=\frac {(e x)^{3/2} \left (-4 \sqrt {x} \left (1-x^2\right )^{3/4}+\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1-x^2}}{-x+\sqrt {1-x^2}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x} \sqrt [4]{1-x^2}}{x+\sqrt {1-x^2}}\right )\right )}{8 x^{3/2}} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.39 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.26, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {135, 262, 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.

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

\(\Big \downarrow \) 135

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 770

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

\(\Big \downarrow \) 755

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

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

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 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

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]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (130) = 260\).

Time = 0.11 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.45 \[ \int \frac {(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx=-\frac {1}{2} \, \sqrt {e x} e {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - \frac {1}{8} \, \sqrt {2} e^{\frac {3}{2}} \arctan \left (\frac {e x^{2} + \sqrt {2} \sqrt {e x} \sqrt {e} {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - e}{e x^{2} - e}\right ) - \frac {1}{8} \, \sqrt {2} e^{\frac {3}{2}} \arctan \left (-\frac {e x^{2} - \sqrt {2} \sqrt {e x} \sqrt {e} {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - e}{e x^{2} - e}\right ) - \frac {1}{16} \, \sqrt {2} e^{\frac {3}{2}} \log \left (\frac {e x^{2} - e \sqrt {x + 1} x \sqrt {-x + 1} + \sqrt {2} \sqrt {e x} \sqrt {e} {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - e}{x^{2} - 1}\right ) + \frac {1}{16} \, \sqrt {2} e^{\frac {3}{2}} \log \left (\frac {e x^{2} - e \sqrt {x + 1} x \sqrt {-x + 1} - \sqrt {2} \sqrt {e x} \sqrt {e} {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - e}{x^{2} - 1}\right ) \] Input:

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

Output:

-1/2*sqrt(e*x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4) - 1/8*sqrt(2)*e^(3/2)*arctan 
((e*x^2 + sqrt(2)*sqrt(e*x)*sqrt(e)*(x + 1)^(3/4)*(-x + 1)^(3/4) - e)/(e*x 
^2 - e)) - 1/8*sqrt(2)*e^(3/2)*arctan(-(e*x^2 - sqrt(2)*sqrt(e*x)*sqrt(e)* 
(x + 1)^(3/4)*(-x + 1)^(3/4) - e)/(e*x^2 - e)) - 1/16*sqrt(2)*e^(3/2)*log( 
(e*x^2 - e*sqrt(x + 1)*x*sqrt(-x + 1) + sqrt(2)*sqrt(e*x)*sqrt(e)*(x + 1)^ 
(3/4)*(-x + 1)^(3/4) - e)/(x^2 - 1)) + 1/16*sqrt(2)*e^(3/2)*log((e*x^2 - e 
*sqrt(x + 1)*x*sqrt(-x + 1) - sqrt(2)*sqrt(e*x)*sqrt(e)*(x + 1)^(3/4)*(-x 
+ 1)^(3/4) - e)/(x^2 - 1))
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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