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

Optimal. Leaf size=244 \[ -\frac{e^{3/2} \log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}-\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{8 \sqrt{2}}+\frac{e^{3/2} \log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}+\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{8 \sqrt{2}}-\frac{e^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt{2}}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}+1\right )}{4 \sqrt{2}}-\frac{1}{2} e \left (1-x^2\right )^{3/4} \sqrt{e x} \]

[Out]

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

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Rubi [A]  time = 0.201608, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {125, 321, 329, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac{e^{3/2} \log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}-\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{8 \sqrt{2}}+\frac{e^{3/2} \log \left (\frac{\sqrt{e} x}{\sqrt{1-x^2}}+\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}+\sqrt{e}\right )}{8 \sqrt{2}}-\frac{e^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt{2}}+\frac{e^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}+1\right )}{4 \sqrt{2}}-\frac{1}{2} e \left (1-x^2\right )^{3/4} \sqrt{e x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 125

Int[((f_.)*(x_))^(p_.)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> 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[m - n, 0] && GtQ[a, 0] && GtQ
[c, 0]

Rule 321

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

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[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 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[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 1165

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

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(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 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[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 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rubi steps

\begin{align*} \int \frac{(e x)^{3/2}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx &=\int \frac{(e x)^{3/2}}{\sqrt [4]{1-x^2}} \, dx\\ &=-\frac{1}{2} e \sqrt{e x} \left (1-x^2\right )^{3/4}+\frac{1}{4} e^2 \int \frac{1}{\sqrt{e x} \sqrt [4]{1-x^2}} \, dx\\ &=-\frac{1}{2} e \sqrt{e x} \left (1-x^2\right )^{3/4}+\frac{1}{2} e \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-\frac{x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )\\ &=-\frac{1}{2} e \sqrt{e x} \left (1-x^2\right )^{3/4}+\frac{1}{2} e \operatorname{Subst}\left (\int \frac{1}{1+\frac{x^4}{e^2}} \, dx,x,\frac{\sqrt{e x}}{\sqrt [4]{1-x^2}}\right )\\ &=-\frac{1}{2} e \sqrt{e x} \left (1-x^2\right )^{3/4}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{e-x^2}{1+\frac{x^4}{e^2}} \, dx,x,\frac{\sqrt{e x}}{\sqrt [4]{1-x^2}}\right )+\frac{1}{4} \operatorname{Subst}\left (\int \frac{e+x^2}{1+\frac{x^4}{e^2}} \, dx,x,\frac{\sqrt{e x}}{\sqrt [4]{1-x^2}}\right )\\ &=-\frac{1}{2} e \sqrt{e x} \left (1-x^2\right )^{3/4}-\frac{e^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}+2 x}{-e-\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\frac{\sqrt{e x}}{\sqrt [4]{1-x^2}}\right )}{8 \sqrt{2}}-\frac{e^{3/2} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}-2 x}{-e+\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\frac{\sqrt{e x}}{\sqrt [4]{1-x^2}}\right )}{8 \sqrt{2}}+\frac{1}{8} e^2 \operatorname{Subst}\left (\int \frac{1}{e-\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\frac{\sqrt{e x}}{\sqrt [4]{1-x^2}}\right )+\frac{1}{8} e^2 \operatorname{Subst}\left (\int \frac{1}{e+\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\frac{\sqrt{e x}}{\sqrt [4]{1-x^2}}\right )\\ &=-\frac{1}{2} e \sqrt{e x} \left (1-x^2\right )^{3/4}-\frac{e^{3/2} \log \left (\sqrt{e}+\frac{\sqrt{e} x}{\sqrt{1-x^2}}-\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}\right )}{8 \sqrt{2}}+\frac{e^{3/2} \log \left (\sqrt{e}+\frac{\sqrt{e} x}{\sqrt{1-x^2}}+\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}\right )}{8 \sqrt{2}}+\frac{e^{3/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt{2}}-\frac{e^{3/2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt{2}}\\ &=-\frac{1}{2} e \sqrt{e x} \left (1-x^2\right )^{3/4}-\frac{e^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt{2}}+\frac{e^{3/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e x}}{\sqrt{e} \sqrt [4]{1-x^2}}\right )}{4 \sqrt{2}}-\frac{e^{3/2} \log \left (\sqrt{e}+\frac{\sqrt{e} x}{\sqrt{1-x^2}}-\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}\right )}{8 \sqrt{2}}+\frac{e^{3/2} \log \left (\sqrt{e}+\frac{\sqrt{e} x}{\sqrt{1-x^2}}+\frac{\sqrt{2} \sqrt{e x}}{\sqrt [4]{1-x^2}}\right )}{8 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0937802, size = 190, normalized size = 0.78 \[ -\frac{(e x)^{3/2} \left (8 \sqrt{x} \left (1-x^2\right )^{3/4}+\sqrt{2} \log \left (\frac{x}{\sqrt{1-x^2}}-\frac{\sqrt{2} \sqrt{x}}{\sqrt [4]{1-x^2}}+1\right )-\sqrt{2} \log \left (\frac{x}{\sqrt{1-x^2}}+\frac{\sqrt{2} \sqrt{x}}{\sqrt [4]{1-x^2}}+1\right )+2 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{x}}{\sqrt [4]{1-x^2}}\right )-2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{x}}{\sqrt [4]{1-x^2}}+1\right )\right )}{16 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((e*x)^(3/2)*(8*Sqrt[x]*(1 - x^2)^(3/4) + 2*Sqrt[2]*ArcTan[1 - (Sqrt[2]*Sqrt[x])/(1 - x^2)^(1/4)] - 2*Sqrt[2]
*ArcTan[1 + (Sqrt[2]*Sqrt[x])/(1 - x^2)^(1/4)] + Sqrt[2]*Log[1 + x/Sqrt[1 - x^2] - (Sqrt[2]*Sqrt[x])/(1 - x^2)
^(1/4)] - Sqrt[2]*Log[1 + x/Sqrt[1 - x^2] + (Sqrt[2]*Sqrt[x])/(1 - x^2)^(1/4)]))/(16*x^(3/2))

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt [4]{1-x}}}{\frac{1}{\sqrt [4]{1+x}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.81109, size = 1299, normalized size = 5.32 \begin{align*} -\frac{1}{2} \, \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + \frac{1}{4} \, \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \arctan \left (-\frac{e^{6} x^{2} - e^{6} + \sqrt{2}{\left (e^{6}\right )}^{\frac{3}{4}} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - \sqrt{2}{\left (e^{6}\right )}^{\frac{3}{4}}{\left (x^{2} - 1\right )} \sqrt{-\frac{e^{3} \sqrt{x + 1} x \sqrt{-x + 1} - \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - \sqrt{e^{6}}{\left (x^{2} - 1\right )}}{x^{2} - 1}}}{e^{6} x^{2} - e^{6}}\right ) + \frac{1}{4} \, \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \arctan \left (\frac{e^{6} x^{2} - e^{6} - \sqrt{2}{\left (e^{6}\right )}^{\frac{3}{4}} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + \sqrt{2}{\left (e^{6}\right )}^{\frac{3}{4}}{\left (x^{2} - 1\right )} \sqrt{-\frac{e^{3} \sqrt{x + 1} x \sqrt{-x + 1} + \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - \sqrt{e^{6}}{\left (x^{2} - 1\right )}}{x^{2} - 1}}}{e^{6} x^{2} - e^{6}}\right ) + \frac{1}{16} \, \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \log \left (-\frac{e^{3} \sqrt{x + 1} x \sqrt{-x + 1} + \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - \sqrt{e^{6}}{\left (x^{2} - 1\right )}}{x^{2} - 1}\right ) - \frac{1}{16} \, \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \log \left (-\frac{e^{3} \sqrt{x + 1} x \sqrt{-x + 1} - \sqrt{2}{\left (e^{6}\right )}^{\frac{1}{4}} \sqrt{e x} e{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - \sqrt{e^{6}}{\left (x^{2} - 1\right )}}{x^{2} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(e*x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4) + 1/4*sqrt(2)*(e^6)^(1/4)*arctan(-(e^6*x^2 - e^6 + sqrt(2)*(e^6)
^(3/4)*sqrt(e*x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4) - sqrt(2)*(e^6)^(3/4)*(x^2 - 1)*sqrt(-(e^3*sqrt(x + 1)*x*sqrt(
-x + 1) - sqrt(2)*(e^6)^(1/4)*sqrt(e*x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4) - sqrt(e^6)*(x^2 - 1))/(x^2 - 1)))/(e^6
*x^2 - e^6)) + 1/4*sqrt(2)*(e^6)^(1/4)*arctan((e^6*x^2 - e^6 - sqrt(2)*(e^6)^(3/4)*sqrt(e*x)*e*(x + 1)^(3/4)*(
-x + 1)^(3/4) + sqrt(2)*(e^6)^(3/4)*(x^2 - 1)*sqrt(-(e^3*sqrt(x + 1)*x*sqrt(-x + 1) + sqrt(2)*(e^6)^(1/4)*sqrt
(e*x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4) - sqrt(e^6)*(x^2 - 1))/(x^2 - 1)))/(e^6*x^2 - e^6)) + 1/16*sqrt(2)*(e^6)^
(1/4)*log(-(e^3*sqrt(x + 1)*x*sqrt(-x + 1) + sqrt(2)*(e^6)^(1/4)*sqrt(e*x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4) - sq
rt(e^6)*(x^2 - 1))/(x^2 - 1)) - 1/16*sqrt(2)*(e^6)^(1/4)*log(-(e^3*sqrt(x + 1)*x*sqrt(-x + 1) - sqrt(2)*(e^6)^
(1/4)*sqrt(e*x)*e*(x + 1)^(3/4)*(-x + 1)^(3/4) - sqrt(e^6)*(x^2 - 1))/(x^2 - 1))

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Sympy [C]  time = 104.32, size = 114, normalized size = 0.47 \begin{align*} - \frac{i e^{\frac{3}{2}}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{5}{8}, - \frac{1}{8} & - \frac{1}{2}, - \frac{1}{4}, 0, 1 \\-1, - \frac{5}{8}, - \frac{1}{2}, - \frac{1}{8}, 0, 0 & \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{x^{2}}} \right )} e^{\frac{i \pi }{4}}}{4 \pi \Gamma \left (\frac{1}{4}\right )} - \frac{e^{\frac{3}{2}}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{5}{4}, - \frac{9}{8}, - \frac{3}{4}, - \frac{5}{8}, - \frac{1}{4}, 1 & \\- \frac{9}{8}, - \frac{5}{8} & - \frac{5}{4}, -1, - \frac{3}{4}, 0 \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{4 \pi \Gamma \left (\frac{1}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*e**(3/2)*meijerg(((-5/8, -1/8), (-1/2, -1/4, 0, 1)), ((-1, -5/8, -1/2, -1/8, 0, 0), ()), exp_polar(-2*I*pi)
/x**2)*exp(I*pi/4)/(4*pi*gamma(1/4)) - e**(3/2)*meijerg(((-5/4, -9/8, -3/4, -5/8, -1/4, 1), ()), ((-9/8, -5/8)
, (-5/4, -1, -3/4, 0)), x**(-2))/(4*pi*gamma(1/4))

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out