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

Optimal. Leaf size=60 \[ \frac{\sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \csc ^{-1}(x)\right |2\right )}{\sqrt [4]{1-x^2}}-\frac{e \left (1-x^2\right )^{3/4}}{\sqrt{e x}} \]

[Out]

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

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Rubi [A]  time = 0.0224061, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {125, 315, 317, 335, 228} \[ \frac{\sqrt [4]{1-\frac{1}{x^2}} \sqrt{e x} E\left (\left .\frac{1}{2} \csc ^{-1}(x)\right |2\right )}{\sqrt [4]{1-x^2}}-\frac{e \left (1-x^2\right )^{3/4}}{\sqrt{e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 315

Int[Sqrt[(c_)*(x_)]/((a_) + (b_.)*(x_)^2)^(1/4), x_Symbol] :> Simp[(c*(a + b*x^2)^(3/4))/(b*Sqrt[c*x]), x] + D
ist[(a*c^2)/(2*b), Int[1/((c*x)^(3/2)*(a + b*x^2)^(1/4)), x], x] /; FreeQ[{a, b, c}, x] && NegQ[b/a]

Rule 317

Int[1/(((c_.)*(x_))^(3/2)*((a_) + (b_.)*(x_)^2)^(1/4)), x_Symbol] :> Dist[(Sqrt[c*x]*(1 + a/(b*x^2))^(1/4))/(c
^2*(a + b*x^2)^(1/4)), Int[1/(x^2*(1 + a/(b*x^2))^(1/4)), x], x] /; FreeQ[{a, b, c}, x] && NegQ[b/a]

Rule 335

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

Rule 228

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(2*EllipticE[(1*ArcSin[Rt[-(b/a), 2]*x])/2, 2])/(a^(1/4)*R
t[-(b/a), 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]

Rubi steps

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

Mathematica [C]  time = 0.004759, size = 25, normalized size = 0.42 \[ \frac{2}{3} x \sqrt{e x} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[e*x]*Hypergeometric2F1[1/4, 3/4, 7/4, x^2])/3

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{ex}{\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)^(1/2)/(1-x)^(1/4)/(1+x)^(1/4),x)

[Out]

int((e*x)^(1/2)/(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{\sqrt{e x}}{{\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)^(1/2)/(1-x)^(1/4)/(1+x)^(1/4),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{e x}{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}}}{x^{2} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 10.0919, size = 105, normalized size = 1.75 \begin{align*} \frac{i \sqrt{e}{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{8}, \frac{3}{8} & 0, \frac{1}{4}, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{8}, 0, \frac{3}{8}, \frac{1}{2}, 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{\sqrt{e}{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{4}, - \frac{5}{8}, - \frac{1}{4}, - \frac{1}{8}, \frac{1}{4}, 1 & \\- \frac{5}{8}, - \frac{1}{8} & - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{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)**(1/2)/(1-x)**(1/4)/(1+x)**(1/4),x)

[Out]

I*sqrt(e)*meijerg(((-1/8, 3/8), (0, 1/4, 1/2, 1)), ((-1/2, -1/8, 0, 3/8, 1/2, 0), ()), exp_polar(-2*I*pi)/x**2
)*exp(I*pi/4)/(4*pi*gamma(1/4)) - sqrt(e)*meijerg(((-3/4, -5/8, -1/4, -1/8, 1/4, 1), ()), ((-5/8, -1/8), (-3/4
, -1/2, -1/4, 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)^(1/2)/(1-x)^(1/4)/(1+x)^(1/4),x, algorithm="giac")

[Out]

Timed out