3.10.14 \(\int \frac {\sqrt {e x}}{\sqrt [4]{1-x} \sqrt [4]{1+x}} \, dx\) [914]

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

[Out]

-e*(-x^2+1)^(3/4)/(e*x)^(1/2)+(1-1/x^2)^(1/4)*(cos(1/2*arccsc(x))^2)^(1/2)/cos(1/2*arccsc(x))*EllipticE(sin(1/
2*arccsc(x)),2^(1/2))*(e*x)^(1/2)/(-x^2+1)^(1/4)

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Rubi [A]
time = 0.02, antiderivative size = 60, normalized size of antiderivative = 1.00, 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 = {126, 321, 323, 342, 234} \begin {gather*} \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}} \end {gather*}

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 126

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[n, m] && GtQ[a, 0] && GtQ[c,
0]

Rule 234

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

Rule 321

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 323

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 342

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]

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 ) \text {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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.02, size = 25, normalized size = 0.42 \begin {gather*} \frac {2}{3} x \sqrt {e x} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {7}{4};x^2\right ) \end {gather*}

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 = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {e x}}{\left (1-x \right )^{\frac {1}{4}} \left (1+x \right )^{\frac {1}{4}}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Sympy [C] Result contains complex when optimal does not.
time = 2.28, size = 105, normalized size = 1.75 \begin {gather*} \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 {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {e\,x}}{{\left (1-x\right )}^{1/4}\,{\left (x+1\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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