3.910 \(\int \frac{1}{\sqrt [4]{1-x} (e x)^{5/2} \sqrt [4]{1+x}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{2 (1-x)^{3/4} (x+1)^{3/4}}{3 e (e x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0038225, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {95} \[ -\frac{2 (1-x)^{3/4} (x+1)^{3/4}}{3 e (e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(1/4)*(e*x)^(5/2)*(1 + x)^(1/4)),x]

[Out]

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

Rule 95

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] /; FreeQ[{a, b, c, d,
e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0
] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [4]{1-x} (e x)^{5/2} \sqrt [4]{1+x}} \, dx &=-\frac{2 (1-x)^{3/4} (1+x)^{3/4}}{3 e (e x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0068246, size = 23, normalized size = 0.77 \[ -\frac{2 x \left (1-x^2\right )^{3/4}}{3 (e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(1/4)*(e*x)^(5/2)*(1 + x)^(1/4)),x]

[Out]

(-2*x*(1 - x^2)^(3/4))/(3*(e*x)^(5/2))

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Maple [A]  time = 0.003, size = 21, normalized size = 0.7 \begin{align*} -{\frac{2\,x}{3} \left ( 1+x \right ) ^{{\frac{3}{4}}} \left ( 1-x \right ) ^{{\frac{3}{4}}} \left ( ex \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-x)^(1/4)/(e*x)^(5/2)/(1+x)^(1/4),x)

[Out]

-2/3*x*(1+x)^(3/4)*(1-x)^(3/4)/(e*x)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (e x\right )^{\frac{5}{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(1/(1-x)^(1/4)/(e*x)^(5/2)/(1+x)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((e*x)^(5/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)), x)

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Fricas [A]  time = 1.55478, size = 74, normalized size = 2.47 \begin{align*} -\frac{2 \, \sqrt{e x}{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}}}{3 \, e^{3} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(1/4)/(e*x)^(5/2)/(1+x)^(1/4),x, algorithm="fricas")

[Out]

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

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Sympy [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(1/(1-x)**(1/4)/(e*x)**(5/2)/(1+x)**(1/4),x)

[Out]

Timed out

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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(1/(1-x)^(1/4)/(e*x)^(5/2)/(1+x)^(1/4),x, algorithm="giac")

[Out]

Timed out