3.912 \(\int \frac{1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx\)

Optimal. Leaf size=76 \[ -\frac{64 \left (1-x^2\right )^{11/4}}{231 e (e x)^{11/2}}+\frac{16 \left (1-x^2\right )^{7/4}}{21 e (e x)^{11/2}}-\frac{2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}} \]

[Out]

(-2*(1 - x^2)^(3/4))/(3*e*(e*x)^(11/2)) + (16*(1 - x^2)^(7/4))/(21*e*(e*x)^(11/2)) - (64*(1 - x^2)^(11/4))/(23
1*e*(e*x)^(11/2))

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Rubi [A]  time = 0.0211333, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {125, 273, 264} \[ -\frac{64 \left (1-x^2\right )^{11/4}}{231 e (e x)^{11/2}}+\frac{16 \left (1-x^2\right )^{7/4}}{21 e (e x)^{11/2}}-\frac{2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - x)^(1/4)*(e*x)^(13/2)*(1 + x)^(1/4)),x]

[Out]

(-2*(1 - x^2)^(3/4))/(3*e*(e*x)^(11/2)) + (16*(1 - x^2)^(7/4))/(21*e*(e*x)^(11/2)) - (64*(1 - x^2)^(11/4))/(23
1*e*(e*x)^(11/2))

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 273

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

Rule 264

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [4]{1-x} (e x)^{13/2} \sqrt [4]{1+x}} \, dx &=\int \frac{1}{(e x)^{13/2} \sqrt [4]{1-x^2}} \, dx\\ &=-\frac{2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}}-\frac{8}{3} \int \frac{\left (1-x^2\right )^{3/4}}{(e x)^{13/2}} \, dx\\ &=-\frac{2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}}+\frac{16 \left (1-x^2\right )^{7/4}}{21 e (e x)^{11/2}}+\frac{32}{21} \int \frac{\left (1-x^2\right )^{7/4}}{(e x)^{13/2}} \, dx\\ &=-\frac{2 \left (1-x^2\right )^{3/4}}{3 e (e x)^{11/2}}+\frac{16 \left (1-x^2\right )^{7/4}}{21 e (e x)^{11/2}}-\frac{64 \left (1-x^2\right )^{11/4}}{231 e (e x)^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.0162375, size = 40, normalized size = 0.53 \[ -\frac{2 \left (1-x^2\right )^{3/4} \left (32 x^4+24 x^2+21\right ) \sqrt{e x}}{231 e^7 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - x)^(1/4)*(e*x)^(13/2)*(1 + x)^(1/4)),x]

[Out]

(-2*Sqrt[e*x]*(1 - x^2)^(3/4)*(21 + 24*x^2 + 32*x^4))/(231*e^7*x^6)

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Maple [A]  time = 0.003, size = 33, normalized size = 0.4 \begin{align*} -{\frac{2\,x \left ( 32\,{x}^{4}+24\,{x}^{2}+21 \right ) }{231} \left ( 1+x \right ) ^{{\frac{3}{4}}} \left ( 1-x \right ) ^{{\frac{3}{4}}} \left ( ex \right ) ^{-{\frac{13}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-x)^(1/4)/(e*x)^(13/2)/(1+x)^(1/4),x)

[Out]

-2/231*x*(1+x)^(3/4)*(32*x^4+24*x^2+21)*(1-x)^(3/4)/(e*x)^(13/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (e x\right )^{\frac{13}{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)^(13/2)/(1+x)^(1/4),x, algorithm="maxima")

[Out]

integrate(1/((e*x)^(13/2)*(x + 1)^(1/4)*(-x + 1)^(1/4)), x)

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Fricas [A]  time = 1.522, size = 108, normalized size = 1.42 \begin{align*} -\frac{2 \,{\left (32 \, x^{4} + 24 \, x^{2} + 21\right )} \sqrt{e x}{\left (x + 1\right )}^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{3}{4}}}{231 \, e^{7} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(1/4)/(e*x)^(13/2)/(1+x)^(1/4),x, algorithm="fricas")

[Out]

-2/231*(32*x^4 + 24*x^2 + 21)*sqrt(e*x)*(x + 1)^(3/4)*(-x + 1)^(3/4)/(e^7*x^6)

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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)**(13/2)/(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)^(13/2)/(1+x)^(1/4),x, algorithm="giac")

[Out]

Timed out