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]

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

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Rubi [A]  time = 0.00, antiderivative size = 30, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.91, size = 25, normalized size = 0.83 \[ -\frac {2 \, \sqrt {e x} {\left (x + 1\right )}^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{3 \, e^{3} x^{2}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e x\right )^{\frac {5}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}}\,{d x} \]

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]

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

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maple [A]  time = 0.00, size = 21, normalized size = 0.70 \[ -\frac {2 \left (x +1\right )^{\frac {3}{4}} \left (-x +1\right )^{\frac {3}{4}} x}{3 \left (e x \right )^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e x\right )^{\frac {5}{2}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}}\,{d x} \]

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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mupad [B]  time = 1.24, size = 36, normalized size = 1.20 \[ -\frac {\sqrt {e\,x}\,\left (\frac {2}{3\,e^3}-\frac {2\,x^2}{3\,e^3}\right )}{x^2\,{\left (1-x\right )}^{1/4}\,{\left (x+1\right )}^{1/4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 65.19, size = 82, normalized size = 2.73 \[ \frac {i {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {11}{8}, \frac {15}{8}, 1 & \frac {3}{2}, \frac {7}{4}, 2 \\1, \frac {11}{8}, \frac {3}{2}, \frac {15}{8}, 2 & 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{2}}} \right )} e^{\frac {i \pi }{4}}}{4 \pi e^{\frac {5}{2}} \Gamma \left (\frac {1}{4}\right )} - \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {3}{4}, \frac {7}{8}, \frac {5}{4}, \frac {11}{8}, \frac {7}{4}, 1 & \\\frac {7}{8}, \frac {11}{8} & \frac {3}{4}, 1, \frac {5}{4}, 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi e^{\frac {5}{2}} \Gamma \left (\frac {1}{4}\right )} \]

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]

I*meijerg(((11/8, 15/8, 1), (3/2, 7/4, 2)), ((1, 11/8, 3/2, 15/8, 2), (0,)), exp_polar(-2*I*pi)/x**2)*exp(I*pi
/4)/(4*pi*e**(5/2)*gamma(1/4)) - meijerg(((3/4, 7/8, 5/4, 11/8, 7/4, 1), ()), ((7/8, 11/8), (3/4, 1, 5/4, 0)),
x**(-2))/(4*pi*e**(5/2)*gamma(1/4))

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