3.1133 \(\int \frac{1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx\)

Optimal. Leaf size=43 \[ \frac{2 x}{3 \sqrt{1-x} \sqrt{x+1}}+\frac{x}{3 (1-x)^{3/2} (x+1)^{3/2}} \]

[Out]

x/(3*(1 - x)^(3/2)*(1 + x)^(3/2)) + (2*x)/(3*Sqrt[1 - x]*Sqrt[1 + x])

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Rubi [A]  time = 0.0047007, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {40, 39} \[ \frac{2 x}{3 \sqrt{1-x} \sqrt{x+1}}+\frac{x}{3 (1-x)^{3/2} (x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(3*(1 - x)^(3/2)*(1 + x)^(3/2)) + (2*x)/(3*Sqrt[1 - x]*Sqrt[1 + x])

Rule 40

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(x*(a + b*x)^(m + 1)*(c + d*x)^(m +
1))/(2*a*c*(m + 1)), x] + Dist[(2*m + 3)/(2*a*c*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /; F
reeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rubi steps

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

Mathematica [A]  time = 0.0064667, size = 23, normalized size = 0.53 \[ -\frac{x \left (2 x^2-3\right )}{3 \left (1-x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x*(-3 + 2*x^2))/(3*(1 - x^2)^(3/2))

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Maple [A]  time = 0.002, size = 23, normalized size = 0.5 \begin{align*} -{\frac{x \left ( 2\,{x}^{2}-3 \right ) }{3} \left ( 1-x \right ) ^{-{\frac{3}{2}}} \left ( 1+x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.992303, size = 34, normalized size = 0.79 \begin{align*} \frac{2 \, x}{3 \, \sqrt{-x^{2} + 1}} + \frac{x}{3 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(5/2)/(1+x)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.58676, size = 85, normalized size = 1.98 \begin{align*} -\frac{{\left (2 \, x^{3} - 3 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1}}{3 \,{\left (x^{4} - 2 \, x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 38.4734, size = 279, normalized size = 6.49 \begin{align*} \begin{cases} - \frac{2 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{3}}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} + \frac{6 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{2}}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} - \frac{3 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} - \frac{\sqrt{-1 + \frac{2}{x + 1}}}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\- \frac{2 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{3}}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} + \frac{6 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{2}}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} - \frac{3 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} - \frac{i \sqrt{1 - \frac{2}{x + 1}}}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*sqrt(-1 + 2/(x + 1))*(x + 1)**3/(12*x + 3*(x + 1)**3 - 12*(x + 1)**2 + 12) + 6*sqrt(-1 + 2/(x +
1))*(x + 1)**2/(12*x + 3*(x + 1)**3 - 12*(x + 1)**2 + 12) - 3*sqrt(-1 + 2/(x + 1))*(x + 1)/(12*x + 3*(x + 1)**
3 - 12*(x + 1)**2 + 12) - sqrt(-1 + 2/(x + 1))/(12*x + 3*(x + 1)**3 - 12*(x + 1)**2 + 12), 2/Abs(x + 1) > 1),
(-2*I*sqrt(1 - 2/(x + 1))*(x + 1)**3/(12*x + 3*(x + 1)**3 - 12*(x + 1)**2 + 12) + 6*I*sqrt(1 - 2/(x + 1))*(x +
1)**2/(12*x + 3*(x + 1)**3 - 12*(x + 1)**2 + 12) - 3*I*sqrt(1 - 2/(x + 1))*(x + 1)/(12*x + 3*(x + 1)**3 - 12*
(x + 1)**2 + 12) - I*sqrt(1 - 2/(x + 1))/(12*x + 3*(x + 1)**3 - 12*(x + 1)**2 + 12), True))

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Giac [B]  time = 1.08233, size = 153, normalized size = 3.56 \begin{align*} \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{192 \,{\left (x + 1\right )}^{\frac{3}{2}}} + \frac{11 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{64 \, \sqrt{x + 1}} - \frac{{\left (4 \, x - 5\right )} \sqrt{x + 1} \sqrt{-x + 1}}{12 \,{\left (x - 1\right )}^{2}} - \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{33 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{192 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-x)^(5/2)/(1+x)^(5/2),x, algorithm="giac")

[Out]

1/192*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) + 11/64*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/12*(4*x - 5)*s
qrt(x + 1)*sqrt(-x + 1)/(x - 1)^2 - 1/192*(x + 1)^(3/2)*(33*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) + 1)/(sqrt(2) -
sqrt(-x + 1))^3