\(\int \frac {1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx\) [1133]

Optimal result

Integrand size = 17, antiderivative size = 43 \[ \int \frac {1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx=\frac {x}{3 (1-x)^{3/2} (1+x)^{3/2}}+\frac {2 x}{3 \sqrt {1-x} \sqrt {1+x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {40, 39} \[ \int \frac {1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx=\frac {2 x}{3 \sqrt {1-x} \sqrt {x+1}}+\frac {x}{3 (1-x)^{3/2} (x+1)^{3/2}} \]

[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 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]

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] /;
FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.56 \[ \int \frac {1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx=\frac {3 x-2 x^3}{3 \left (1-x^2\right )^{3/2}} \]

[In]

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

[Out]

(3*x - 2*x^3)/(3*(1 - x^2)^(3/2))

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.53

method	result	size
gosper	\(-\frac {x \left (2 x^{2}-3\right )}{3 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}\)	\(23\)
default	\(\frac {1}{3 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {1}{\sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}-\frac {2 \sqrt {1-x}}{3 \left (1+x \right )^{\frac {3}{2}}}-\frac {2 \sqrt {1-x}}{3 \sqrt {1+x}}\)	\(57\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx=-\frac {{\left (2 \, x^{3} - 3 \, x\right )} \sqrt {x + 1} \sqrt {-x + 1}}{3 \, {\left (x^{4} - 2 \, x^{2} + 1\right )}} \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 7.26 (sec) , antiderivative size = 280, normalized size of antiderivative = 6.51 \[ \int \frac {1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx=\begin {cases} - \frac {2 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3}}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} + \frac {6 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} - \frac {3 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} - \frac {\sqrt {-1 + \frac {2}{x + 1}}}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {2 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3}}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} + \frac {6 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} - \frac {3 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} - \frac {i \sqrt {1 - \frac {2}{x + 1}}}{12 x + 3 \left (x + 1\right )^{3} - 12 \left (x + 1\right )^{2} + 12} & \text {otherwise} \end {cases} \]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.58 \[ \int \frac {1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx=\frac {2 \, x}{3 \, \sqrt {-x^{2} + 1}} + \frac {x}{3 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (31) = 62\).

Time = 0.32 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.63 \[ \int \frac {1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx=\frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{192 \, {\left (x + 1\right )}^{\frac {3}{2}}} + \frac {11 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{64 \, \sqrt {x + 1}} - \frac {{\left (4 \, x - 5\right )} \sqrt {x + 1} \sqrt {-x + 1}}{12 \, {\left (x - 1\right )}^{2}} - \frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {33 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{192 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} \]

[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

Mupad [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx=\frac {3\,x\,\sqrt {1-x}-2\,x^3\,\sqrt {1-x}}{\left (3\,x+3\right )\,{\left (x-1\right )}^2\,\sqrt {x+1}} \]

[In]

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

[Out]

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