\(\int \frac {1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx\) [1122]

Optimal result

Integrand size = 17, antiderivative size = 42 \[ \int \frac {1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx=\frac {1}{3 (1-x)^{3/2} \sqrt {1+x}}+\frac {2 x}{3 \sqrt {1-x} \sqrt {1+x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 42, 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 = {47, 39} \[ \int \frac {1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx=\frac {2 x}{3 \sqrt {1-x} \sqrt {x+1}}+\frac {1}{3 (1-x)^{3/2} \sqrt {x+1}} \]

[In]

Int[1/((1 - x)^(5/2)*(1 + x)^(3/2)),x]

[Out]

1/(3*(1 - x)^(3/2)*Sqrt[1 + x]) + (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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx=\frac {1+2 x-2 x^2}{3 (1-x)^{3/2} \sqrt {1+x}} \]

[In]

Integrate[1/((1 - x)^(5/2)*(1 + x)^(3/2)),x]

[Out]

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

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.60

method	result	size
gosper	\(-\frac {2 x^{2}-2 x -1}{3 \left (1-x \right )^{\frac {3}{2}} \sqrt {1+x}}\)	\(25\)
default	\(\frac {1}{3 \left (1-x \right )^{\frac {3}{2}} \sqrt {1+x}}+\frac {2}{3 \sqrt {1-x}\, \sqrt {1+x}}-\frac {2 \sqrt {1-x}}{3 \sqrt {1+x}}\)	\(44\)
risch	\(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{2}-2 x -1\right )}{3 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right ) \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\)	\(51\)

[In]

int(1/(1-x)^(5/2)/(1+x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.29 \[ \int \frac {1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx=\frac {x^{3} - x^{2} - {\left (2 \, x^{2} - 2 \, x - 1\right )} \sqrt {x + 1} \sqrt {-x + 1} - x + 1}{3 \, {\left (x^{3} - x^{2} - x + 1\right )}} \]

[In]

integrate(1/(1-x)^(5/2)/(1+x)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.19 (sec) , antiderivative size = 160, normalized size of antiderivative = 3.81 \[ \int \frac {1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx=\begin {cases} - \frac {2 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 12 x + 3 \left (x + 1\right )^{2}} + \frac {6 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{- 12 x + 3 \left (x + 1\right )^{2}} - \frac {3 \sqrt {-1 + \frac {2}{x + 1}}}{- 12 x + 3 \left (x + 1\right )^{2}} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {2 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 12 x + 3 \left (x + 1\right )^{2}} + \frac {6 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{- 12 x + 3 \left (x + 1\right )^{2}} - \frac {3 i \sqrt {1 - \frac {2}{x + 1}}}{- 12 x + 3 \left (x + 1\right )^{2}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/(1-x)**(5/2)/(1+x)**(3/2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx=\frac {2 \, x}{3 \, \sqrt {-x^{2} + 1}} - \frac {1}{3 \, {\left (\sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} \]

[In]

integrate(1/(1-x)^(5/2)/(1+x)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (30) = 60\).

Time = 0.31 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.60 \[ \int \frac {1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx=\frac {\sqrt {2} - \sqrt {-x + 1}}{8 \, \sqrt {x + 1}} - \frac {{\left (5 \, x - 7\right )} \sqrt {x + 1} \sqrt {-x + 1}}{12 \, {\left (x - 1\right )}^{2}} - \frac {\sqrt {x + 1}}{8 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}} \]

[In]

integrate(1/(1-x)^(5/2)/(1+x)^(3/2),x, algorithm="giac")

[Out]

1/8*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 1/12*(5*x - 7)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^2 - 1/8*sqrt(x + 1)
/(sqrt(2) - sqrt(-x + 1))

Mupad [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx=\frac {2\,x\,\sqrt {1-x}+\sqrt {1-x}-2\,x^2\,\sqrt {1-x}}{3\,{\left (x-1\right )}^2\,\sqrt {x+1}} \]

[In]

int(1/((1 - x)^(5/2)*(x + 1)^(3/2)),x)

[Out]

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