\(\int \frac {\sqrt {1-x}}{(1+x)^{5/2}} \, dx\) [1130]

Optimal result

Integrand size = 17, antiderivative size = 20 \[ \int \frac {\sqrt {1-x}}{(1+x)^{5/2}} \, dx=-\frac {(1-x)^{3/2}}{3 (1+x)^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {37} \[ \int \frac {\sqrt {1-x}}{(1+x)^{5/2}} \, dx=-\frac {(1-x)^{3/2}}{3 (x+1)^{3/2}} \]

[In]

Int[Sqrt[1 - x]/(1 + x)^(5/2),x]

[Out]

-1/3*(1 - x)^(3/2)/(1 + x)^(3/2)

Rule 37

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] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-x}}{(1+x)^{5/2}} \, dx=-\frac {(1-x)^{3/2}}{3 (1+x)^{3/2}} \]

[In]

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

[Out]

-1/3*(1 - x)^(3/2)/(1 + x)^(3/2)

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75

method	result	size
gosper	\(-\frac {\left (1-x \right )^{\frac {3}{2}}}{3 \left (1+x \right )^{\frac {3}{2}}}\)	\(15\)
default	\(-\frac {2 \sqrt {1-x}}{3 \left (1+x \right )^{\frac {3}{2}}}+\frac {\sqrt {1-x}}{3 \sqrt {1+x}}\)	\(30\)
risch	\(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{2}-2 x +1\right )}{3 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}} \sqrt {-\left (-1+x \right ) \left (1+x \right )}}\)	\(44\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (14) = 28\).

Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.85 \[ \int \frac {\sqrt {1-x}}{(1+x)^{5/2}} \, dx=-\frac {x^{2} - \sqrt {x + 1} {\left (x - 1\right )} \sqrt {-x + 1} + 2 \, x + 1}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.30 \[ \int \frac {\sqrt {1-x}}{(1+x)^{5/2}} \, dx=\begin {cases} \frac {\sqrt {-1 + \frac {2}{x + 1}}}{3} - \frac {2 \sqrt {-1 + \frac {2}{x + 1}}}{3 \left (x + 1\right )} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\\frac {i \sqrt {1 - \frac {2}{x + 1}}}{3} - \frac {2 i \sqrt {1 - \frac {2}{x + 1}}}{3 \left (x + 1\right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (14) = 28\).

Time = 0.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.90 \[ \int \frac {\sqrt {1-x}}{(1+x)^{5/2}} \, dx=-\frac {2 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} + 2 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{3 \, {\left (x + 1\right )}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (14) = 28\).

Time = 0.31 (sec) , antiderivative size = 89, normalized size of antiderivative = 4.45 \[ \int \frac {\sqrt {1-x}}{(1+x)^{5/2}} \, dx=\frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}}{24 \, {\left (x + 1\right )}^{\frac {3}{2}}} - \frac {\sqrt {2} - \sqrt {-x + 1}}{8 \, \sqrt {x + 1}} + \frac {{\left (x + 1\right )}^{\frac {3}{2}} {\left (\frac {3 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{24 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{3}} \]

[In]

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

[Out]

1/24*(sqrt(2) - sqrt(-x + 1))^3/(x + 1)^(3/2) - 1/8*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) + 1/24*(x + 1)^(3/2)*
(3*(sqrt(2) - sqrt(-x + 1))^2/(x + 1) - 1)/(sqrt(2) - sqrt(-x + 1))^3

Mupad [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60 \[ \int \frac {\sqrt {1-x}}{(1+x)^{5/2}} \, dx=\frac {x\,\sqrt {1-x}-\sqrt {1-x}}{\left (3\,x+3\right )\,\sqrt {x+1}} \]

[In]

int((1 - x)^(1/2)/(x + 1)^(5/2),x)

[Out]

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