3.1130 \(\int \frac{\sqrt{1-x}}{(1+x)^{5/2}} \, dx\)

Optimal. Leaf size=20 \[ -\frac{(1-x)^{3/2}}{3 (x+1)^{3/2}} \]

[Out]

-(1 - x)^(3/2)/(3*(1 + x)^(3/2))

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Rubi [A]  time = 0.0015594, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {37} \[ -\frac{(1-x)^{3/2}}{3 (x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x]/(1 + x)^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.0040988, size = 20, normalized size = 1. \[ -\frac{(1-x)^{3/2}}{3 (x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1 - x)^(3/2)/(3*(1 + x)^(3/2))

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Maple [A]  time = 0.002, size = 15, normalized size = 0.8 \begin{align*} -{\frac{1}{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-x)^(1/2)/(1+x)^(5/2),x)

[Out]

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

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Maxima [B]  time = 0.982584, size = 51, normalized size = 2.55 \begin{align*} -\frac{2 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x^{2} + 2 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{3 \,{\left (x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 1.85103, size = 99, normalized size = 4.95 \begin{align*} -\frac{x^{2} - \sqrt{x + 1}{\left (x - 1\right )} \sqrt{-x + 1} + 2 \, x + 1}{3 \,{\left (x^{2} + 2 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 2.41654, size = 65, normalized size = 3.25 \begin{align*} \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{2}{\left |{x + 1}\right |} > 1 \\\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Giac [B]  time = 1.07845, size = 120, normalized size = 6. \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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