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

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

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Rubi [A]  time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {(1-x)^{3/2}}{3 (x+1)^{3/2}} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} -\frac {(1-x)^{3/2}}{3 (x+1)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(1 - x)^(3/2)/(1 + x)^(3/2)

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IntegrateAlgebraic [A]  time = 0.07, size = 20, normalized size = 1.00 \begin {gather*} -\frac {(1-x)^{3/2}}{3 (x+1)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(1 - x)^(3/2)/(1 + x)^(3/2)

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fricas [B]  time = 1.30, size = 37, normalized size = 1.85 \begin {gather*} -\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 {gather*}

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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giac [B]  time = 0.71, size = 89, normalized size = 4.45 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.00, size = 15, normalized size = 0.75 \begin {gather*} -\frac {\left (-x +1\right )^{\frac {3}{2}}}{3 \left (x +1\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.32, size = 38, normalized size = 1.90 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.26, size = 32, normalized size = 1.60 \begin {gather*} \frac {x\,\sqrt {1-x}-\sqrt {1-x}}{\left (3\,x+3\right )\,\sqrt {x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 1.69, size = 65, normalized size = 3.25 \begin {gather*} \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 {gather*}

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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