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

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

[Out]

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

-1/3*(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*} \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 = 0.04, size = 20, normalized size = 1.00 \begin {gather*} -\frac {(1-x)^{3/2}}{3 (1+x)^{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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 2.93, size = 66, normalized size = 3.30 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\left (-1+x\right ) \sqrt {\frac {1-x}{1+x}}}{3 \left (1+x\right )},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},\frac {-2 I \sqrt {1-\frac {2}{1+x}}}{3 \left (1+x\right )}+\frac {I \sqrt {1-\frac {2}{1+x}}}{3}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Piecewise[{{(-1 + x) Sqrt[(1 - x) / (1 + x)] / (3 (1 + x)), 1 / Abs[1 + x] > 1 / 2}}, -2 I Sqrt[1 - 2 / (1 + x
)] / (3 (1 + x)) + I Sqrt[1 - 2 / (1 + x)] / 3]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(29\) vs. \(2(14)=28\).
time = 0.16, size = 30, normalized size = 1.50

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	\(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{2}-2 x +1\right )}{3 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (14) = 28\).
time = 0.26, 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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (14) = 28\).
time = 0.29, 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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Sympy [A]
time = 0.95, size = 66, normalized size = 3.30 \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 {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} \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)), 1/Abs(x + 1) > 1/2), (I*sqrt(1 - 2/(x
+ 1))/3 - 2*I*sqrt(1 - 2/(x + 1))/(3*(x + 1)), True))

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Giac [A]
time = 0.01, size = 41, normalized size = 2.05 \begin {gather*} -\frac {\sqrt {-x+1} \sqrt {-x+1} \sqrt {-x+1} \sqrt {x+1}}{3 \left (x+1\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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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