3.1120 \(\int \frac{1}{\sqrt{1-x} (1+x)^{3/2}} \, dx\)

Optimal. Leaf size=18 \[ -\frac{\sqrt{1-x}}{\sqrt{x+1}} \]

[Out]

-(Sqrt[1 - x]/Sqrt[1 + x])

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Rubi [A]  time = 0.0015644, antiderivative size = 18, 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} \[ -\frac{\sqrt{1-x}}{\sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 - x]*(1 + x)^(3/2)),x]

[Out]

-(Sqrt[1 - x]/Sqrt[1 + x])

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

Mathematica [A]  time = 0.0034147, size = 18, normalized size = 1. \[ -\frac{\sqrt{1-x}}{\sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - x]/Sqrt[1 + x])

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Maple [A]  time = 0.002, size = 15, normalized size = 0.8 \begin{align*} -{\sqrt{1-x}{\frac{1}{\sqrt{1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.55, size = 22, normalized size = 1.22 \begin{align*} -\frac{\sqrt{-x^{2} + 1}}{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)/(x + 1)

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Fricas [A]  time = 1.7723, size = 61, normalized size = 3.39 \begin{align*} -\frac{x + \sqrt{x + 1} \sqrt{-x + 1} + 1}{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x + sqrt(x + 1)*sqrt(-x + 1) + 1)/(x + 1)

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Sympy [A]  time = 1.07564, size = 29, normalized size = 1.61 \begin{align*} \begin{cases} - \sqrt{-1 + \frac{2}{x + 1}} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\- i \sqrt{1 - \frac{2}{x + 1}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.07944, size = 58, normalized size = 3.22 \begin{align*} \frac{\sqrt{2} - \sqrt{-x + 1}}{2 \, \sqrt{x + 1}} - \frac{\sqrt{x + 1}}{2 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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