3.1111 \(\int \frac{1}{(1-x)^{3/2} \sqrt{1+x}} \, dx\)

Optimal. Leaf size=17 \[ \frac{\sqrt{x+1}}{\sqrt{1-x}} \]

[Out]

Sqrt[1 + x]/Sqrt[1 - x]

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Rubi [A]  time = 0.0031938, antiderivative size = 17, 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} \[ \frac{\sqrt{x+1}}{\sqrt{1-x}} \]

Antiderivative was successfully verified.

[In]

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

Mathematica [A]  time = 0.0031577, size = 17, normalized size = 1. \[ \frac{\sqrt{x+1}}{\sqrt{1-x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 + x]/Sqrt[1 - x]

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Maple [A]  time = 0.002, size = 14, 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)^(3/2)/(1+x)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.48572, size = 22, normalized size = 1.29 \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)^(3/2)/(1+x)^(1/2),x, algorithm="maxima")

[Out]

-sqrt(-x^2 + 1)/(x - 1)

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Fricas [A]  time = 1.76067, size = 59, normalized size = 3.47 \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)^(3/2)/(1+x)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.04886, size = 29, normalized size = 1.71 \begin{align*} \begin{cases} \frac{1}{\sqrt{-1 + \frac{2}{x + 1}}} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\- \frac{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)**(3/2)/(1+x)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.08491, size = 26, normalized size = 1.53 \begin{align*} -\frac{\sqrt{x + 1} \sqrt{-x + 1}}{x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(x + 1)*sqrt(-x + 1)/(x - 1)