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

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

[Out]

(1 + x)^(3/2)/(3*(1 - x)^(3/2))

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Rubi [A]  time = 0.0017524, 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} \[ \frac{(x+1)^{3/2}}{3 (1-x)^{3/2}} \]

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 = 0.0040198, size = 20, normalized size = 1. \[ \frac{(x+1)^{3/2}}{3 (1-x)^{3/2}} \]

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 = 0.003, 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 = 1.48774, 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 = 1.49898, size = 89, normalized size = 4.45 \begin{align*} \frac{x^{2} +{\left (x + 1\right )}^{\frac{3}{2}} \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 + (x + 1)^(3/2)*sqrt(-x + 1) - 2*x + 1)/(x^2 - 2*x + 1)

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Sympy [A]  time = 2.99195, size = 61, normalized size = 3.05 \begin{align*} \begin{cases} \frac{i \left (x + 1\right )^{\frac{3}{2}}}{3 \sqrt{x - 1} \left (x + 1\right ) - 6 \sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\- \frac{\left (x + 1\right )^{\frac{3}{2}}}{3 \sqrt{1 - x} \left (x + 1\right ) - 6 \sqrt{1 - x}} & \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((I*(x + 1)**(3/2)/(3*sqrt(x - 1)*(x + 1) - 6*sqrt(x - 1)), Abs(x + 1)/2 > 1), (-(x + 1)**(3/2)/(3*sq
rt(1 - x)*(x + 1) - 6*sqrt(1 - x)), True))

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Giac [A]  time = 1.70521, size = 26, normalized size = 1.3 \begin{align*} \frac{{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1}}{3 \,{\left (x - 1\right )}^{2}} \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]

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