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

Optimal. Leaf size=18 \[ -\frac{(x+1)^{3/2}}{3 (x-1)^{3/2}} \]

[Out]

-(1 + x)^(3/2)/(3*(-1 + x)^(3/2))

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Rubi [A]  time = 0.0013852, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {37} \[ -\frac{(x+1)^{3/2}}{3 (x-1)^{3/2}} \]

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 = 0.0080133, size = 18, normalized size = 1. \[ -\frac{(x+1)^{3/2}}{3 (x-1)^{3/2}} \]

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 = 0.001, size = 13, normalized size = 0.7 \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 = 1.03217, size = 46, normalized size = 2.56 \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 = 1.533, size = 89, normalized size = 4.94 \begin{align*} -\frac{{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{x - 1} + x^{2} - 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 + 1)^(3/2)*sqrt(x - 1) + x^2 - 2*x + 1)/(x^2 - 2*x + 1)

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Sympy [A]  time = 5.4403, size = 61, normalized size = 3.39 \begin{align*} \begin{cases} - \frac{\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{i \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((-(x + 1)**(3/2)/(3*sqrt(x - 1)*(x + 1) - 6*sqrt(x - 1)), Abs(x + 1)/2 > 1), (I*(x + 1)**(3/2)/(3*sq
rt(1 - x)*(x + 1) - 6*sqrt(1 - x)), True))

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Giac [A]  time = 1.35924, size = 16, normalized size = 0.89 \begin{align*} -\frac{{\left (x + 1\right )}^{\frac{3}{2}}}{3 \,{\left (x - 1\right )}^{\frac{3}{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)/(x - 1)^(3/2)