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]

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

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \[ -\frac {(x+1)^{3/2}}{3 (x-1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(1 + x)^(3/2)/(-1 + x)^(3/2)

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fricas [B]  time = 1.12, size = 31, normalized size = 1.72 \[ -\frac {{\left (x + 1\right )}^{\frac {3}{2}} \sqrt {x - 1} + x^{2} - 2 \, x + 1}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \]

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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giac [A]  time = 1.27, size = 12, normalized size = 0.67 \[ -\frac {{\left (x + 1\right )}^{\frac {3}{2}}}{3 \, {\left (x - 1\right )}^{\frac {3}{2}}} \]

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)

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maple [A]  time = 0.00, size = 13, normalized size = 0.72 \[ -\frac {\left (x +1\right )^{\frac {3}{2}}}{3 \left (x -1\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.48, size = 34, normalized size = 1.89 \[ -\frac {2 \, \sqrt {x^{2} - 1}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {\sqrt {x^{2} - 1}}{3 \, {\left (x - 1\right )}} \]

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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mupad [B]  time = 1.08, size = 27, normalized size = 1.50 \[ -\frac {x\,\sqrt {x+1}+\sqrt {x+1}}{\left (3\,x-3\right )\,\sqrt {x-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)^(1/2)/(x - 1)^(5/2),x)

[Out]

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

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sympy [A]  time = 1.58, size = 63, normalized size = 3.50 \[ \begin {cases} - \frac {\left (x + 1\right )^{\frac {3}{2}}}{3 \sqrt {x - 1} \left (x + 1\right ) - 6 \sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\- \frac {i \left (x + 1\right )^{\frac {3}{2}}}{- 3 \sqrt {1 - x} \left (x + 1\right ) + 6 \sqrt {1 - x}} & \text {otherwise} \end {cases} \]

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*
sqrt(1 - x)*(x + 1) + 6*sqrt(1 - x)), True))

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