Optimal. Leaf size=18 \[ -\frac{(x+1)^{3/2}}{3 (x-1)^{3/2}} \]
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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.
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Rule 37
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.
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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.
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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.
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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.
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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.
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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.
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