Optimal. Leaf size=41 \[ -\frac{\sqrt{1-x}}{3 \sqrt{x+1}}-\frac{\sqrt{1-x}}{3 (x+1)^{3/2}} \]
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Rubi [A] time = 0.0043043, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {45, 37} \[ -\frac{\sqrt{1-x}}{3 \sqrt{x+1}}-\frac{\sqrt{1-x}}{3 (x+1)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-x} (1+x)^{5/2}} \, dx &=-\frac{\sqrt{1-x}}{3 (1+x)^{3/2}}+\frac{1}{3} \int \frac{1}{\sqrt{1-x} (1+x)^{3/2}} \, dx\\ &=-\frac{\sqrt{1-x}}{3 (1+x)^{3/2}}-\frac{\sqrt{1-x}}{3 \sqrt{1+x}}\\ \end{align*}
Mathematica [A] time = 0.0111907, size = 23, normalized size = 0.56 \[ -\frac{\sqrt{1-x} (x+2)}{3 (x+1)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 18, normalized size = 0.4 \begin{align*} -{\frac{2+x}{3}\sqrt{1-x} \left ( 1+x \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49294, size = 51, normalized size = 1.24 \begin{align*} -\frac{\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 [A] time = 1.92189, size = 101, normalized size = 2.46 \begin{align*} -\frac{2 \, x^{2} +{\left (x + 2\right )} \sqrt{x + 1} \sqrt{-x + 1} + 4 \, x + 2}{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 = 4.23274, size = 65, normalized size = 1.59 \begin{align*} \begin{cases} - \frac{\sqrt{-1 + \frac{2}{x + 1}}}{3} - \frac{\sqrt{-1 + \frac{2}{x + 1}}}{3 \left (x + 1\right )} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\- \frac{i \sqrt{1 - \frac{2}{x + 1}}}{3} - \frac{i \sqrt{1 - \frac{2}{x + 1}}}{3 \left (x + 1\right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.07142, size = 120, normalized size = 2.93 \begin{align*} \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{48 \,{\left (x + 1\right )}^{\frac{3}{2}}} + \frac{3 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{16 \, \sqrt{x + 1}} - \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{9 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{48 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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