Optimal. Leaf size=41 \[ -\frac{2 (1-x)^{3/2}}{3 (x+1)^{3/2}}+\frac{2 \sqrt{1-x}}{\sqrt{x+1}}+\sin ^{-1}(x) \]
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Rubi [A] time = 0.0055739, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {47, 41, 216} \[ -\frac{2 (1-x)^{3/2}}{3 (x+1)^{3/2}}+\frac{2 \sqrt{1-x}}{\sqrt{x+1}}+\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 47
Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{(1-x)^{3/2}}{(1+x)^{5/2}} \, dx &=-\frac{2 (1-x)^{3/2}}{3 (1+x)^{3/2}}-\int \frac{\sqrt{1-x}}{(1+x)^{3/2}} \, dx\\ &=-\frac{2 (1-x)^{3/2}}{3 (1+x)^{3/2}}+\frac{2 \sqrt{1-x}}{\sqrt{1+x}}+\int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-\frac{2 (1-x)^{3/2}}{3 (1+x)^{3/2}}+\frac{2 \sqrt{1-x}}{\sqrt{1+x}}+\int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-\frac{2 (1-x)^{3/2}}{3 (1+x)^{3/2}}+\frac{2 \sqrt{1-x}}{\sqrt{1+x}}+\sin ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0538163, size = 49, normalized size = 1.2 \[ \frac{-8 x^2+4 x+4}{3 \sqrt{1-x} (x+1)^{3/2}}-2 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 73, normalized size = 1.8 \begin{align*} -{\frac{8\,{x}^{2}-4\,x-4}{3}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) } \left ( 1+x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{1-x}}}}+{\arcsin \left ( x \right ) \sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.5003, size = 89, normalized size = 2.17 \begin{align*} -\frac{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{3 \,{\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} - \frac{2 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x^{2} + 2 \, x + 1\right )}} + \frac{7 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x + 1\right )}} + \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.81969, size = 188, normalized size = 4.59 \begin{align*} \frac{2 \,{\left (2 \, x^{2} + 2 \,{\left (2 \, x + 1\right )} \sqrt{x + 1} \sqrt{-x + 1} - 3 \,{\left (x^{2} + 2 \, x + 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 4 \, x + 2\right )}}{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 [C] time = 5.09716, size = 126, normalized size = 3.07 \begin{align*} \begin{cases} \frac{8 \sqrt{-1 + \frac{2}{x + 1}}}{3} - \frac{4 \sqrt{-1 + \frac{2}{x + 1}}}{3 \left (x + 1\right )} + i \log{\left (\frac{1}{x + 1} \right )} + i \log{\left (x + 1 \right )} + 2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\\frac{8 i \sqrt{1 - \frac{2}{x + 1}}}{3} - \frac{4 i \sqrt{1 - \frac{2}{x + 1}}}{3 \left (x + 1\right )} + i \log{\left (\frac{1}{x + 1} \right )} - 2 i \log{\left (\sqrt{1 - \frac{2}{x + 1}} + 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 = 1.1396, size = 138, normalized size = 3.37 \begin{align*} \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{12 \,{\left (x + 1\right )}^{\frac{3}{2}}} - \frac{5 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{4 \, \sqrt{x + 1}} + \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{15 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{12 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} + 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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