Optimal. Leaf size=61 \[ \frac{2 \sqrt{x+1}}{15 \sqrt{1-x}}+\frac{2 \sqrt{x+1}}{15 (1-x)^{3/2}}+\frac{\sqrt{x+1}}{5 (1-x)^{5/2}} \]
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Rubi [A] time = 0.0078398, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {45, 37} \[ \frac{2 \sqrt{x+1}}{15 \sqrt{1-x}}+\frac{2 \sqrt{x+1}}{15 (1-x)^{3/2}}+\frac{\sqrt{x+1}}{5 (1-x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{(1-x)^{7/2} \sqrt{1+x}} \, dx &=\frac{\sqrt{1+x}}{5 (1-x)^{5/2}}+\frac{2}{5} \int \frac{1}{(1-x)^{5/2} \sqrt{1+x}} \, dx\\ &=\frac{\sqrt{1+x}}{5 (1-x)^{5/2}}+\frac{2 \sqrt{1+x}}{15 (1-x)^{3/2}}+\frac{2}{15} \int \frac{1}{(1-x)^{3/2} \sqrt{1+x}} \, dx\\ &=\frac{\sqrt{1+x}}{5 (1-x)^{5/2}}+\frac{2 \sqrt{1+x}}{15 (1-x)^{3/2}}+\frac{2 \sqrt{1+x}}{15 \sqrt{1-x}}\\ \end{align*}
Mathematica [A] time = 0.0093584, size = 30, normalized size = 0.49 \[ \frac{\sqrt{x+1} \left (2 x^2-6 x+7\right )}{15 (1-x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.4 \begin{align*}{\frac{2\,{x}^{2}-6\,x+7}{15}\sqrt{1+x} \left ( 1-x \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4832, size = 86, normalized size = 1.41 \begin{align*} -\frac{\sqrt{-x^{2} + 1}}{5 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac{2 \, \sqrt{-x^{2} + 1}}{15 \,{\left (x^{2} - 2 \, x + 1\right )}} - \frac{2 \, \sqrt{-x^{2} + 1}}{15 \,{\left (x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86757, size = 139, normalized size = 2.28 \begin{align*} \frac{7 \, x^{3} - 21 \, x^{2} -{\left (2 \, x^{2} - 6 \, x + 7\right )} \sqrt{x + 1} \sqrt{-x + 1} + 21 \, x - 7}{15 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 35.3623, size = 301, normalized size = 4.93 \begin{align*} \begin{cases} \frac{2 \left (x + 1\right )^{2}}{15 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{2} - 60 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right ) + 60 \sqrt{-1 + \frac{2}{x + 1}}} - \frac{10 \left (x + 1\right )}{15 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{2} - 60 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right ) + 60 \sqrt{-1 + \frac{2}{x + 1}}} + \frac{15}{15 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{2} - 60 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right ) + 60 \sqrt{-1 + \frac{2}{x + 1}}} & \text{for}\: \frac{2}{\left |{x + 1}\right |} > 1 \\- \frac{2 i \left (x + 1\right )^{2}}{15 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{2} - 60 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right ) + 60 \sqrt{1 - \frac{2}{x + 1}}} + \frac{10 i \left (x + 1\right )}{15 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{2} - 60 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right ) + 60 \sqrt{1 - \frac{2}{x + 1}}} - \frac{15 i}{15 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{2} - 60 \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right ) + 60 \sqrt{1 - \frac{2}{x + 1}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08045, size = 39, normalized size = 0.64 \begin{align*} -\frac{{\left (2 \,{\left (x + 1\right )}{\left (x - 4\right )} + 15\right )} \sqrt{x + 1} \sqrt{-x + 1}}{15 \,{\left (x - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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