Optimal. Leaf size=61 \[ \frac{2 (x+1)^{5/2}}{315 (1-x)^{5/2}}+\frac{2 (x+1)^{5/2}}{63 (1-x)^{7/2}}+\frac{(x+1)^{5/2}}{9 (1-x)^{9/2}} \]
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Rubi [A] time = 0.0080453, 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} \[ \frac{2 (x+1)^{5/2}}{315 (1-x)^{5/2}}+\frac{2 (x+1)^{5/2}}{63 (1-x)^{7/2}}+\frac{(x+1)^{5/2}}{9 (1-x)^{9/2}} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{(1+x)^{3/2}}{(1-x)^{11/2}} \, dx &=\frac{(1+x)^{5/2}}{9 (1-x)^{9/2}}+\frac{2}{9} \int \frac{(1+x)^{3/2}}{(1-x)^{9/2}} \, dx\\ &=\frac{(1+x)^{5/2}}{9 (1-x)^{9/2}}+\frac{2 (1+x)^{5/2}}{63 (1-x)^{7/2}}+\frac{2}{63} \int \frac{(1+x)^{3/2}}{(1-x)^{7/2}} \, dx\\ &=\frac{(1+x)^{5/2}}{9 (1-x)^{9/2}}+\frac{2 (1+x)^{5/2}}{63 (1-x)^{7/2}}+\frac{2 (1+x)^{5/2}}{315 (1-x)^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0117375, size = 30, normalized size = 0.49 \[ \frac{(x+1)^{5/2} \left (2 x^2-14 x+47\right )}{315 (1-x)^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 25, normalized size = 0.4 \begin{align*}{\frac{2\,{x}^{2}-14\,x+47}{315} \left ( 1+x \right ) ^{{\frac{5}{2}}} \left ( 1-x \right ) ^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00336, size = 232, normalized size = 3.8 \begin{align*} \frac{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{3 \,{\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} + \frac{2 \, \sqrt{-x^{2} + 1}}{9 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{63 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{105 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac{2 \, \sqrt{-x^{2} + 1}}{315 \,{\left (x^{2} - 2 \, x + 1\right )}} - \frac{2 \, \sqrt{-x^{2} + 1}}{315 \,{\left (x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6927, size = 224, normalized size = 3.67 \begin{align*} \frac{47 \, x^{5} - 235 \, x^{4} + 470 \, x^{3} - 470 \, x^{2} -{\left (2 \, x^{4} - 10 \, x^{3} + 21 \, x^{2} + 80 \, x + 47\right )} \sqrt{x + 1} \sqrt{-x + 1} + 235 \, x - 47}{315 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09394, size = 39, normalized size = 0.64 \begin{align*} -\frac{{\left (2 \,{\left (x + 1\right )}{\left (x - 8\right )} + 63\right )}{\left (x + 1\right )}^{\frac{5}{2}} \sqrt{-x + 1}}{315 \,{\left (x - 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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