Optimal. Leaf size=101 \[ \frac{8 (x+1)^{3/2}}{3465 (1-x)^{3/2}}+\frac{8 (x+1)^{3/2}}{1155 (1-x)^{5/2}}+\frac{4 (x+1)^{3/2}}{231 (1-x)^{7/2}}+\frac{4 (x+1)^{3/2}}{99 (1-x)^{9/2}}+\frac{(x+1)^{3/2}}{11 (1-x)^{11/2}} \]
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Rubi [A] time = 0.0186567, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {45, 37} \[ \frac{8 (x+1)^{3/2}}{3465 (1-x)^{3/2}}+\frac{8 (x+1)^{3/2}}{1155 (1-x)^{5/2}}+\frac{4 (x+1)^{3/2}}{231 (1-x)^{7/2}}+\frac{4 (x+1)^{3/2}}{99 (1-x)^{9/2}}+\frac{(x+1)^{3/2}}{11 (1-x)^{11/2}} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\sqrt{1+x}}{(1-x)^{13/2}} \, dx &=\frac{(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac{4}{11} \int \frac{\sqrt{1+x}}{(1-x)^{11/2}} \, dx\\ &=\frac{(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac{4 (1+x)^{3/2}}{99 (1-x)^{9/2}}+\frac{4}{33} \int \frac{\sqrt{1+x}}{(1-x)^{9/2}} \, dx\\ &=\frac{(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac{4 (1+x)^{3/2}}{99 (1-x)^{9/2}}+\frac{4 (1+x)^{3/2}}{231 (1-x)^{7/2}}+\frac{8}{231} \int \frac{\sqrt{1+x}}{(1-x)^{7/2}} \, dx\\ &=\frac{(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac{4 (1+x)^{3/2}}{99 (1-x)^{9/2}}+\frac{4 (1+x)^{3/2}}{231 (1-x)^{7/2}}+\frac{8 (1+x)^{3/2}}{1155 (1-x)^{5/2}}+\frac{8 \int \frac{\sqrt{1+x}}{(1-x)^{5/2}} \, dx}{1155}\\ &=\frac{(1+x)^{3/2}}{11 (1-x)^{11/2}}+\frac{4 (1+x)^{3/2}}{99 (1-x)^{9/2}}+\frac{4 (1+x)^{3/2}}{231 (1-x)^{7/2}}+\frac{8 (1+x)^{3/2}}{1155 (1-x)^{5/2}}+\frac{8 (1+x)^{3/2}}{3465 (1-x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.017413, size = 40, normalized size = 0.4 \[ \frac{(x+1)^{3/2} \left (8 x^4-56 x^3+180 x^2-364 x+547\right )}{3465 (1-x)^{11/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 35, normalized size = 0.4 \begin{align*}{\frac{8\,{x}^{4}-56\,{x}^{3}+180\,{x}^{2}-364\,x+547}{3465} \left ( 1+x \right ) ^{{\frac{3}{2}}} \left ( 1-x \right ) ^{-{\frac{11}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04257, size = 232, normalized size = 2.3 \begin{align*} \frac{2 \, \sqrt{-x^{2} + 1}}{11 \,{\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{99 \,{\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac{4 \, \sqrt{-x^{2} + 1}}{693 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac{4 \, \sqrt{-x^{2} + 1}}{1155 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac{8 \, \sqrt{-x^{2} + 1}}{3465 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{8 \, \sqrt{-x^{2} + 1}}{3465 \,{\left (x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53397, size = 279, normalized size = 2.76 \begin{align*} \frac{547 \, x^{6} - 3282 \, x^{5} + 8205 \, x^{4} - 10940 \, x^{3} + 8205 \, x^{2} +{\left (8 \, x^{5} - 48 \, x^{4} + 124 \, x^{3} - 184 \, x^{2} + 183 \, x + 547\right )} \sqrt{x + 1} \sqrt{-x + 1} - 3282 \, x + 547}{3465 \,{\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, 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 = 1.15099, size = 57, normalized size = 0.56 \begin{align*} \frac{{\left (4 \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 10\right )} + 99\right )}{\left (x + 1\right )} - 231\right )}{\left (x + 1\right )} + 1155\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1}}{3465 \,{\left (x - 1\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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