Optimal. Leaf size=61 \[ \frac{2 (x+1)^{3/2}}{105 (1-x)^{3/2}}+\frac{2 (x+1)^{3/2}}{35 (1-x)^{5/2}}+\frac{(x+1)^{3/2}}{7 (1-x)^{7/2}} \]
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Rubi [A] time = 0.0082057, 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} \[ \frac{2 (x+1)^{3/2}}{105 (1-x)^{3/2}}+\frac{2 (x+1)^{3/2}}{35 (1-x)^{5/2}}+\frac{(x+1)^{3/2}}{7 (1-x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\sqrt{1+x}}{(1-x)^{9/2}} \, dx &=\frac{(1+x)^{3/2}}{7 (1-x)^{7/2}}+\frac{2}{7} \int \frac{\sqrt{1+x}}{(1-x)^{7/2}} \, dx\\ &=\frac{(1+x)^{3/2}}{7 (1-x)^{7/2}}+\frac{2 (1+x)^{3/2}}{35 (1-x)^{5/2}}+\frac{2}{35} \int \frac{\sqrt{1+x}}{(1-x)^{5/2}} \, dx\\ &=\frac{(1+x)^{3/2}}{7 (1-x)^{7/2}}+\frac{2 (1+x)^{3/2}}{35 (1-x)^{5/2}}+\frac{2 (1+x)^{3/2}}{105 (1-x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0114946, size = 30, normalized size = 0.49 \[ \frac{(x+1)^{3/2} \left (2 x^2-10 x+23\right )}{105 (1-x)^{7/2}} \]
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}-10\,x+23}{105} \left ( 1+x \right ) ^{{\frac{3}{2}}} \left ( 1-x \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.03258, size = 128, normalized size = 2.1 \begin{align*} \frac{2 \, \sqrt{-x^{2} + 1}}{7 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{35 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac{2 \, \sqrt{-x^{2} + 1}}{105 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{2 \, \sqrt{-x^{2} + 1}}{105 \,{\left (x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52901, size = 181, normalized size = 2.97 \begin{align*} \frac{23 \, x^{4} - 92 \, x^{3} + 138 \, x^{2} +{\left (2 \, x^{3} - 8 \, x^{2} + 13 \, x + 23\right )} \sqrt{x + 1} \sqrt{-x + 1} - 92 \, x + 23}{105 \,{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 87.1587, size = 568, normalized size = 9.31 \begin{align*} \begin{cases} \frac{2 i \left (x + 1\right )^{\frac{9}{2}}}{105 \sqrt{x - 1} \left (x + 1\right )^{4} - 840 \sqrt{x - 1} \left (x + 1\right )^{3} + 2520 \sqrt{x - 1} \left (x + 1\right )^{2} - 3360 \sqrt{x - 1} \left (x + 1\right ) + 1680 \sqrt{x - 1}} - \frac{18 i \left (x + 1\right )^{\frac{7}{2}}}{105 \sqrt{x - 1} \left (x + 1\right )^{4} - 840 \sqrt{x - 1} \left (x + 1\right )^{3} + 2520 \sqrt{x - 1} \left (x + 1\right )^{2} - 3360 \sqrt{x - 1} \left (x + 1\right ) + 1680 \sqrt{x - 1}} + \frac{63 i \left (x + 1\right )^{\frac{5}{2}}}{105 \sqrt{x - 1} \left (x + 1\right )^{4} - 840 \sqrt{x - 1} \left (x + 1\right )^{3} + 2520 \sqrt{x - 1} \left (x + 1\right )^{2} - 3360 \sqrt{x - 1} \left (x + 1\right ) + 1680 \sqrt{x - 1}} - \frac{70 i \left (x + 1\right )^{\frac{3}{2}}}{105 \sqrt{x - 1} \left (x + 1\right )^{4} - 840 \sqrt{x - 1} \left (x + 1\right )^{3} + 2520 \sqrt{x - 1} \left (x + 1\right )^{2} - 3360 \sqrt{x - 1} \left (x + 1\right ) + 1680 \sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\- \frac{2 \left (x + 1\right )^{\frac{9}{2}}}{105 \sqrt{1 - x} \left (x + 1\right )^{4} - 840 \sqrt{1 - x} \left (x + 1\right )^{3} + 2520 \sqrt{1 - x} \left (x + 1\right )^{2} - 3360 \sqrt{1 - x} \left (x + 1\right ) + 1680 \sqrt{1 - x}} + \frac{18 \left (x + 1\right )^{\frac{7}{2}}}{105 \sqrt{1 - x} \left (x + 1\right )^{4} - 840 \sqrt{1 - x} \left (x + 1\right )^{3} + 2520 \sqrt{1 - x} \left (x + 1\right )^{2} - 3360 \sqrt{1 - x} \left (x + 1\right ) + 1680 \sqrt{1 - x}} - \frac{63 \left (x + 1\right )^{\frac{5}{2}}}{105 \sqrt{1 - x} \left (x + 1\right )^{4} - 840 \sqrt{1 - x} \left (x + 1\right )^{3} + 2520 \sqrt{1 - x} \left (x + 1\right )^{2} - 3360 \sqrt{1 - x} \left (x + 1\right ) + 1680 \sqrt{1 - x}} + \frac{70 \left (x + 1\right )^{\frac{3}{2}}}{105 \sqrt{1 - x} \left (x + 1\right )^{4} - 840 \sqrt{1 - x} \left (x + 1\right )^{3} + 2520 \sqrt{1 - x} \left (x + 1\right )^{2} - 3360 \sqrt{1 - x} \left (x + 1\right ) + 1680 \sqrt{1 - x}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07231, size = 39, normalized size = 0.64 \begin{align*} \frac{{\left (2 \,{\left (x + 1\right )}{\left (x - 6\right )} + 35\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1}}{105 \,{\left (x - 1\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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