Optimal. Leaf size=63 \[ \frac{2 (x+1)^{5/2}}{3 (1-x)^{3/2}}-\frac{10 (x+1)^{3/2}}{3 \sqrt{1-x}}-5 \sqrt{1-x} \sqrt{x+1}+5 \sin ^{-1}(x) \]
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Rubi [A] time = 0.0101658, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {47, 50, 41, 216} \[ \frac{2 (x+1)^{5/2}}{3 (1-x)^{3/2}}-\frac{10 (x+1)^{3/2}}{3 \sqrt{1-x}}-5 \sqrt{1-x} \sqrt{x+1}+5 \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{(1+x)^{5/2}}{(1-x)^{5/2}} \, dx &=\frac{2 (1+x)^{5/2}}{3 (1-x)^{3/2}}-\frac{5}{3} \int \frac{(1+x)^{3/2}}{(1-x)^{3/2}} \, dx\\ &=-\frac{10 (1+x)^{3/2}}{3 \sqrt{1-x}}+\frac{2 (1+x)^{5/2}}{3 (1-x)^{3/2}}+5 \int \frac{\sqrt{1+x}}{\sqrt{1-x}} \, dx\\ &=-5 \sqrt{1-x} \sqrt{1+x}-\frac{10 (1+x)^{3/2}}{3 \sqrt{1-x}}+\frac{2 (1+x)^{5/2}}{3 (1-x)^{3/2}}+5 \int \frac{1}{\sqrt{1-x} \sqrt{1+x}} \, dx\\ &=-5 \sqrt{1-x} \sqrt{1+x}-\frac{10 (1+x)^{3/2}}{3 \sqrt{1-x}}+\frac{2 (1+x)^{5/2}}{3 (1-x)^{3/2}}+5 \int \frac{1}{\sqrt{1-x^2}} \, dx\\ &=-5 \sqrt{1-x} \sqrt{1+x}-\frac{10 (1+x)^{3/2}}{3 \sqrt{1-x}}+\frac{2 (1+x)^{5/2}}{3 (1-x)^{3/2}}+5 \sin ^{-1}(x)\\ \end{align*}
Mathematica [C] time = 0.0080356, size = 37, normalized size = 0.59 \[ \frac{8 \sqrt{2} \, _2F_1\left (-\frac{5}{2},-\frac{3}{2};-\frac{1}{2};\frac{1-x}{2}\right )}{3 (1-x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 84, normalized size = 1.3 \begin{align*}{\frac{3\,{x}^{3}-31\,{x}^{2}-11\,x+23}{-3+3\,x}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }}}{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}}+5\,{\frac{\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }\arcsin \left ( x \right ) }{\sqrt{1-x}\sqrt{1+x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.4886, size = 134, normalized size = 2.13 \begin{align*} -\frac{{\left (-x^{2} + 1\right )}^{\frac{5}{2}}}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1} - \frac{5 \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}{3 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac{10 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{35 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x - 1\right )}} + 5 \, \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.579, size = 205, normalized size = 3.25 \begin{align*} -\frac{23 \, x^{2} +{\left (3 \, x^{2} - 34 \, x + 23\right )} \sqrt{x + 1} \sqrt{-x + 1} + 30 \,{\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 46 \, x + 23}{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 [B] time = 14.2402, size = 576, normalized size = 9.14 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09, size = 59, normalized size = 0.94 \begin{align*} -\frac{{\left ({\left (3 \, x - 37\right )}{\left (x + 1\right )} + 60\right )} \sqrt{x + 1} \sqrt{-x + 1}}{3 \,{\left (x - 1\right )}^{2}} + 10 \, \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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