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 = 0.01, antiderivative size = 63, normalized size of antiderivative = 1.00, 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 41
Rule 47
Rule 50
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*}
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Mathematica [C] time = 0.01, 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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fricas [A] time = 0.43, size = 75, normalized size = 1.19 \[ -\frac {23 \, x^{2} + {\left (3 \, x^{2} - 34 \, x + 23\right )} \sqrt {x + 1} \sqrt {-x + 1} + 30 \, {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - 46 \, x + 23}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 44, normalized size = 0.70 \[ -\frac {{\left ({\left (3 \, x - 37\right )} {\left (x + 1\right )} + 60\right )} \sqrt {x + 1} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}^{2}} + 10 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 84, normalized size = 1.33 \[ \frac {5 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{\sqrt {x +1}\, \sqrt {-x +1}}+\frac {\left (3 x^{3}-31 x^{2}-11 x +23\right ) \sqrt {\left (x +1\right ) \left (-x +1\right )}}{3 \left (x -1\right ) \sqrt {-\left (x +1\right ) \left (x -1\right )}\, \sqrt {-x +1}\, \sqrt {x +1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.97, size = 99, normalized size = 1.57 \[ -\frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1} - \frac {5 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {10 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {35 \, \sqrt {-x^{2} + 1}}{3 \, {\left (x - 1\right )}} + 5 \, \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (x+1\right )}^{5/2}}{{\left (1-x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.47, size = 576, normalized size = 9.14 \[ \begin {cases} \frac {30 i \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} - \frac {15 \pi \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} - \frac {60 i \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} + \frac {30 \pi \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} + \frac {3 i \left (x + 1\right )^{15}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} - \frac {40 i \left (x + 1\right )^{14}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} + \frac {60 i \left (x + 1\right )^{13}}{- 3 \sqrt {x - 1} \left (x + 1\right )^{\frac {27}{2}} + 6 \sqrt {x - 1} \left (x + 1\right )^{\frac {25}{2}}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\\frac {30 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} - \frac {60 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} + \frac {3 \left (x + 1\right )^{15}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} - \frac {40 \left (x + 1\right )^{14}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} + \frac {60 \left (x + 1\right )^{13}}{3 \sqrt {1 - x} \left (x + 1\right )^{\frac {27}{2}} - 6 \sqrt {1 - x} \left (x + 1\right )^{\frac {25}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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