Optimal. Leaf size=65 \[ \frac {2 (x+1)^{5/2}}{\sqrt {1-x}}+\frac {5}{2} \sqrt {1-x} (x+1)^{3/2}+\frac {15}{2} \sqrt {1-x} \sqrt {x+1}-\frac {15}{2} \sin ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 65, 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} \[ \frac {2 (x+1)^{5/2}}{\sqrt {1-x}}+\frac {5}{2} \sqrt {1-x} (x+1)^{3/2}+\frac {15}{2} \sqrt {1-x} \sqrt {x+1}-\frac {15}{2} \sin ^{-1}(x) \]
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)^{3/2}} \, dx &=\frac {2 (1+x)^{5/2}}{\sqrt {1-x}}-5 \int \frac {(1+x)^{3/2}}{\sqrt {1-x}} \, dx\\ &=\frac {5}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {2 (1+x)^{5/2}}{\sqrt {1-x}}-\frac {15}{2} \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx\\ &=\frac {15}{2} \sqrt {1-x} \sqrt {1+x}+\frac {5}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {2 (1+x)^{5/2}}{\sqrt {1-x}}-\frac {15}{2} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {15}{2} \sqrt {1-x} \sqrt {1+x}+\frac {5}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {2 (1+x)^{5/2}}{\sqrt {1-x}}-\frac {15}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {15}{2} \sqrt {1-x} \sqrt {1+x}+\frac {5}{2} \sqrt {1-x} (1+x)^{3/2}+\frac {2 (1+x)^{5/2}}{\sqrt {1-x}}-\frac {15}{2} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [C] time = 0.01, size = 35, normalized size = 0.54 \[ \frac {8 \sqrt {2} \, _2F_1\left (-\frac {5}{2},-\frac {1}{2};\frac {1}{2};\frac {1-x}{2}\right )}{\sqrt {1-x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 58, normalized size = 0.89 \[ \frac {{\left (x^{2} + 7 \, x - 24\right )} \sqrt {x + 1} \sqrt {-x + 1} + 30 \, {\left (x - 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 24 \, x - 24}{2 \, {\left (x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 42, normalized size = 0.65 \[ \frac {{\left ({\left (x + 6\right )} {\left (x + 1\right )} - 30\right )} \sqrt {x + 1} \sqrt {-x + 1}}{2 \, {\left (x - 1\right )}} - 15 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 77, normalized size = 1.18 \[ -\frac {15 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{2 \sqrt {x +1}\, \sqrt {-x +1}}-\frac {\left (x^{3}+8 x^{2}-17 x -24\right ) \sqrt {\left (x +1\right ) \left (-x +1\right )}}{2 \sqrt {-\left (x +1\right ) \left (x -1\right )}\, \sqrt {-x +1}\, \sqrt {x +1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 56, normalized size = 0.86 \[ -\frac {x^{3}}{2 \, \sqrt {-x^{2} + 1}} - \frac {4 \, x^{2}}{\sqrt {-x^{2} + 1}} + \frac {17 \, x}{2 \, \sqrt {-x^{2} + 1}} + \frac {12}{\sqrt {-x^{2} + 1}} - \frac {15}{2} \, \arcsin \relax (x) \]
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 \[ \int \frac {{\left (x+1\right )}^{5/2}}{{\left (1-x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.76, size = 139, normalized size = 2.14 \[ \begin {cases} 15 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} + \frac {i \left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {x - 1}} + \frac {5 i \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {x - 1}} - \frac {15 i \sqrt {x + 1}}{\sqrt {x - 1}} & \text {for}\: \frac {\left |{x + 1}\right |}{2} > 1 \\- 15 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )} - \frac {\left (x + 1\right )^{\frac {5}{2}}}{2 \sqrt {1 - x}} - \frac {5 \left (x + 1\right )^{\frac {3}{2}}}{2 \sqrt {1 - x}} + \frac {15 \sqrt {x + 1}}{\sqrt {1 - x}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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