Optimal. Leaf size=87 \[ \frac {35}{8} \sqrt {1-x} \sqrt {1+x}+\frac {35}{24} (1-x)^{3/2} \sqrt {1+x}+\frac {7}{12} (1-x)^{5/2} \sqrt {1+x}+\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {35}{8} \sin ^{-1}(x) \]
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Rubi [A]
time = 0.01, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {52, 41, 222}
\begin {gather*} \frac {1}{4} \sqrt {x+1} (1-x)^{7/2}+\frac {7}{12} \sqrt {x+1} (1-x)^{5/2}+\frac {35}{24} \sqrt {x+1} (1-x)^{3/2}+\frac {35}{8} \sqrt {x+1} \sqrt {1-x}+\frac {35}{8} \sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 41
Rule 52
Rule 222
Rubi steps
\begin {align*} \int \frac {(1-x)^{7/2}}{\sqrt {1+x}} \, dx &=\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {7}{4} \int \frac {(1-x)^{5/2}}{\sqrt {1+x}} \, dx\\ &=\frac {7}{12} (1-x)^{5/2} \sqrt {1+x}+\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {35}{12} \int \frac {(1-x)^{3/2}}{\sqrt {1+x}} \, dx\\ &=\frac {35}{24} (1-x)^{3/2} \sqrt {1+x}+\frac {7}{12} (1-x)^{5/2} \sqrt {1+x}+\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {35}{8} \int \frac {\sqrt {1-x}}{\sqrt {1+x}} \, dx\\ &=\frac {35}{8} \sqrt {1-x} \sqrt {1+x}+\frac {35}{24} (1-x)^{3/2} \sqrt {1+x}+\frac {7}{12} (1-x)^{5/2} \sqrt {1+x}+\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {35}{8} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {35}{8} \sqrt {1-x} \sqrt {1+x}+\frac {35}{24} (1-x)^{3/2} \sqrt {1+x}+\frac {7}{12} (1-x)^{5/2} \sqrt {1+x}+\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {35}{8} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {35}{8} \sqrt {1-x} \sqrt {1+x}+\frac {35}{24} (1-x)^{3/2} \sqrt {1+x}+\frac {7}{12} (1-x)^{5/2} \sqrt {1+x}+\frac {1}{4} (1-x)^{7/2} \sqrt {1+x}+\frac {35}{8} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 63, normalized size = 0.72 \begin {gather*} \frac {\sqrt {1+x} \left (160-241 x+113 x^2-38 x^3+6 x^4\right )}{24 \sqrt {1-x}}+\frac {35}{4} \tan ^{-1}\left (\frac {\sqrt {1+x}}{\sqrt {1-x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 14.28, size = 157, normalized size = 1.80 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-210 \text {ArcCosh}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]-163 \sqrt {-1+x} \left (1+x\right )^{\frac {3}{2}}-6 \sqrt {-1+x} \left (1+x\right )^{\frac {7}{2}}+50 \sqrt {-1+x} \left (1+x\right )^{\frac {5}{2}}+279 \sqrt {-1+x} \sqrt {1+x}\right )}{24},\text {Abs}\left [1+x\right ]>2\right \}\right \},\frac {-605 \left (1+x\right )^{\frac {3}{2}}}{24 \sqrt {1-x}}-\frac {31 \left (1+x\right )^{\frac {7}{2}}}{12 \sqrt {1-x}}+\frac {\left (1+x\right )^{\frac {9}{2}}}{4 \sqrt {1-x}}+\frac {35 \text {ArcSin}\left [\frac {\sqrt {2} \sqrt {1+x}}{2}\right ]}{4}+\frac {263 \left (1+x\right )^{\frac {5}{2}}}{24 \sqrt {1-x}}+\frac {93 \sqrt {1+x}}{4 \sqrt {1-x}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.14, size = 85, normalized size = 0.98
method | result | size |
risch | \(\frac {\left (6 x^{3}-32 x^{2}+81 x -160\right ) \sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{24 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {35 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\) | \(82\) |
default | \(\frac {\left (1-x \right )^{\frac {7}{2}} \sqrt {1+x}}{4}+\frac {7 \left (1-x \right )^{\frac {5}{2}} \sqrt {1+x}}{12}+\frac {35 \left (1-x \right )^{\frac {3}{2}} \sqrt {1+x}}{24}+\frac {35 \sqrt {1-x}\, \sqrt {1+x}}{8}+\frac {35 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 56, normalized size = 0.64 \begin {gather*} -\frac {1}{4} \, \sqrt {-x^{2} + 1} x^{3} + \frac {4}{3} \, \sqrt {-x^{2} + 1} x^{2} - \frac {27}{8} \, \sqrt {-x^{2} + 1} x + \frac {20}{3} \, \sqrt {-x^{2} + 1} + \frac {35}{8} \, \arcsin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 52, normalized size = 0.60 \begin {gather*} -\frac {1}{24} \, {\left (6 \, x^{3} - 32 \, x^{2} + 81 \, x - 160\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {35}{4} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 16.46, size = 197, normalized size = 2.26 \begin {gather*} \begin {cases} - \frac {i \sqrt {x - 1} \left (x + 1\right )^{\frac {7}{2}}}{4} + \frac {25 i \sqrt {x - 1} \left (x + 1\right )^{\frac {5}{2}}}{12} - \frac {163 i \sqrt {x - 1} \left (x + 1\right )^{\frac {3}{2}}}{24} + \frac {93 i \sqrt {x - 1} \sqrt {x + 1}}{8} - \frac {35 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {35 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} + \frac {\left (x + 1\right )^{\frac {9}{2}}}{4 \sqrt {1 - x}} - \frac {31 \left (x + 1\right )^{\frac {7}{2}}}{12 \sqrt {1 - x}} + \frac {263 \left (x + 1\right )^{\frac {5}{2}}}{24 \sqrt {1 - x}} - \frac {605 \left (x + 1\right )^{\frac {3}{2}}}{24 \sqrt {1 - x}} + \frac {93 \sqrt {x + 1}}{4 \sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 107, normalized size = 1.23 \begin {gather*} 2 \left (\left (\left (\frac {1}{8} \sqrt {-x+1} \sqrt {-x+1}+\frac {7}{24}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {35}{48}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {35}{16}\right ) \sqrt {-x+1} \sqrt {x+1}-\frac {35}{4} \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-x\right )}^{7/2}}{\sqrt {x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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